Problem of the Week 1245

Goldilocs

Charlie has 128 identical-looking gold-colored coins; all are counterfeit except one.

He calls Alice into his office and seats her at a table containing two 8 × 8 boards, with a coin on each of the 128 squares, each coin showing a head or tail as he chooses.

He tells Alice which coin is made of gold. Alice can then turn at most 4 coins upside down, replacing them on their squares, but her inversions must all be on the first row of the left-hand board. She then leaves the room.

Bob enters and is seated in the same position, facing the boards. He may take one of the 128 coins.

Find a strategy that allows Bob to always take the gold coin.

As usual, Alice and Bob know the protocol in advance and can plan a strategy, but cannot communicate after Alice enters the room.

Note: If Alice could invert up to 7 coins, then she could arrange the first 7 squares of the left-board's first row to be any of 2^7 = 128 numbers. This problem is surprising in that it can be done with at most 4 flips by Alice. I'll also add that the solution is quite short and no computation was necessary.

Source: Peter Saltzman, Jim Tilley, and Stan Wagon. ("Goldilocs" abbreviates "Gold Locations.")