^ I think you're overcomplicating it. No matter how I figure it, putting half the coins on the table and flipping them works.

Let's say you have 20 coins in all - 10 heads, 10 tails on the floor to begin with.

Pick up 1 head and 9 tails, leaving 9 heads and 1 tail on the floor. Of course, when you flip the ones you picked up, you have the same on the table.

Pick up 2 heads and 8 tails - same thing happens. It leaves 8 heads and 2 tails on the floor, and that's what you get on the table when you flip them.

Without actually running through it for every possible combination, it seems clear this will happen every time.