Stone Piles
There are a few piles of stones. Take turns removing any number of stones from one pile; whoever takes the last stone wins. Play the computer and discover how to spot a position that always wins.
Your turn. Tap a stone to choose how many to take.
How to play
Tap a stone in the pile you want to take from. That stone and every stone above it are chosen. Then press "Take". You can take from only one pile each turn, and at least one stone. Whoever takes the last stone wins. "Hint" gives an idea, then a binary table (numbers written with only 0 and 1), then a concrete move. "Auto-solve" lets the computer play the winning strategy from here and explain why.
Learn more
Write each pile size in binary and count the 1s in each column. If every column has an even number of 1s, the player about to move loses (this "all columns even" condition is the same as the XOR of the pile sizes being 0). Any move makes some column odd. And from any position with an odd column, you can shrink one pile to make every column even again. So if you keep handing your opponent an all-even position, you will take the last stone. This page checks the claim by exhaustive search over every position with up to four piles of up to six stones: only the XOR-zero positions are losses.