On Fortune's Wheel

The brigands went their rather weightier way, laden with Brian's loot, after a bit of last-minute nastiness over the treasure. The leader of the band asserted that he should get a double share, since he was the leader. Such greed pushed his second-in-command into challenging him, and the end result was a new leader, who in turn asked a double share. The others weren't exactly happy about it, but he pointed out that they would all get the same size of portion as they would have before, since there was one brigand less to demand a share.

Brian and his bodyguards watched all this with some trepidation, but the new leader honored the old's promises. He cut their bonds and returned the letter; he did not give their weapons back, but was generous enough to leave them their clothes and boots. However, the brigands did leave the trio in a bit of a fix. They were in the middle of nowhere with no horses, no weapons, and no money.

Fortunately this last turned out not to be quite true. Brian has seen the previous leader stash a little loot in a hollow tree when no one (else) was looking; since the leader had subsequently been butchered, the treasure was still there. Unfortunately it was not a very large treasure; not enough to get three people to Matthew's kingdom.

The trio set out again, and before long came to a town. Brian hoped to use this opportunity to grow his capital a bit. He went into the local pub and, after enjoying a mess of stew, challenged a couple of the other men to a bet. He said that he would give them five-to-one odds that they couldn't beat him at a simple game. They were naturally suspicious about the good odds, but he breezily explained that it was just because he was familiar with the game. To show his honesty, he would let them go first if they wanted to.

The game's rules were as follows:

There were sixteen game pieces arranged in rows of odd length:

0

000

00000

0000000

The players would take turns removing any number of pieces (1+) from any one of the rows (note that only the odd-numbered horizontal rows counted as rows; the vertical and diagonal rows were not considered rows). Whoever removed the final piece lost, and would have to pay up.

After losing quite a bit of money to Brian, someone demanded that he be allowed to go second; Brian agreed, but the man still lost. Having played everyone in the pub a few times (everyone wanted to be the one to beat this upstart), Brian politely took his leave with quadruple his previous capital. He left them all wondering whether there had, indeed, been a trick of some sort, and how he could beat them all so consistently.

Was there a trick? Was Brian just taking advantage of their unfamiliarity with the game, or could he guarantee a win under certain conditions?

Solution