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 Clock 
Puzzle 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 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 
set of the Clock Puzzle display. This is always possible given the color coding 
of the different numbers. One can "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.

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 
by Square Enix Co., Ltd. with all rights reserved. This text has been prepared 
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.



                               TABLE OF CONTENTS

[A] THE BASIC ALGORITHM - Presents the execution steps for the reference 
solution approach of this document.
 
[B] EXAMPLE APPLICATION - Records the analysis steps and results produced by 
use of the Basic Algorithm to solve a typical (eight-element) Clock Puzzle.

[C] SHORTCUT CONSIDERATIONS - Introduces the idea of looking at a Clock Puzzle 
from varying viewpoints to be able to construct alternate solution methods.
 
[D] AN ALTERNATIVE APPROACH - Introduces a second Clock Puzzle solution method 
that is competitive with respect to the Basic Algorithm's solution efficiency.

[E] GENERAL PRINCIPLES - 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



                            [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 for 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. If a number is found to lack a predecessor, it is the starting point 
of the "solution" pathway(s) 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 defines a "loop" 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] AN ALTERNATE APPROACH

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. 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*---> 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*---> 3+---> 3 ---> 4+---> 2 ---> 3+ 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.

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 could range from a simple three-element 
solution exercise (three 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 (the 
corresponding displays for the EXAMPLE APPLICATION are not included in this 
document due to the difficult nature of producing the necessary ASCII artwork) 
to define a visual record.


                          [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 are 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 means 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 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 means 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 example 
situations described attempt to provide insights about this.

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 the most convenient 
way to solve a thirteen-element Clock Puzzle.