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Having been through 50 something battles on my play through, I've only had a single one and I believe "critical strike" can only occur when you're at a disadvantage, as these are the only times I've heard of other people getting them. Basically the critical strike is a thing that can kick in when you're on the losing side to still allow you to win, even when you have fewer soldiers. When mine happened, I was at a disadvantage and had been brought down to only a few hundred less than my opponent. I don't know what the limits are for it, but I assume you have to be on the losing side for it to kick in. As for whether or not it can happen to you, my experience would say no.
I have had it happen twice. Both times had all the following true:
* Have one win and one loss. * Be on the third battle. * Be just a few hundred Miis below the opponent after the evaluation.
It then glowed and Critical Striked for me.
You know, I think I had that similar set up. I remember being worried I was going to lose that fight because I won the first, then lost the second and new "crap, all I have is paper and he had his scissors left" but the critical strike was kind enough to hit.
Well, I managed to get it. I was facing an army more than 1.5 times my size, lost the 1st round, won the 2nd, and then I had just barely more soldiers for the third round but was at a paper covers rock disadvantage, and then the critical strike kicked in to save my bacon.
I just got a critical hit, It happened on the third round and I had less units and was at a disadvantage in the weapon triangle.
Now the fun part. The crit not only reduces enemies by at least half (I didn't pay attention how much.) It nullifies any advantages and bonuses the enemy might have. Case in point, I had a paper unit against a scissor berserker and won with a crit by base numbers.