Take as many androids as you like, but only from one row. Take the last one and you win.
There is no cleverness in it and no search tree. Write each row's count in binary, XOR them all together, and the result — the "Nim-sum" — tells you everything. If it's zero when it's your turn, you lose against anyone who knows this. If it's anything else, there is a move that makes it zero, and taking that move hands the losing position to your opponent.
So "plays properly" is the whole strategy: compute the Nim-sum, find the row where taking androids drives it to zero, take them. It never needs to think ahead, because the arithmetic already did.
The opening position here is 3, 5 and 7. That XORs to 1, not zero — which means you are the one holding the won position, and you move first. If the computer beats you on the proper setting, it's because you handed it the zero.
The 1978 original this site is named after is Leo Christopherson's, and so are his androids. This is written from scratch. The rules of Nim belong to nobody. The story is on the front page.