## Clock Puzzle Guide by AZorro007

Version: 1.0 | Updated: 02/20/13 | Printable Version | Search Guide | Bookmark Guide

```                            CLOCK PUZZLE RESOLUTION
FINAL FANTASY XIII-2
Version 1.0
by
AZorro007

This note will describe simple methods that can be applied to "solve" any of
"The Hands of Time" (Clock) Puzzles that can be presented to a player during
Final Fantasy XIII-2 game play.

Application of the methods will require use of a writing implement (e.g., a
pencil), a few sheets of paper and perseverance by the investigator. The need
for perseverance is emphasized due to the fact that puzzle solution typically
involves a somewhat tedious process of elimination to accomplish a careful,
exhaustive examination of pathways through the maze posed by the puzzle.

The presentation is designed to appeal to those wishing to resolve encountered
Clock Puzzles without recourse to "automatic" solution-producing mechanisms
available from other sources.

If carefully applied, the methods are "sure fire" and, if used to completely
explore any given Clock Puzzle, will yield every solution pathway for any
attempted puzzle. The methods discussed include the optional use of "shortcuts"
to limit computational requirements. Shortcuts may be of particular interest to
those seeking a single pathway through a Clock Puzzle maze.

The presentation that follows assumes that the reader is familiar with the
Final Fantasy XIII-2 Clock Puzzle concept. A series of numbers are arranged in
a circle to define a display reminiscent of a clock face. As one "rests" on one
of the numbers, movement must be to a next number with a "distance" in terms of
"steps around the clock face" equal to the amount indicated by the occupied
number. The task is to find a pathway starting from one of the numbers that
travels though the entire number set while occupying each once and only once.

The discussion also assumes that the reader has been able to examine the Clock
Puzzle to be resolved to determine the positioning and sequence of the element
(number) set of the Clock Puzzle display. This is always possible given the
color coding of the different numbers.

One can always "sneak up" on a puzzle without starting the countdown timer and
see most of the numbers. For a few numbers, only the bottom part of the number
and circle containing it may be visible. Color coding plus experience will
permit identification of those numbers to expose the complete element set as
well. (Reader Trevor Nelson reminds that, by making use of the "Eyes of the
Goddess" Fragment Skill, the default camera view can be managed to be able to
"sneak up" and scan a puzzle to see all of the clock face numbers.)

The content of this document is derived solely from game play experience.

The material of this note is copyright 2013 by its author; Robert E. Fricks
(email: zorro007@pacbell.net). Final Fantasy XIII-2 is copyright 2011 and 2012
for exclusive posting on www.GameFAQs.com solely for the support of web site
users. Reproduction of single copies of this document for personal use is
authorized. All other uses, particularly those for financial gain or
commercial purpose, are not authorized.

[A] THE BASIC ALGORITHM. . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Presents the execution steps for the Reference Solution Approach of this
document.

[B] EXAMPLE APPLICATION. . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Records the analysis steps and results produced by use of the Basic Algorithm
to solve a typical (eight-element) Clock Puzzle.

[C] SHORTCUT CONSIDERATIONS. . . . . . . . . . . . . . . . . . . . . . . . . 8
Introduces the idea of looking at a Clock Puzzle from varying viewpoints to be
able to construct alternate solution methods.

[D] USE OF SPECIAL FEATURES. . . . . . . . . . . . . . . . . . . . . . . . .10
Introduces a "high efficiency" Clock Puzzle solution method applicable to
puzzles having a particular special feature.

[E] GENERAL PRINCIPLES . . . . . . . . . . . . . . . . . . . . . . . . . . .13
Gathers and organizes the ideas displayed in prior sections as a collection of
concepts to be applied to solve Clock Puzzles.

[E1] Solution Preliminaries
[E2] "Pure" Solution Approaches
[E3] "Pure" Shortcuts
[E4] "Combination" Shortcuts

[F] SOLUTION GRAPHICS. . . . . . . . . . . . . . . . . . . . . . . . . . . .19
Introduces the "creative" use of diagrams to expose internal puzzle structures
to as means to "quickly" accomplish logic-driven puzzle solutions.

[F1] Puzzle Principles
[F2] Puzzle Structures
[F3] Solution Approach
[F4] Example Resolution

[G] EXAMPLE PUZZLES. . . . . . . . . . . . . . . . . . . . . . . . . . . . .28
Presents a selection of Clock Puzzle displays encountered during game play and
identified solutions for each.

[G1]  Five-Element Puzzle with Five (5) Identified Solutions
[G2]  Six-Element Puzzle with One (1) Identified Solution
[G3]  Seven-Element Puzzle with Five (5) Identified Solutions
[G4]  Eight-Element Puzzle with Seven (7) Identified Solutions
[G5]  Nine-Element Puzzle with Twenty-Four (24) Identified Solutions
[G6]  Ten-Element Puzzle with Four (4) Identified Solutions
[G7]  Ten-Element Puzzle with Four (4) Identified Solutions
[G8]  Eleven-Element Puzzle with Five (5) Identified Solutions
[G9]  Twelve-element Puzzle with Eighteen (18) Identified Solutions
[G10] Thirteen-Element Puzzle with Eight (8) Identified Solutions

[A] THE BASIC ALGORITHM

The basic approach to be considered rests upon a "backward" progression
through generated path alternatives to identify and evaluate all possible
forward pathways through the Clock Puzzle maze.

After the Clock Puzzle display has been constructed, the number of "backward"
pathway examination steps required by the algorithm will be equal to the
quantity of numbers appearing in the Clock Puzzle display.

STEP 0: Construct a Clock Puzzle display as a "circular" sequence of numbers
that define the puzzle of interest.

STEP 1: Make a sequenced list of the numbers appearing in the Clock Puzzle
display. If duplicate numbers appear, tag the individuals within each
duplicate set so that they will be distinguishable from one another.

STEP 2: For each number of the sequenced list, determine its predecessor
number (or numbers if more than one predecessor exists). As this is done make
a list of the ordered pairs of numbers (successor <--- predecessor) that
result. Note that a complete set of ordered pairs of numbers will total twice
the quantity of numbers presented by the puzzle minus the quantity of numbers
presented by the puzzle having a single successor number.  If a number is
found to lack a predecessor, it is the starting point of the "solution" path
or the "solution" pathways through the Clock Puzzle maze.

STEP 3: Evaluate the results of STEP 2 to not carry forward into STEP 3 any
identified number sequence that cannot be "expanded" by the addition of a
predecessor number to define an ordered sequence of three numbers. (If a pair
is found to lack a predecessor, it may be the starting sequence of a solution
pathway through the Clock Puzzle maze.)

For each ordered sequence of two numbers that remain, use the results of STEP
2 to expand the pair into a triplet (or set of triplets) of ordered numbers by
adding the predecessor number (or numbers if more than one predecessor has
been identified). As this is done, list the ordered triplets of numbers
(successor <--- predecessor1 <--- predecessor2) that result.

STEP 4: Evaluate the results of STEP 3. Any sequence that shows the same
number more than once does not present a viable solution path segment and need
not be carried forward to STEP 4. Also, do not carry forward into STEP 4 any
identified number sequence that cannot be "expanded" by the addition of a
predecessor number to define an ordered sequence of four numbers. (If a
triplet is found to lack a predecessor it may be a three-element starting
sequence of a solution pathway through the Clock Puzzle maze.)

For the ordered sequences of three numbers that remain, use the results of
STEP 2 to expand the triplet into a quartet (or set of quartets) of ordered
numbers by adding a predecessor number (or numbers if there is more than one
predecessor). As this is done, make a list of the ordered quartets of numbers
(successor <--- predecessor1 <--- predecessor2 <--- predecessor3) that result.

STEP 5: Apply the logic used in STEP 4 to form five number chains. Continue
the evaluation and chain expansion process until the Clock Puzzle is "solved"
by production of one or more element strings having member elements equal in
quantity to the number of elements appearing on the clock face of the Clock
Puzzle under investigation.

If there are N numbers that appear on the "clock" face, exactly N steps within
the algorithm will be required to produce all of the solution pathways for the
puzzle.

[B] EXAMPLE APPLICATION

STEP 0: Consider the displayed Clock Puzzle which appeared during game play.

3
4       2

1           4

4       3
2

STEP 1: Eight potential end points identified.

3
2
4
3+
2+
4+
1
4*

STEP 2: Thirteen potential end-of-pathway sequence pairs identified.

3 <--- 3+
2 <--- 4+
4 <--- 2+ Pair not expandable
3+<--- 3
3+<--- 2
3+<--- 4*
2+ None   Starting Point
4+<--- 1
4+<--- 3
1 <--- 4
1 <--- 3+
1 <--- 2+ Pair not expandable
4*<--- 1
4*<--- 2

STEP 3: Twenty-one potential end-of-pathway sequence triplets identified by
using the results of STEP 2 to expand the two-element sequences. Note that
since STEP 2 has shown that 2+ is the pathway starting point, two pathway
pairs are eliminated from further consideration as expansion is not possible.

3 <--- 3+<--- 3  Duplicate Number appears
3 <--- 3+<--- 2
3 <--- 3+<--- 4*
2 <--- 4+<--- 1
2 <--- 4+<--- 3
3+<--- 3 <--- 3+ Duplicate Number appears
3+<--- 2 <--- 4+
3+<--- 4*<--- 1
3+<--- 4*<--- 2
4+<--- 1 <--- 4
4+<--- 1 <--- 3+
4+<--- 1 <--- 2+ Triplet not expandable
4+<--- 3 <--- 3+
1 <--- 4 <--- 2+ Triplet not expandable
1 <--- 3+<--- 3
1 <--- 3+<--- 2
1 <--- 3+<--- 4*
4*<--- 1 <--- 4
4*<--- 1 <--- 3+
4*<--- 1 <--- 2+ Triplet not expandable
4*<--- 2 <--- 4+

STEP 4: Thirty potential end-of-pathway sequence quartets identified by using
the results of STEP 2 to expand accepted three-element sequences. Note that
STEP 3 shows two triplets with internal loops (duplicate numbers) and three
triplets that are not expandable. Those five sequences are not carried forward
for consideration in the STEP 4 candidate sequence list generation process.

3 <--- 3+<--- 2 <--- 4+
3 <--- 3+<--- 4*<--- 1
3 <--- 3+<--- 4*<--- 2
2 <--- 4+<--- 1 <--- 4
2 <--- 4+<--- 1 <--- 3+
2 <--- 4+<--- 1 <--- 2+ Not Expandable
2 <--- 4+<--- 3 <--- 3+
3+<--- 2 <--- 4+<--- 1
3+<--- 2 <--- 4+<--- 3
3+<--- 4*<--- 1 <--- 4
3+<--- 4*<--- 1 <--- 3+ Duplicate Number
3+<--- 4*<--- 1 <--- 2+ Not Expandable
3+<--- 4*<--- 2 <--- 4+
4+<--- 1 <--- 4 <--- 2+ Not Expandable
4+<--- 1 <--- 3+<--- 3
4+<--- 1 <--- 3+<--- 2
4+<--- 1 <--- 3+<--- 4*
4+<--- 3 <--- 3+<--- 3  Duplicate Number
4+<--- 3 <--- 3+<--- 2
4+<--- 3 <--- 3+<--- 4*
1 <--- 3+<--- 3 <--- 3+ Duplicate Number
1 <--- 3+<--- 2 <--- 4+
1 <--- 3+<--- 4*<--- 1  Duplicate Number
1 <--- 3+<--- 4*<--- 2
4*<--- 1 <--- 4 <--- 2+ Not Expandable
4*<--- 1 <--- 3+<--- 3
4*<--- 1 <--- 3+<--- 2
4*<--- 1 <--- 3+<--- 4* Duplicate Number
4*<--- 2 <--- 4+<--- 1
4*<--- 2 <--- 4+<--- 3

STEP 5: Thirty-six potential five-element, end-of-pathway sequences identified
by using the results of STEP 2 to expand accepted four-element sequences. Note
that STEP 4 shows five quartets with internal loops (duplicate numbers) and
four quartets that are not expandable. Those nine sequences are not carried
forward for consideration in the STEP 5 five-element candidate sequence list
generation process.

3 <--- 3+<--- 2 <--- 4+ <--- 1
3 <--- 3+<--- 2 <--- 4+ <--- 3  Duplicate Number
3 <--- 3+<--- 4*<--- 1  <--- 2+ Not Expandable
3 <--- 3+<--- 4*<--- 1  <--- 3+ Duplicate Number
3 <--- 3+<--- 4*<--- 1  <--- 4
3 <--- 3+<--- 4*<--- 2  <--- 4+
2 <--- 4+<--- 1 <--- 4  <--- 2+ Not Expandable
2 <--- 4+<--- 1 <--- 3+ <--- 3
2 <--- 4+<--- 1 <--- 3+ <--- 2  Duplicate Number
2 <--- 4+<--- 1 <--- 3+ <--- 4*
2 <--- 4+<--- 3 <--- 3+ <--- 3  Duplicate Number
2 <--- 4+<--- 3 <--- 3+ <--- 2  Duplicate Number
2 <--- 4+<--- 3 <--- 3+ <--- 4*
3+<--- 2 <--- 4+<--- 1  <--- 2+ Not Expandable
3+<--- 2 <--- 4+<--- 1  <--- 3+ Duplicate Number
3+<--- 2 <--- 4+<--- 1  <--- 4
3+<--- 2 <--- 4+<--- 3  <--- 3+ Duplicate Number
3+<--- 4*<--- 1 <--- 4  <--- 2+ Not Expandable
3+<--- 4*<--- 2 <--- 4+ <--- 1
3+<--- 4*<--- 2 <--- 4+ <--- 3
4+<--- 1 <--- 3+<--- 3  <--- 3+ Duplicate Number
4+<--- 1 <--- 3+<--- 2  <--- 4+ Duplicate Number
4+<--- 1 <--- 3+<--- 4* <--- 1  Duplicate Number
4+<--- 1 <--- 3+<--- 4* <--- 2
4+<--- 3 <--- 3+<--- 2  <--- 4+ Duplicate Number
4+<--- 3 <--- 3+<--- 4* <--- 1
4+<--- 3 <--- 3+<--- 4* <--- 2
1 <--- 3+<--- 2 <--- 4+ <--- 1  Duplicate Number
1 <--- 3+<--- 2 <--- 4+ <--- 3
1 <--- 3+<--- 4*<--- 2  <--- 4+
4*<--- 1 <--- 3+<--- 3  <--- 3+ Duplicate Number
4*<--- 1 <--- 3+<--- 2  <--- 4+
4*<--- 2 <--- 4+<--- 1  <--- 2+ Not Expandable
4*<--- 2 <--- 4+<--- 1  <--- 3+
4*<--- 2 <--- 4+<--- 1  <--- 4
4*<--- 2 <--- 4+<--- 3  <--- 3+

STEP 6: Thirty-three potential six-element, end-of-pathway sequences
identified by using the results of STEP 2 to expand accepted five-element
sequences. Note that STEP 5 shows thirteen quintets with internal loops
(duplicate numbers) and five quintets that are not expandable. Those eighteen
sequences are not carried forward for consideration in the STEP 6 six-element
candidate sequence list generation process.

3 <--- 3+<--- 2 <--- 4+ <--- 1  <--- 2+ Not Expandable
3 <--- 3+<--- 2 <--- 4+ <--- 1  <--- 3+ Duplicate Number
3 <--- 3+<--- 2 <--- 4+ <--- 1  <--- 4
3 <--- 3+<--- 4*<--- 1  <--- 4  <--- 2+ Not Expandable
3 <--- 3+<--- 4*<--- 2  <--- 4+ <--- 1
3 <--- 3+<--- 4*<--- 2  <--- 4+ <--- 3  Duplicate Number
2 <--- 4+<--- 1 <--- 3+ <--- 3  <--- 3+ Duplicate Number
2 <--- 4+<--- 1 <--- 3+ <--- 4* <--- 1  Duplicate Number
2 <--- 4+<--- 1 <--- 3+ <--- 4* <--- 2  Duplicate Number
2 <--- 4+<--- 3 <--- 3+ <--- 4* <--- 1
2 <--- 4+<--- 3 <--- 3+ <--- 4* <--- 2  Duplicate Number
3+<--- 2 <--- 4+<--- 1  <--- 4  <--- 2+ Not Expandable
3+<--- 4*<--- 2 <--- 4+ <--- 1  <--- 2+ Not Expandable
3+<--- 4*<--- 2 <--- 4+ <--- 1  <--- 3+ Duplicate Number
3+<--- 4*<--- 2 <--- 4+ <--- 1  <--- 4
3+<--- 4*<--- 2 <--- 4+ <--- 3  <--- 3+ Duplicate Number
4+<--- 1 <--- 3+<--- 4* <--- 2  <--- 4+ Duplicate Number
4+<--- 3 <--- 3+<--- 4* <--- 1  <--- 2+ Not Expandable
4+<--- 3 <--- 3+<--- 4* <--- 1  <--- 3+ Duplicate Number
4+<--- 3 <--- 3+<--- 4* <--- 1  <--- 4
4+<--- 3 <--- 3+<--- 4* <--- 2  <--- 4+ Duplicate Number
1 <--- 3+<--- 2 <--- 4+ <--- 3  <--- 3+ Duplicate Number
1 <--- 3+<--- 4*<--- 2  <--- 4+ <--- 1  Duplicate Number
1 <--- 3+<--- 4*<--- 2  <--- 4+ <--- 3
4*<--- 1 <--- 3+<--- 2  <--- 4+ <--- 1  Duplicate Number
4*<--- 1 <--- 3+<--- 2  <--- 4+ <--- 3
4*<--- 2 <--- 4+<--- 1  <--- 3+ <--- 3
4*<--- 2 <--- 4+<--- 1  <--- 3+ <--- 2  Duplicate Number
4*<--- 2 <--- 4+<--- 1  <--- 3+ <--- 4* Duplicate Number
4*<--- 2 <--- 4+<--- 1  <--- 4  <--- 2+ Not Expandable
4*<--- 2 <--- 4+<--- 3  <--- 3+ <--- 3  Duplicate Number
4*<--- 2 <--- 4+<--- 3  <--- 3+ <--- 2  Duplicate Number
4*<--- 2 <--- 4+<--- 3  <--- 3+ <--- 4* Duplicate Number

STEP 7: Twelve potential seven-element, end-of-pathway sequences identified by
using the results of STEP 2 to expand eight "accepted" six-element sequences.
Note that STEP 6 shows nineteen sextets with internal loops (duplicate
numbers) and six sextets that are not expandable. Those twenty-five sequences
are not carried forward for consideration in the STEP 7 seven-element
candidate sequence list generation process.

3 <--- 3+<--- 2 <--- 4+ <--- 1  <--- 4 <--- 2+ Not Expandable
3 <--- 3+<--- 4*<--- 2  <--- 4+ <--- 1 <--- 2+ Not Expandable
3 <--- 3+<--- 4*<--- 2  <--- 4+ <--- 1 <--- 3+ Duplicate Number
3 <--- 3+<--- 4*<--- 2  <--- 4+ <--- 1 <--- 4
2 <--- 4+<--- 3 <--- 3+ <--- 4* <--- 1 <--- 2+ Not Expandable
2 <--- 4+<--- 3 <--- 3+ <--- 4* <--- 1 <--- 3+ Duplicate Number
2 <--- 4+<--- 3 <--- 3+ <--- 4* <--- 1 <--- 4
3+<--- 4*<--- 2 <--- 4+ <--- 1  <--- 4 <--- 2+ Not Expandable
4+<--- 3 <--- 3+<--- 4* <--- 1  <--- 4 <--- 2+ Not Expandable
1 <--- 3+<--- 4*<--- 2  <--- 4+ <--- 3 <--- 3+ Duplicate Number
4*<--- 1 <--- 3+<--- 2  <--- 4+ <--- 3 <--- 3+ Duplicate Number
4*<--- 2 <--- 4+<--- 1  <--- 3+ <--- 3 <--- 3+ Duplicate Number

STEP 8: Two eight-element solution pathways identified by using the results of
STEP 2 to expand two "accepted" seven element sequences. Note that STEP 7
shows five septets with internal loops (duplicate numbers) and five septets
that are not expandable. Those ten sequences are not carried forward for
consideration in the STEP 8 eight-element solution path definition process.

3 <--- 3+<--- 4*<--- 2  <--- 4+ <--- 1 <--- 4  <--- 2+
2 <--- 4+<--- 3 <--- 3+ <--- 4* <--- 1 <--- 4  <--- 2+

[C] SHORTCUT CONSIDERATIONS

Given the magnitude of the task that faces the puzzle solver when using the
Basic Algorithm, it is perhaps natural to ask, "Is there not a better way?"

The answer to that question is a qualified "maybe" if "shortcuts" constructed
based upon a knowledge of the sequence construction principles employed by
basic algorithm are pursued. Note however the details of a successful shortcut
approach will depend upon the nature of the challenge presented by the Clock
Puzzle under consideration.

Generally, shortcuts will be based upon a combination of "creative" use of
"initial" products produced by the basic algorithm plus careful consideration
of any "special features" presented by the Clock Puzzle of interest.

A comprehensive discussion of possible shortcut methods would require a
comprehensive examination of a large set of Clock Puzzle solutions. This
discussion will limit itself to a re examination of the example Clock Puzzle
as a means of suggesting approaches to shortcut development. More generally,
shortcut definition will only be limited by the creativity of the puzzle
solver and the characteristics of the challenge presented by the puzzle.

To begin, consider the proposition that, for the example puzzle, the Basic
Algorithm views the number 8 as being defined as 1+1+1+1+1+1+1+1. This so
since the Basic Algorithm starts with a single number and, at each step, adds
one additional number to form an ordered sequence longer by one unit. When an
ordered sequence of eight numbers has been found, the puzzle has been solved.

There are many alternate ways of viewing the number 8; other viewpoints
include 2+2+2+2, 2+3+3 and 4+4. "Shortcut" methods based upon the alternate
views can be conceived and used to solve the example puzzle.

Given the view is 2+2+2+2, one thought might be to apply the approach of the
Basic Algorithm to add two sequenced numbers to any candidate solution
sequence at each step. This could be done for the example problem by using the
triplets found in STEP 3 and the pairs displayed in STEP 2. The first number
of each triplet would be used as an "overlap" connector to key back to
matching pairs shown in STEP 2 to form quartets and then used again to key
back to matching quartets to define sextets. One more step using triplets to
key back to sextets would yield the puzzle solution.

The 2+3+3 view could be used to develop a puzzle solution by adding two Step 3
triplets to "qualified" number pairs displayed by the results of Step 2. In
the "direct" approach triplet overlap would not be used to key back the
triplets to the "correct" pairs Shown in Step 2. Instead, the successor -
predecessor relationships displayed by the pairs listed in Step 2 would be
used to combine (concatenate) a pair and a triplet to form a candidate five-
element number sequence. (One alternate would be to use an "overlap" connector
based upon use of four element sequences listed in STEP 4.) Combining the
five-element number sequences with the appropriate triplets would then produce
the solutions to the puzzle.

The 4+4 view admits a "one-pass" solution process that combines two of the
four-element products of STEP 4 through use of the precedence relationships
displayed by the pairings shown in STEP 2 to produce the puzzle solution. The
analysis is simplified by the "special feature" exhibited by the puzzle under
study: STEP 1 has identified the starting point for the solution path(s) of
the puzzle. The "shortcut" solution method is as shown.

STEP A: Scan the STEP 4 product list to select all quartets having the known
starting point as the final element of the ordered sequence of four numbers.

2 <--- 4+<--- 1 <--- 2+
3+<--- 4*<--- 1 <--- 2+
4+<--- 1 <--- 4 <--- 2+
4*<--- 1 <--- 4 <--- 2+

STEP B: List all other STEP 4 products not showing Duplicate Numbers in their
sequence of four numbers. They are potential ending sequences for the
identified candidate beginning sequences.

3 <--- 3+<--- 2 <--- 4+
3 <--- 3+<--- 4*<--- 1
3 <--- 3+<--- 4*<--- 2
2 <--- 4+<--- 1 <--- 4
2 <--- 4+<--- 1 <--- 3+
2 <--- 4+<--- 3 <--- 3+
3+<--- 2 <--- 4+<--- 1
3+<--- 2 <--- 4+<--- 3
3+<--- 4*<--- 1 <--- 4
3+<--- 4*<--- 2 <--- 4+
4+<--- 1 <--- 3+<--- 3
4+<--- 1 <--- 3+<--- 2
4+<--- 1 <--- 3+<--- 4*
4+<--- 3 <--- 3+<--- 2
4+<--- 3 <--- 3+<--- 4*
1 <--- 3+<--- 2 <--- 4+
1 <--- 3+<--- 4*<--- 2
4*<--- 1 <--- 3+<--- 3
4*<--- 1 <--- 3+<--- 2
4*<--- 2 <--- 4+<--- 1
4*<--- 2 <--- 4+<--- 3

STEP C: Evaluate each four-element sequence of STEP A for potential
combination (concatenation) with one or more of the sequences shown in STEP B.
Combination is possible only if: (i) the leftmost number of a considered
sequence from STEP A is a predecessor for the rightmost number of the
considered four-element sequence of STEP B and (ii) no duplicate numbers will
be seen in the eight-element sequence that results if the two quartets are
combined.

First STEP A Quartet: 2 is a predecessor for 3+ and 4* only. Contained
numbers are 2, 4+ and 1. The two STEP B quartets having 3+ in the
"connection" position contain duplicate numbers. The two STEP B quartets
having 4* in the connection position contain duplicate numbers. The first
STEP A quartet is not on a solution path.

Second STEP A Quartet: 3+ is a predecessor for 3 and 1 only. Contained
numbers are 3+, 4* and 1. The four STEP B quartets having 3 in the
connection position contain duplicate numbers. The three STEP B quartets
having 1 in the connection position contain duplicate numbers. The
second STEP A quartet is not on a solution path.

Third STEP A Quartet: 4+ is a predecessor for 2 only. Contained numbers
are 4+, 1 and 4. Of the five STEP B quartets having 2 in the connection
position, only the first (3<--- 3+<--- 4*<--- 2) does not contain
duplicate numbers. The third STEP A quartet is a starting four-element
set for a solution pathway.

Fourth STEP A Quartet: 4* is a predecessor for 3+ only. Contained numbers
are 4*, 1 and 4. Of the two STEP B quartets having 3+ in the connection
position, one (2<--- 4+<--- 3<--- 3+) does not contain duplicate
numbers. The fourth STEP A quartet is a starting four-element set for a
solution pathway.

STEP D: Combine the quartets to define the puzzle solution pathway(s)

3 <--- 3+<--- 4*<--- 2  <--- 4+<--- 1 <--- 4 <--- 2+
2 <--- 4+<--- 3 <--- 3+ <--- 4*<--- 1 <--- 4 <--- 2+

[D] USE OF SPECIAL FEATURES

Any systematic path elimination method can be used to resolve Clock Puzzles.
Rather than "begin at the end" and work "backwards" as described by the Basic
Algorithm, it may seem more natural to "begin at the start" and work forwards.

An algorithm based upon this latter method would begin by assuming all clock
number positions to be candidates for a starting point designation. The
algorithm would continue to use the identified starting point candidates to
proceed to work through Clock Puzzle maze pathway alternatives and identify a
solution path or pathways. The step-by-step details of such a "forward"
solution method would be much like those displayed for the Basic "backward"
Algorithm.

In the abstract, it would seem that the amount of effort required under either
approach would be roughly the same. The reality of any given situation may be
different as puzzle characteristics may favor the application of one
alternative over the other.

For example, if a single number of the Clock Puzzle set can be identified as
the unique starting point for all solution pathways, it would seem that the
forward solution approach might produce results with less effort than the
backward solution approach applied to the same puzzle. (In the general case,
certain identification of a unique ending point for all solution pathways will
not be possible.) The example Clock Puzzle provides means to test this
hypothesis by using the identified "starting point" number from STEP 1 of the
"backward" approach to begin the forward solution process.

A step-by-step description of the resulting solution process for the example
Clock Puzzle follows.

STEP 0: Consider the displayed Clock Puzzle which appeared during game play.

3
4       2

1           4

4       3
2

STEP 1: Eight potential starting points identified.

3
2
4
3+
2+
4+
1
4*

STEP 2: Examine potential starting points for predecessors. Since there is one
number without any predecessors, a unique starting point has been identified.

3 <--- 3+
2 <--- 4+
4 <--- 2+
3+<--- 3
3+<--- 2
3+<--- 4*
2+ None   Starting Point
4+<--- 1
4+<--- 3
1 <--- 4
1 <--- 3+
1 <--- 2+
4*<--- 1
4*<--- 2

STEP 3: Use the candidate starting point list to establish successor pairs for
use in the pathway candidate elimination process. For this "symmetric" puzzle
some elements have a single successor.

3 ---> 3+
3 ---> 4+
2 ---> 3+
2 ---> 4*
4 ---> 1
3+---> 3
3+---> 1
2+---> 4
2+---> 1
4+---> 2
1 ---> 4+
1 ---> 4*
4*---> 3+

STEP 4: Use the identified starting point and STEP 3 successor pair sequences
to build candidate two-element path segments.

2+---> 4
2+---> 1

STEP 5: Use STEP 3 successor pairs to and the STEP 4 path segments to build
candidate three-element path segments.

2+---> 4 ---> 1
2+---> 1 ---> 4+
2+---> 1 ---> 4*

STEP 6: Use STEP 3 successor pairs and STEP 5 path segments to build candidate
four-element path segments.

2+---> 4 ---> 1 ---> 4+
2+---> 4 ---> 1 ---> 4*
2+---> 1 ---> 4+---> 2
2+---> 1 ---> 4*---> 3+

STEP 7: Use STEP 3 successor pairs and STEP 6 path segments to build candidate
five-element path segments.

2+---> 4 ---> 1 ---> 4+---> 2
2+---> 4 ---> 1 ---> 4*---> 3+
2+---> 1 ---> 4+---> 2 ---> 3+
2+---> 1 ---> 4+---> 2 ---> 4*
2+---> 1 ---> 4*---> 3+---> 3
2+---> 1 ---> 4*---> 3+---> 1  Duplicate Number

STEP 8: Use STEP 3 successor pairs and STEP 7 path segments that do not
contain duplicate numbers to build candidate six-element path segments. (Five-
element path segments containing duplicate numbers need not be considered
further by this search.)

2+---> 4 ---> 1 ---> 4+---> 2 ---> 3+
2+---> 4 ---> 1 ---> 4+---> 2 ---> 4*
2+---> 4 ---> 1 ---> 4*---> 3+---> 3
2+---> 4 ---> 1 ---> 4*---> 3+---> 1  Duplicate Number
2+---> 1 ---> 4+---> 2 ---> 3+---> 3
2+---> 1 ---> 4+---> 2 ---> 3+---> 1  Duplicate Number
2+---> 1 ---> 4+---> 2 ---> 4*---> 3+
2+---> 1 ---> 4*---> 3+---> 3 ---> 3+ Duplicate Number
2+---> 1 ---> 4*---> 3+---> 3 ---> 4+

STEP 9: Use STEP 3 successor pairs and STEP 8 path segments that do not
contain duplicate numbers to build candidate seven-element path segments. (Six
element path segments containing duplicate numbers need not be considered
further in this search.)

2+---> 4 ---> 1 ---> 4+---> 2 ---> 3+---> 3
2+---> 4 ---> 1 ---> 4+---> 2 ---> 3+---> 1  Duplicate Number
2+---> 4 ---> 1 ---> 4+---> 2 ---> 4*---> 3+
2+---> 4 ---> 1 ---> 4*---> 3+---> 3 ---> 3+ Duplicate Number
2+---> 4 ---> 1 ---> 4*---> 3+---> 3 ---> 4+
2+---> 1 ---> 4+---> 2 ---> 3+---> 3 ---> 3+ Duplicate Number
2+---> 1 ---> 4+---> 2 ---> 4*---> 3+---> 4+ Duplicate Number
2+---> 1 ---> 4+---> 2 ---> 4*---> 3+---> 3
2+---> 1 ---> 4+---> 2 ---> 4*---> 3+---> 1  Duplicate Number
2+---> 1 ---> 4*---> 3+---> 3 ---> 4+---> 2

STEP 10: Use STEP 3 successor pairs and STEP 9 path segments that do not
contain duplicate numbers to build the eight-element candidate solution path
set. (Seven-element path segments containing duplicate numbers need not be
considered further in this search.)

2+---> 4 ---> 1 ---> 4+---> 2 ---> 3+---> 3 ---> 3+ Duplicate Number
2+---> 4 ---> 1 ---> 4+---> 2 ---> 3+---> 3 ---> 4+ Duplicate Number
2+---> 4 ---> 1 ---> 4+---> 2 ---> 4*---> 3+---> 3
2+---> 4 ---> 1 ---> 4+---> 2 ---> 4*---> 3+---> 1  Duplicate Number
2+---> 4 ---> 1 ---> 4*---> 3+---> 3 ---> 4+---> 2
2+---> 1 ---> 4+---> 2 ---> 4*---> 3+---> 3 ---> 3+ Duplicate Number
2+---> 1 ---> 4+---> 2 ---> 4*---> 3+---> 3 ---> 4+ Duplicate Number
2+---> 1 ---> 4*---> 3+---> 3 ---> 4+---> 2 ---> 3+ Duplicate Number
2+---> 1 ---> 4*---> 3+---> 3 ---> 4+---> 2 ---> 4* Duplicate Number

STEP 11: Evaluate the eight-element solution path candidates to identify the
Clock Puzzle solution paths.

2+---> 4 ---> 1 ---> 4+---> 2 ---> 4*---> 3+---> 3
2+---> 4 ---> 1 ---> 4*---> 3+---> 3 ---> 4+---> 2

[E] GENERAL PRINCIPLES

The discussion of this section is intended to elevate the solution technique
coverage provided by previous address of a specific eight-element Clock Puzzle
example to produce general guidelines that may be applied to solve any Clock
Puzzle that may be presented by the Final Fantasy XIII-2 game.

As a practical matter, the overriding principle to be recognized is that a
typical player will be interested in finding ONE solution to any given Clock
Puzzle NOT ALL solutions. The algorithmic discussion presented in this guide
represents means for finding all solutions of any given puzzle. Even so, the
principles presented will be useful to those interested in obtaining a single
solution as well; just apply the suggested ideas until a desirable solution
has been found.

A sophisticated reader may be able to thoroughly understand and apply the
solution principles contained in this document by a single read of the text in
this section. Perhaps more typical will be the reader who will find it useful
to refer back to the example detailed presentations to reinforce the meaning
intent of the "high-level" statements made here and to realize the full
benefit of the presentation.

Clock Puzzles presented by the game can range from a simpler five-element
solution exercise (five numbers are presented to the player on the "face" of
the "clock") to a significantly more complex challenge involving the
identification of at least one solution pathway through the maze represented
by the appearance of thirteen elements (numbers) on the face of the clock.

It is the nature of Clock Puzzles that the complexity of the potential
solution path set increases at a disproportionately high rate as the number of
clock-face elements (numbers) to be addressed increases. The exact
relationship between the increases is unknown. It does seem safe to conclude
that to say "doubling the number of elements (numbers) to be addressed by the
player more than doubles the amount of effort required to resolve the puzzle"
understates the effect of an increase in clock face elements.

The magnitude of the task faced to solve the more complex puzzles that could
be presented by the game leads one to have interest in "shortcut" methods that
have potential for reducing the number of operations needed to produce a
solution.

To keep the discussion "real" the general solution principles are presented
with reference to a thirteen-element (thirteen numbers appear on the clock
face) Clock Puzzle rather than through use of mathematical abstraction.

Observe that successful application of any of the described processes will
require careful attention to detail and the preparation and maintenance of a
complete record of the decisions made to produce and eliminate candidate
solution pathways.

The form of the record could be a tabulation (such as shown in EXAMPLE
APPLICATION) of alternatives considered. Another option would be an organized
display of a series of clock face diagrams annotated by lines drawn between
numbers (elements) with arrowheads used to indicate movement direction to
produce a visual record. (The corresponding displays for the EXAMPLE
APPLICATION are not included in this document due to the difficult nature of
producing the necessary ASCII artwork.)

[E1] Solution Preliminaries

If needed, formal Clock Puzzle solution should be accomplished before an
attempt to play through the puzzle is made. This because, for at least some
puzzle situations, if an attempt to navigate the maze is made and fails, the
player will find that the Clock Puzzle has changed when a retry is attempted.

Thus the first action in any solution attempt should be a careful, noninvasive
examination of the presented puzzle to produce an accurate sketch of the Clock
Face showing the position of all (thirteen) presented numeric elements.

[E2] "Pure" Solution Approaches

This work discusses use of two "Pure" Solution Approaches. There is a
"backward-looking" method and a "forward-looking" method.

The "backward-looking" method begins by assuming that all of the (thirteen)
presented elements are candidate end points of a solution path through the
maze. It continues by identifying the possible predecessor elements for each
of the assumed end point elements to define the complete set of successor-
predecessor pairs (also called "precedence pairs" in the text) for the puzzle
at hand. Solution of the puzzle continues by using the identified successor-
predecessor pairs to (where possible) add a single element to each successor-
predecessor pair to form triplet number sets. Solution is completed by
continuing use of the successor-predecessor pairs through the number of cycles
necessary to produce at least one ordered number string in which no particular
clock face element appears more than once and whose length is equal to the
number of elements presented by the clock face of the puzzle (thirteen).

The "forward-looking" method begins by assuming that all of the (thirteen)
presented elements are candidate starting points of a solution path through
the maze. It continues by identifying the possible successor elements for each
of the assumed starting point elements (there will always be one or two) to
define the complete set of predecessor-successor (precedence) pairs for the
puzzle at hand. Solution of the puzzle continues by using the identified
predecessor-successor pairs to add a single element to each predecessor-
successor pair to form triplet number sets. Solution is completed by
continuing use of the predecessor-successor pairs through the number of cycles
necessary to produce at least one ordered number string in which no particular
clock face element appears more than once and whose length is equal to the
number of elements presented by the clock face of the puzzle (thirteen).

[E3] "Pure" Shortcuts

"Pure Shortcut" is here defined as a solution method obtained solely by using
procedural techniques derived from one and only one of the two "Pure" Solution
Approaches. Many such shortcuts can be conceived.

Each "Pure" Solution Approach produces thirteen-element solution sequences by
starting with a single element and investigating single element additions
until a satisfactory thirteen-element pathway is found. As discussed here,
each "Pure" Shortcut is defined in this context by combining multiple element
groups (each containing more than one element) to produce a satisfactory
thirteen-element pathway.

This text will represent the class of "Pure Shortcut" solutions by overview of
two example attempts at finding a solution for the generic thirteen-element
Clock Puzzle. It is thought that this discussion will be adequate to reveal
the key application principles associated with this class of solution methods.

Consider a situation in which decision had been made to solve a thirteen-
element Clock Puzzle by using a "Pure" Solution Approach to produce "half" of
a thirteen-element solution sequence followed by the use of shortcut logic to
arrive at a complete solution. Additions that suggest the two options to be
considered can be described as 6 + 6 + 1 and 5 + 5 + 3.

The first example begins from a view of thirteen as six plus one plus six (13
= 6 + 1 + 6) to set the intended structure of the first "Pure Shortcut"
solution process example.

This implication of the selected structure is that the information needed to
complete this shortcut solution attempt will be collected by using a "Pure"
Solution Approach to produce the feasible set of all candidate six-element
solution path sequences. As a part of the process that produces the six-
element sequences, the useful feasible set of three-element sequences will
also be generated.

Solution path development can follow by completing two steps that make use of
the generated element sequence information.

Step 1 is to list all "feasible" six-element pairs. A "feasible" pair will not
have the same element in both members of the pair and will be formed with
recognition of the directionality of the six-element sequence for each pair
member.

Step 2 is to use the members of the feasible set of three-element sequences to
attempt to connect feasible pair members. Directionality of the three-element
connectors must be considered by the connection attempts.

Each three-element sequence has potential to add the thirteenth element needed
to complete a solution path as the destination position of its middle element.
The two end elements provide a simple connection-feasibility check.

For "success" both end point elements of a candidate triplet must "overlap"
identical elements in one of the members of a six-element pair. Additionally
the middle element must not duplicate any element that appears in the two six-
element sequences to be combined.

Each successful connection produces a thirteen-element Clock Puzzle solution
path.

The second example begins from a view of thirteen as five plus three plus five
(13 = 5 + 3 + 5) to set the intended structure of the intended shortcut
solution process example.

Observe that this view trades a reduction in the length of the generated
sequences needed to execute the shortcut process for an increase in logical
complexity associated with establishing thirteen-element sequences by
combining two five-element sequences with one three-element sequence.

This implication of the selected structure is that the information needed to
complete this shortcut solution attempt will be collected by using a "Pure"
Solution Approach to produce the feasible set of all candidate five-element
solution path sequences. As a part of the process that produces the five-
element sequences, the feasible set of two-element sequences and the feasible
set of three-element sequences will also be generated.

Solution path development can follow by completing two steps that make use of
the generated element sequence information.

Step 1 is to list all "feasible" five-element pairs. A "feasible" pair will
not have the same element in both members of the pair and will be formed with
recognition of the directionality of the five-element sequence for each pair
member.

Step 2 is to use the members of the feasible set of two-element sequences to
guide attempts to connect candidate pairs of two five-element pathway segments
by use of the candidate three-element path segments. Directionality of the
candidate segments must be considered by the connection process.

A "complete" connection attempt will require consideration of the alternatives
"allowed" by the "precedence pair set" (the feasible set of two element
sequences) for two connections between candidate three-element sequences and
both members of each candidate five-element sequence pair.

For "success" the three "combined" (concatenated) segments must only contain
one copy of each of the thirteen elements presented by the Clock Puzzle.

Each successful connection produces a thirteen-element Clock Puzzle solution
path.

[E4] "Combination" Shortcuts

"Combination Shortcut" is here defined as a solution method obtained by
combining procedural techniques drawn from both of the two "Pure" Solution
Approaches. Quite a large number such shortcuts can be created.

A "Combination" solution approach will produce thirteen-element solution
sequences by starting with one of the two "Pure" Solution Approaches and
accomplishing a switch to the alternate "Pure" Solution Approach as the
solution process is continued.

Conceptually, multiple switches between the two "pure" processes could be a
part of a "Combination Shortcut" solution approach. However, each switch will
increase the complexity of the decision process that must be applied to
complete the solution. Therefore, as a practical matter, this text will
present a few "Combination Shortcut" concepts involving simple switch
situations only.

The basic idea is to traverse a Clock Puzzle maze by starting at both of its
"ends" to produce a "meeting" in its middle to identify a solution pathway.

Making use of "Combination" Shortcuts can be a tricky business. The four

The first example situation to be considered can arise as a natural
consequence of initiating a solution of a (thirteen-element) Clock Puzzle
using the "backward-looking" "Pure" Solution Approach.

Execution of the second step of the "backward-looking" process creates the set
of "ordered" element pairs that might be the last two elements of a solution
path. A side product of this effort will be the identification of any one of
the thirteen clock face elements that has no predecessor element. Assuming
that the puzzle has a solution, and since each element must be a part of some
solution path, the conclusion must be that the identified element is the
starting point for all solution paths through the Clock Puzzle maze.

At this point it seems natural to switch to the "forward-looking" process to
complete the puzzle solution. After all, identification of the unique starting
point for solution paths has just reduced the number of alternative paths that
must be considered under the forward-looking method by a factor of thirteen!
(Concurrently, the immediate impact on the number of alternative paths that
must be considered under the "backward-looking" method is not as significant.)

More generally, the possible existence of a unique starting point in any posed
puzzle suggests that the puzzle solver might wish to begin any puzzle solution
investigation by taking the trouble to execute the first two steps of the
"backward-looking" process. The potential for reward from using a "simplified"
solution method involving application of the forward-looking "Pure" Solution
Approach starting from the identified starting point will always be present.

As a second example consider a situation where the (thirteen-element) Clock
Puzzle does not have a unique starting point element. Absent any additional
information it might be concluded that use of either "Pure" Solution Approach
would produce results with an equal amount of effort.

Indeed that may be the case but it is also possible that the (unknown)
internal structure of the puzzle could favor the application of one of the two
"Pure" Solution Approaches over the other.

The puzzle solver could just pick one of the two methods and proceed to a
solution. Alternatively the puzzle solver could adopt an iterative approach
involving the performance of a step of each "Pure" process, comparing the
results of each and then making a choice of whether to continue iterating,
combine the intermediate products generated, or settle on one of the two
"Pure" Solution Approaches to complete the solution.

It seems possible that an "uneven" application of the two "Pure" processes
plus combination of intermediate products will minimize the amount of effort
required to produce a solution. For the forward-looking process, the candidate
segment list is expanded by a maximum of one at each trial (there is at most
two successor elements for each expansion candidate). As the "backward-looking
process potentially will see more than two predecessor elements for successor
elements, its candidate segment list expansion might be greater than that seen
under the forward-looking method.

In this context, hypothetically one possible derived solution space view of
the thirteen-element Clock Puzzle could be represented by "eight plus five"
(13 = 8 + 5). That is, the hypothetical solution method applied was to first
generate all possible eight-element pathway sequences using the forward-
looking process and to also generate all possible five-element sequences using
the backward-looking process. Solution pathways were then defined by combining
eight-element pathways with five-element pathways using the techniques
introduced in the discussion of the "Pure" Shortcut section.

The third example situation is defined by consideration of features of
particular interest drawn from the first and second example situations. The
point of interest is complexity introduced when multiple switches (in this
case there will be two) are used in a "Combination" Shortcut solution method.

Suppose the backward-looking process has been applied to a thirteen-element
Clock Puzzle to produce the identification of a unique solution pathway
starting point. Also suppose that the forward-looking process has used the
identified starting point to create a candidate set of eight-element pathway
sequences. Finally suppose that, by using an iterative process such as
described under the second example situation as a decision aid, a decision has
been made to complete the solution by combining eight-element sequences
generated by the forward-looking process with five-element sequences generated
by the backward-looking process. That is, once again, the achieved solution
view is "eight plus five" (13 = 8 + 5).

As before the combination techniques described in the "Pure" Shortcut section
apply but, one must recognize what has been done in achieving the solution
view to properly apply them. The "precedence pairs" used to investigate
possible connections must be comprehensive. Thus the precedence pairs found by
beginning the forward-looking process from the identified starting point must
be discarded as useless in this application (there will be at most two of
them). On the other hand, the precedence pairs generated by the backward
solution process have comprehensive coverage of the possible connection
options. It is those pairs that must be used in the search for solution paths.

The fourth example situation recognizes the fact that, given that a backward-
looking process and a forward-looking process exists, there is opportunity for
solution of a thirteen-element Clock Puzzle by making use of candidate four-
element sequences "only".

Begin the solution by using the backward-looking process to generate its
feasible four-element sequence candidates. Likewise use the forward-looking
process to generate its feasible four-element sequence candidates.

For this latter effort, the sequence generation MUST start from the complete
set of thirteen elements even if it is known that a unique stating point
exists. The set of four-element sequences realized by using only a single
starting point as a basis for generation will not provide complete exposure of
candidate pathway options.

Continue solution development by arranging the four-element sequences produced
by the "backward-looking" method in pairs such that precedence relationships
are preserved and the "left-end" element of one sequence is identical to the
"right-end" element of a second sequence but that, otherwise, no duplicate
elements are seen. Do likewise for the four-element sequences generated by the
"forward-looking method.

Combine the individual four-element sequences of each such pair by using their
common element to join them at the overlap. Each four-element pair becomes a
single seven-element sequence.

Follow up this step by pairing the seven-element sequences generated from the
"forward-looking" process with the seven-element sequences generated from the
"backward-looking" process. Directionality of the sequences must be considered
as the pairing is accomplished.

Construct pairs that have the "left-end" element of one seven-element sequence
identical to the "right-end" element of a second seven element sequence and
which do not otherwise exhibit duplicate clock face elements.

Complete the solution by using the overlap element to join the two seven-
element sequences and define a thirteen-element solution pathway.

This neat and simple "Combination Shortcut" method may be one convenient way
to solve a thirteen-element Clock Puzzle.

[F] SOLUTION GRAPHICS

Prior discussion has focused upon presentation of techniques closely
associated with a Reference Solution Approach to provide the reader with a
guaranteed-to-be-successful puzzle analysis method and to establish a
technically accurate basis for Clock Puzzle resolution strategy development.

It is the intent of this section to suggest to the reader that "creative",
intuition-based solution approaches incorporating use of simple graphical
decision aids, decision tree analysis records, intermediate conclusions based
upon auxiliary logical analyses and involving "trial-and-error" examination of
candidate solution alternatives can be of benefit.

The discussion focus also shifts from having at least a passing interest in
finding all solutions to a given puzzle to one of finding a single solution
with as little effort as practical. The methods described can be used to
accomplish an exhaustive examination of any given Clock Puzzle for the purpose
of finding all possible solutions if desired.

At the conclusion of these introductory remarks the presentation will proceed

A section on basic Clock Puzzle characteristics is immediately followed by a
description of certain Clock Puzzle internal structures which can be used to
assist solution definition.

An example solution approach making use of the basic characteristics and
internal structures in the context of Clock Puzzle graphic diagrams is then
revealed.

A description of an example application of the suggested solution approach and
remarks on possible excursions concludes the topical coverage.

As a key feature of the solution approach described will be the use of Clock
Puzzle diagrams at each step of the to-be-described (example) process, the
title chosen for this text section has been used to reflect that emphasis.

There is an unfortunate drawback to making the choice to introduce Clock
Puzzle graphics as a key ingredient of the (example) solution process.

It has not been possible to produce the desired drawings to support a clear
graphical presentation of the proposed method using ASCII format artwork.

As a consequence, for the attempted puzzle, a reference graphic will be
presented with other graphics involved in its solution only described by
description provided by these pages by taking pencil and paper in hand and
using the descriptive paragraphs to produce any graphics thought to be needed
to understand the example discussion but found absent from the presentation.

The alternative seemed to be to not present the example solution approach
attended in these pages. In view of the great potential of this class of
methods for relatively "painless" Clock Puzzle resolution, it was felt that
such an omission would do the reader of this guide a significant disservice.

[F1] Puzzle Principles

Certain facts can be taken as "givens" independent of the Clock Puzzle being
attempted. The set of such facts that will serve as the foundation for the
solution process to be described are stipulated by the list following.

1. A Clock Puzzle diagram consists of a "circular" display of numbers termed
the "elements" of the Clock Puzzle.

2. Clock Puzzle graphics are constructed by adding "valid" directed line
segments connecting two Clock Puzzle elements to a Clock Puzzle diagram.

3. A "valid" directed line segment is one that starts at a chosen
"predecessor" element and terminates at a "successor" element determined by
moving around the Clock "face" a number of positions corresponding to the
number displayed by the chosen predecessor element.

4. Every chosen predecessor element will have one or two successor elements.

5. The total number of successor elements for a Clock Puzzle will, at most, be
twice the number of elements appearing on the clock face.

6. A "solution" is a set of directed line segments which "connect" all puzzle
elements without incorporating any element more than once.

7. All Final Fantasy XIII-2 Clock Puzzles will have at least one solution.

8. A solution for a Clock Puzzle diagram with N elements will have exactly N-1
directed line segments. (e.g., a solution "path" for a Clock Puzzle diagram
with 8 elements will be composed of 7 directed line segments.)

[F2] Puzzle Structures

There are facts that can be taken as "givens" in addition to those of the
"Puzzle Principles" list. These facts seem not to be obvious from an initial
inspection of a general Clock Puzzle but can be inferred from the evaluation
of the internal structure of a variety of Clock Puzzle appearances.

1.  A Clock Puzzle element may exhibit potential connections with none, one or
multiple predecessor elements.

2.  A Clock Puzzle element with no predecessor element is the unique starting
point for all solution pathways for the containing Clock Puzzle.

3.  A Clock Puzzle element having one unique predecessor element that is not
also a unique predecessor element for another element may be the successor
element in a pair of elements that define a directed line segment that is
part of the solution pathway of the containing Clock Puzzle.

4.  A Clock Puzzle having one unique predecessor element that is not also a
unique predecessor element for another element may be a starting point
element for a solution pathway of the containing Clock Puzzle.

5.  If two Clock Puzzle elements have the same, unique predecessor element,
one of the two presumed successor elements must be the starting point of a
solution pathway for the containing Clock Puzzle.

6.  If two Clock Puzzle elements have the same, unique predecessor element,
One of the two presumed successor elements must be the successor element
for a directed line segment starting at the predecessor element that is a
directed line segment on the solution pathway of the Clock Puzzle.

7.  A Clock Puzzle element with two unique predecessor elements will be the
successor element of a directed line segment originating at one of the two
predecessor elements and that is a part of the solution pathway for the
containing Clock Puzzle.

8.  If two Clock Puzzle elements have the same unique successor Clock Puzzle
element, one of the two presumed predecessor elements must be the end
point for a solution pathway of the containing Clock Puzzle.

9.  If two Clock Puzzle elements have the same unique successor Clock Puzzle
element, one of the two presumed predecessor elements must be the starting
point for a directed line segment to the successor element that is a part
of the solution pathway of the containing Clock Puzzle.

10. A solution pathway starting point element will show a single successor
element in the Clock Puzzle solution pathway description.

11. A solution pathway ending point element will have a single predecessor
element in the Clock Puzzle solution pathway description.

12. Other than starting or ending point elements, all elements will show a
single predecessor element and a single successor element in the Clock
Puzzle solution pathway description.

[F3] Solution Approach

The thrust of the chosen approach is to use the Puzzle Principles and Puzzle
Structures facts as the basis for an iterative examination and revision of
Clock Puzzle diagrams to find a logical basis for solution pathway directed-
line-segment identifications.

As much as is possible, choice conclusions based upon logical examination of
the alternatives presented by the Clock Puzzle diagrams should be used to
continue the solution process.

Once the available information upon which to base logic-driven choices have
been exhausted, resort must be made to informed "guesses" as to what the next
step in the solution search must be. Choices made in this manner should be
recorded in a "Decision Tree" structure that comprises a graphical record of
"uncertain" decisions made as part of the examination history.

If all the choices made lead to the desired solution result, the decision tree
may be discarded. If a more exhaustive examination of the subject Clock Puzzle
is of interest or if no solution is found, the decision tree may be used to
avoid duplication of effort in a continued examination of the Clock Puzzle.

STEP 0: Annotate a graphical representation of the Clock Puzzle display with
"tags" that serve to uniquely distinguish each of the clock "face" elements
from all other clock face elements.

STEP 1: For each element of the clock face, determine the possible predecessor
elements (or numbers if more than one predecessor exists). As this is done
make a list of the ordered pairs of elements (successor <--- predecessor) that
result.

STEP 2: Confirm completeness of the list of element pairs by verifying the
count is twice the number of elements appearing on the clock face less the
number of elements appearing in the list that have only one successor element.

STEP 3: Decompose the Clock Puzzle by constructing displays showing potential
directed solution path links between "similar" pairs of elements. For example:

Graphic 0P: Precedence pair links for any element with no predecessor

Graphic 1P: Precedence pair links for any element with one predecessor

Graphic 1S: Precedence pair inks for any element with one successor

Graphic 2P: Precedence pair links for any element with two predecessors

Graphic 3P: Precedence pair links for any element with three predecessors

Graphic 4P: Precedence pair links for any element with four predecessors

STEP 4: Identify possibilities for combining individual Clock Puzzle displays
into one or more composite displays. Consider presented link directionalities
and recognize constraints imposed by observation of Puzzle Principles to

STEP 5: Use the Puzzle Structures rules to assist identification of link
additions and/or deletions required for solution pathway feasibility.

STEP 6: Produce composite displays that incorporate the identified changes
required and document the current status of the solution effort.

STEP 7: If, after a very thorough examination, a solution is not evident from
the composite Clock Puzzle displays created, develop a decision tree structure
showing the "unsupported" choices (i.e., "guesses") that must be made in order
to continue the solution process.

STEP 8: Move "down" the decision tree by recording the "first" necessary
choice made and displaying the results of that choice as a new composite Clock
Puzzle diagrams updated to show the impacts of the choice.

STEP 9: Repeat STEP 4 through STEP 8 until either a solution is found or the
decision possibilities are exhausted.

STEP 10: If no solution results as a result of one or more of "unsupported"
decisions made, review the decision tree record to continue the Clock Puzzle
examination using alternate decision choices starting with STEP 4.

[F4] Example Resolution

The example eight-element Clock Puzzle used to introduce the Reference
Solution Approach has been chosen as the example application of the just
introduced example process.  Among other things, this may enable reader
appreciation of the advantage of viewing any given clock Puzzle from more than
one perspective when attempting its solution.

STEP 0: Puzzle element uniqueness established.

3                                3
4       2                        4*      2
1           4   - - - - - - ->   1           4

4       3                        4+      3+
2                                2+

STEP 1: Precedence pair determination completed.

3 <--- 3+

2 <--- 4+ (Twice)

4 <--- 2+

3+<--- 3
3+<--- 2
3+<--- 4* (Twice)

2+ None

4+<--- 3
4+<--- 1

1 <--- 4  (Twice)
1 <--- 3+
1 <--- 2+

4*<--- 2
4*<--- 1

STEP 2: Four this eight-element puzzle, a maximum of sixteen precedence pair
appearances is expected. Since three elements show a single successor element,
the maximum expected number of precedence pair appearances becomes thirteen.
The number of precedence pairs seen in the developed list is in agreement with
the expectation.

STEP 3: Five graphic displays decompose the Clock Puzzle into useful visual
aids that assist puzzle solution (The reader is asked to use pencil and paper
to draw those based upon the written descriptions provided).

Graphic 0P: Shows two precedence pair links for the single element
having no potential predecessor element (2+---> 4 and
2+---> 1).

Graphic 1P: Shows three precedence pair links; one for each element
having a single, unique potential predecessor element
(3+--->3, 2+---> 4 and 4+--->2 ).

Graphic 1S: Shows three precedence pair links; one for each element
having a single, unique potential successor element
(4 --->1 , 4+--->2  and 4*--->3+).

Graphic 2P: Shows two precedence pair links for each of the two
elements having exactly two potential predecessor elements
(3 --->4+ and 1 --->4+) plus (2 --->4* and 1 --->4*).

Graphic 3P: Shows three precedence pair links for each of the two
elements having exactly three potential predecessor
elements (3 --->3+, 2 --->3+ and 4*--->3+) plus
(4 --->1 , 3+--->1  and 2+--->1 ).

Graphic 4P: Not required for this puzzle.

STEP 4: Inspection of the constructed solution graphics suggests that
examination of potential logical combinations of the simpler figures to form
composite link structures is a reasonable step toward solution definition.

Simple overlay of the links shown in the first four Graphics seems easy to do
but the result leaves directionality conflicts and predecessor pair
ambiguities to be resolved.  Immediate introduction of the link alternatives
suggested by Graphic 3P seems to complicate rather than resolve the
difficulties.  An obvious solution pathway seems not to be in evidence from
immediate combination.

STEP 5: Additional conclusions supporting definition of viable solution
alternatives can be made if the constructed solution graphics are evaluated
using recognized internal Puzzle Structure features.

Graphic 0P: The 2+ element is the unique starting point for all
solution pathways. Therefore, either 2+---> 4 or 2+---> 1
must be the initial directed line segment of any viable
solution pathway candidate.

Graphic 1P: Since 2+ is the unique starting point for all solution
pathways, all displayed precedence pair links (3+---> 3,
2+---> 4 and 4+--->2 )are valid directional line segments
of solution pathways. The 2+---> 1 solution pathway
starting point directional line segment link alternative
shown in Graphic 0 has been eliminated.

Graphic 1S: Since 4 is the second element of the solution pathway and
definitely not a solution path end point, the displayed
4 ---> 1 precedence pair link is the second directed line
segment of the solution pathway. The 4+--->2 precedence
pair link has been determined to be a solution pathway
directed line segment by the analysis of Graphic 1P. The
role of the 4*--->3+ link is yet to be determined.

Graphic 2P: Since 1 is the third element of the solution pathway, its
successor will be either 4+ or 4*.  If 4+, the link
3 --->4+ is not a viable solution pathway directed line
segment candidate. If 4*, the link 2 --->4* is not a viable
solution pathway directed line segment candidate.

Graphic 3P: Two precedence pair links for element 3+ remain as
candidates. The 3 --->3+ precedence pair link was
eliminated as a result of the discussion of Graphic 1P.
Predecessor pair links 2+---> 1 and 3+---> 1 are eliminated
by the directionality imposed by the position occupied by
element 1 in the partially-defined solution pathway.

STEP 6: A single composite diagram that reflects the conclusions of STEP 5
seems to provide sufficient information to support STEP 7 investigation.

The constructed diagram displays a two-segment start for a solution pathway as
2+--->4 --->1, known solution path directed line segments 3+--->3  and 4+--->2
as well as potential solution path directed line segment 4*--->3+. Known
solution path segments can be represented by solid, dark connection lines and
potential solution path segments by dashed, dark connection lines.

The constructed diagram also incorporates key "uncertain" link candidates.
Candidate connections for element 1 can be represented by dotted connection
links 1 --->4+ and 1 --->4*. One of the two possible predecessor pair
connections with element 3+ (4*--->3+) is a potential solution path directed
line segment. The other (2 --->3+) can be presented as a very light line
segment drawn between the involved elements.

STEP 7: Examination of the diagram suggests four issues that need attention in
order that a solution be achieved; a solution pathway continuation starting
with a directed line segment starting from element 1 needs to be determined; a
decision as to the use of potential pathway segment 4*--->3+ needs to be made;
a solution pathway directed line segment ending at element 3+ must be found,
and; the end point of the solution pathway must be determined.

Having exhausted the logic-driven choices that produce immediate solution
pathway directed line segment identifications for this puzzle; it is time to
set up a decision tree plan to continue solution definition. To keep things as
simple as possible, the chosen plan structure recognizes each pending
selection as one or more "either-or" decision points displayed as "forks" in a
continuous hierarchical path structure exhibiting successive two-legged forks
to record the succession of decisions to be examined. The purpose is that all
required decisions are recognized and, as decisions are made and results
obtained, that a complete record is kept.

From the various possibilities that exist, the following decision choices were
considered for ordered examination to enable continuation of the solution
search:

First decision point  - Element 4* is either an end point for a solution
pathway or is not such an element.

Second decision point - The successor element for element 1 is either
element 4* or it is element 4+.

Third decision point  - The predecessor element for element 3+ is either
element 2 or element 4*

Fourth decision point - The solution path end point is either element 3
or element 2

Once again, the reader is requested to supply the graphic corresponding to the
text description of the decision tree examination if needed.

STEP 8: Begin by assuming that element 4* in an end point for a solution path.

With reference to the composite diagram produced under STEP 6, the immediate
impacts are to remove the candidate solution path directed line segment
4*--->3+ from consideration and establish the predecessor pair link 1 --->4+
as the third directed line segment of the solution pathway.

In turn this requires the fifth element on the solution pathway to be the
element 2 .

Since element 2 is the fifth element on the solution pathway, it is required
to connect to a successor sixth element. The revised diagram shows that
element 3+ and element 4* are the two potential successor candidates for
element 2 .

However, connection to element 4* (as presumed endpoint of the solution
pathway) yields a solution pathway of five segments rather than the required
seven. Also, connection to element 3+ produces a six-segment pathway with no
immediate connection to the assumed end point element 4*.

Clearly the decision to make element 4* the final element of the solution
pathway was not one that produces a puzzle solution.

Therefore, label that branch of the decision tree structure as unrewarding and
proceed by consideration of the alternate choice for element 4*.

STEP 4: Update the composite drawing of STEP 6 by recognizing the 4*--->3+
predecessor pair link to be a directed line segment of the solution pathway.

STEP 5: Element 3+ is now "completely covered" as an element of the solution
pathway. This construction means that the candidate 2 --->3+ predecessor pair
link displayed in the composite drawing of STEP 6 does not define a viable
solution path directed line segment link candidate.

STEP 6: Update the current composite drawing by recognizing that the
predecessor pair candidate 2 --->3+ link should not appear in the diagram.

STEP 7: Move down the decision tree to the second decision point which
concerns making a choice of the successor element for element 1, the third
element on the puzzle solution path.

STEP 8: Select 4* as the solution path successor element for element 1.

A five-segment candidate solution path starting with element 2+ and ending
with element 3 is the immediate result.

As element 3 has only one successor element available to it (element 4+), the
sixth candidate solution path segment must be 3 --->4+.

Element 4+ begins a predecessor pair connection that is known to be one of the
directed line segments of the puzzle solution.

With the addition of this connection, a solution to the puzzle has been found.

2+--->4 --->1 --->4*--->3+--->3--->4+--->2

The discussion of this example and the underlying process will be concluded by
offering a retrospective perspective on characteristics of the presentation.

First and foremost, the reader should recognize that application of the
Graphics-based method described is much simpler and quicker than might be
inferred by a player tasked with the chore of completely understanding its
intent by examination of a detailed explanation such as has been presented.

A little practice using the techniques tool kit suggested should have the
player solving complex puzzles in a few hours at most. With total familiarity
of puzzle structures, use of a rapidly-produced series of sketches should
allow individual solutions to be found fairly quickly.

Secondly, the reader should recognize that creativity is the key to efficient
puzzle solution. The solution approach just presented is, in a real sense,
just an appetizer for player definition of an approach more suited to that
player's puzzle solution style.

Although not emphasized in the presentation, the described solution approach
has used a mix of Reference Solution Approach examination concepts, forward
and backward definition of candidate link structures, shortcuts, trial and
error alternatives investigation and solution graphics to achieve its end. The
interested reader should not hesitate to employ a "mix and match" techniques
usage strategy to create an efficient puzzle solution method for personal use.

In the example, there is a point at which at which the follow-on directed line
segment for the 4 --->1  solution pathway must be determined. Rather than use
the method presented in the example, one could choose to apply the Reference
Solution Approach  "forward looking" basic algorithm to a six element puzzle
with element 1 as a starting point and elements 2+ and 4 removed from
consideration to complete solution.

At that same point, inspection of the updated composite diagram suggests that
the solution path has three of the remaining elements (3, 2 and 4*) as
possible end points. One could choose to proceed to a solution under the
Reference Solution Approach by using a "backward looking" basic algorithm
starting with at one of the possible end points, disregarding known solution
path elements 4 and 2+, and seeking a connection with element 1 as the last
known point of the solution path.

Indeed, the solution approach possibilities are many, even for a simple puzzle
such as that posed by the example. Certainly the reader is not required to
slavishly follow any of the approaches presented by this guide to solve
presented Clock Puzzles. The cited methods are available for use if desired
but there is also opportunity for application of player ingenuity to create a
"better" solution method tailored to each individual puzzle solving situation.

[G] EXAMPLE PUZZLES

The section of the guide records a selection of Clock Puzzles encountered
during play of the Final Fantasy XIII-2 game. The displays seek to cover the
complexity range of Clock Puzzles the average player my find during game play.

It is intended that a quick review of the Clock Puzzle set provide the reader
with a sense of the challenge that must be met to accomplish Clock Puzzle
resolution in the general case.

Puzzle detail and solutions have been provided so that a reader concerned with
Clock Puzzle resolution may use the examples to practice puzzle solution
techniques. The choice of the solution technique is, of course, the choice of
the individual attempting resolution. Techniques suggested by the writings in
this guide have possible application. Techniques independently conceived and
developed by the user of this guide certainly are appropriate for use as well.

Individual examples comprise three information items: first, a Clock Puzzle
problem statement as an ASCII figure that attempts to reproduce the diagram
presented by the Final Fantasy XIII-2 game during play; second, an annotated
Clock Puzzle diagram providing a unique reference designation for each of the
individual elements of the Clock Puzzle "face" and; third, a description of
the known solutions for the Clock Puzzle under scrutiny.

The solutions listed have been found by application of the Reference Solution
Approach detailed in this guide. This as the intent has been to provide as
many solutions for each puzzle as is possible. However, although the solutions
listed are indeed solutions, there is no guarantee that all possible solutions
have been recognized in every example.

The presentation uses the naming and display conventions introduced in the
earlier sections of this guide. So, although the intent has been to make the
provided material self-explanatory, should difficulty in comprehension of
intent arise, review of the prior Clock Puzzle resolution example discussion
of earlier sections may help clarify the situation.

[G1] Five-Element Puzzle with Five (5) Solutions

2                           2
2       2  - - - - - - ->   2#      2+

1   2                       1   2*

Identified Solutions

1. Start 2 -->2*-->2#-->2+-->1  End

2. Start 2 -->1 -->2*-->2#-->2+ End

3. Start 2+-->1 -->2#-->2*-->2  End

4. Start 2+-->2#-->2*-->2 -->1  End

5. Start 2#-->2+-->1 -->2*-->2  End

[G2] Six-Element Puzzle with One (1) Identified Solution

3                            3

2       3     Tags Added     2       3+
- - - - - - ->
1       1                    1*      1

1                            1+

Identified Solutions

1. Start 3 -->1+-->1 -->3+-->1*-->2  End

[G3] Seven-Element Puzzle with Five (5) Identified Solutions

3                            3
1       1     Tags Added     1+      1
- - - - - - ->
3       3                    3*      3+

2   2                        2+  2

Identified Solutions

1. Start 2 -->1 -->3 -->2+-->3+-->1+-->3* End

2. Start 2 -->1 -->3 -->2+-->1+-->3*-->3+ End

3. Start 2 -->3*-->1 -->3+-->1+-->3 -->2+ End

4. Start 2+-->3+-->1+-->3 -->2 -->3*-->1  End

5. Start 2+-->3+-->1+-->3*-->1 -->3 -->2  End

[G4] Eight-Element Puzzle with Seven (7) Identified Solutions

1                                1
3       2                        3*      2
1           3   - - - - - - ->   1*          3

1       3                        1+      3+
2                                2+

Identified Solutions

1. Start 1 -->2 -->3+-->1*-->1+-->2+-->3 -->3* End

2. Start 1 -->2 -->3+-->1*-->3*-->3 -->1+-->2+ End

3. Start 1 -->2 -->3+-->1*-->3*-->2+-->3 -->1+ End

4. Start 2 -->3+-->1 -->3*-->3 -->1+-->2+-->1* End

5. Start 2 -->3+-->1 -->3*-->2+-->3 -->1+-->1* End

6. Start 3+-->1 -->2 -->3*-->3 -->1+-->2+-->1* End

7. Start 3+-->1 -->2 -->3*-->2+-->3 -->1+-->1* End

[G5] Nine-Element Puzzle with Twenty-Four (24) Identified Solutions

4                              4
4       1                      4#       1
1          4   - - - - - - ->   1+         4+

2        2                      2+       2
3  4                            3  4*

Identified Solutions

1. Start 2 -->1 -->4 -->3 -->4+-->1+-->2+-->4#-->4* End

2. Start 2 -->1 -->4+-->1+-->2+-->4#-->4*-->4 -->3  End

3. Start 4*-->4 -->3 -->4+-->1+-->2+-->4#-->2 -->1  End

4. Start 3 -->4+-->1+-->2+-->4#-->2 -->1 -->4 -->4* End

5. Start 2+-->4*-->4 -->3 -->4+-->1+-->4#-->2 -->1  End

6. Start 1+-->4#-->2 -->1 -->4 -->3 -->4+-->2+-->4* End

---->4#---->2 ---->1 ---->4 ---->3 --->
|                                       |
7.-15.   |        A Loop of Nine Solutions       |
|                                       |
<---- <----4*<----2+<----1+<----4+<----

---->4#---->2 ---->1 ---->4+---->1+--->
|                                       |
16.-24.   |        B Loop of Nine Solutions       |
|                                       |
<---- <----3 <----4 <----4*<----2+<----

[G6] Ten-Element Puzzle with Four (4) Identified Solutions

5                                     5
3       1                             3        1

2             1       Tags Added      2+              1+
- - - - - - ->
4             2                       4+              2

1       4                             1#        4
1                                      1*

Identified Solutions

1. Start 3 ->1+->1 ->5 ->1*->4 ->2+->1#->4+->2  End

2. Start 3 ->1+->2 ->1 ->5 ->1*->4 ->2+->1#->4+ End

3. Start 3 ->1+->2 ->1*->4 ->2+->1#->4+->1 ->5  End

4. Start 3 ->1#->4+->1 ->1+->2 ->1*->4 ->2+->5  End

[G7] Ten-Element Puzzle with Four (4) Identified Solutions

2                                     2
4       4                             4+       4

5             1       Tags Added      5+              1
- - - - - - ->
1             1                       1*              1+

2       5                             2+        5
3                                      3

Identified Solutions

1. Start 2 ->1 ->4 ->1*->2+->5 ->4+->3 ->5+->1+ End

2. Start 2 ->1 ->4 ->1*->2+->5+->1+->5 ->4+->3  End

3. Start 2 ->5+->1+->1 ->4 ->1*->2+->5 ->4+->3  End

4. Start 2 ->5+->1+->5 ->4+->3 ->1 ->4 ->1*->2+ End

[G8] Eleven-Element Puzzle with Five (5) Identified Solutions

4                                    4
3       4                            3+      4+

2             1                       2*           1
2               4    - - - - - - ->   2+             4*

5             5                       5+           5

3       2                             3       2

Identified Solutions

1. Start 2 ->4*->5+->1 ->4+->2+->3 ->2*->4 ->5 ->3+ End

2. Start 2 ->4*->3+->5+->1 ->4+->2+->3 ->2*->4 ->5  End

3. Start 2 ->5+->4+->2+->3 ->2*->4 ->5 ->3+->1 ->4* End

4. Start 2+->3 ->2*->4 ->5 ->3+->1 ->4+->2 ->4*->5+ End

5. Start 2+->3 ->2*->4 ->5 ->3+->5+->1 ->4+->2 ->4* End

[G9] Twelve-Element Puzzle with Eighteen (18) Identified Solutions

6                                     6
2         1                           2*        1

6               5                      6+              5
4                 4   - - - - - - ->   4*                4

1               2                      1+              2

4         5                           4+         5+
2                                      2+

Identified Solutions

1. Start 1 ->6 ->2+->1+->4+->4 ->2*->4*->5+->6+->2 ->5  End

2. Start 5 ->4*->5+->6+->2 ->2+->1+->4+->4 ->2*->1 ->6  End

3. Start 4 ->4+->2*->1 ->6 ->2+->1+->4*->5+->6+->2 ->5  End

4. Start 5+->6+->2 ->5 ->4*->1 ->6 ->2+->1+->4+->4 ->2* End

5. Start 1+->4+->4 ->2*->1 ->6 ->2+->2 ->5 ->4*->5+->6+ End

6. Start 6+->2 ->5 ->4*->5+->6 ->2+->1+->4+->4 ->2*->1  End

---->6 --->2+--->1+--->4*--->5+--->6+--->
|                                        |
7.-18.   |       A Loop of Twelve Solutions       |
|                                        |
<----1 <---2*<---4 <---4+<---5 <---2 <----

[G10] Thirteen-Element Puzzle with Eight (8) Identified Solutions

2                                     2
2         5                           2+         5

3               5                      3+              5+
4                 1   - - - - - - ->   4                 1

5               1                      5#              1+

3          6                           3           6
5   6                                  5*  6+

Identified Solutions

1. Start 5 ->5#->1+->1 ->5+->5*->2+->4 ->6+->2 ->3+->3 ->6  End

2. Start 6+->2 ->3+->3 ->6 ->2+->4 ->5 ->5#->1+->1 ->5+->5* End

3. Start 5*->5+->4 ->6+->2 ->3+->3 ->6 ->2+->5 ->5#->1+->1  End

4. Start 5*->2+->5 ->5#->1+->1 ->5+->4 ->6+->2 ->3+->3 ->6  End

5. Start 3 ->6 ->3+->5 ->5#->1+->1 ->5+->5*->2+->4 ->6+->2  End

6. Start 3 ->6 ->2+->4 ->6+->2 ->3+->5 ->5#->1+->1 ->5+->5* End

7. Start 5#->1+->1 ->5+->5*->2+->4 ->5 ->6+->2 ->3+->3 ->6  End

8. Start 4 ->6+->2 ->3+->3 ->6 ->2+->5 ->5#->1+->1 ->5+->5* End
```