[GGG]

Spoiler!

We know that we can take only two actions - split into groups and flip groups
We know that we can do each of these actions once because doing more won’t gain us info
We know we need two groups of 50 so the first split into two groups of 50 and the flip one of the groups. These are the only actions we can take therefore it should be correct.

Testing it So if we have 50 heads and split into two groups of 50. One group say has 40 heads and 10 tails and the other group then has 10 heads and 40 tails we flip one and you get 10 tails and 40 heads in each groups. It works because you don’t care the number of heads or tails just that they are equal. So it’s just x heads and 50-x tails in one pile and x tails and 50-x heads so flip and they match no matter what split the coins are.

That’s neat.