# Who Wins At Horsie-Push?

Allan, Brian, and Charlie had a chess set and an afternoon with nothing to do. A chess tournament was suggested but to three towering intellects such as these, regular chess is a frivolous, juvenile pursuit. Accordingly, they set themselves to play some high-stakes horsie-push.

For those unfamiliar with the sport, the rules are as follows. The squares on a chessboard are numbered in sequence from one to sixty-four. The players each place a potato chip in a dish beside the board. These are the stakes: winner takes all. The first player (Allan, because it is his board) takes a knight and places it on any one of the first five squares. The next player, Brian, then must advance the knight at least one but no more than five squares in a straight line. If Allan's first move was to place the knight on square three, then Brian must move it to one of 4, 5, 6, 7, or 8. Charlie, who has funny looking ears and a yellow tooth, goes last following the same one-to-five squares rule as Brian. Play proceeds until a player is able to move the knight to the final square where he wins the game and three disks of salty deep fried starch.

But on this day Charlie had an objection. "It's not fair. You guys are just going to gang up on me. "

"Don't be a wimp," countered Allan.

"If you don't play fair, I'm going to take my bag of chips and go home."

This was a forceful argument. The source from which all the wagers were drawn was Charlie's bag of chips. Compromise became the rule of the day and it was agreed that Charlie could move the knight from one up to nine squares when it was his turn.

Assuming perfect co-operation between Allan and Brian and the unlikely event of best play on both sides, who will win the game?

• "the first five squares" meaning numbers 1 through 5? And advancing isn't a knight's move, but just moving along the number sequence? Commented Mar 10, 2016 at 19:36
• That's right. They just push it straight forward. I will edit. Commented Mar 10, 2016 at 19:38
• I love the way you phrased this - I nearly lost my composure in the middle of class!
– Deusovi
Commented Mar 10, 2016 at 20:08

Here's the situation:

Charlie can win if he gets a move with the knight at spot 55 or higher. Therefore, Charlie will win if he can get the knight to spot 53. Allan and Brian can only move the knight 10 spots between them (up to 63), but must move the knight at least 2 spots (up to 55). If Charlie has to put the knight at spot 54 or higher, he loses and Brian wins.

Working backwards...

Moving back another 11 spots, that means Charlie can win if he gets the knight to spot 42. Similarly to the last statement, Allan and Brian have to move it at least 2 spots (up to 44) which would allow Charlie to get it to 53. But they can't move it more than 10 spots (up to 52), so Charlie will definitely be able to get it to 53.

Keep going...

We can now see that Charlie will win if he can get the knight to 31, 20 or 9, by the same logic as the previous steps. So Allan would need to place the knight in a spot where Charlie couldn't get it there.

Therefore...

Allan places the knight on spot 5. Brian moves it to spot 9. Charlie must move it off of 9, but can only move it as far as 18. Thus it will be between 10 and 18. Allan and Brian can combine to move the knight to 20, thus after Charlie's next move it will be between 21 and 29. Allan and Brian keep cooperating to place the knight on the spots Charlie would need to get it to in order to win - 31, 42 then 53. On each of Charlie's moves, he has to move it to within reach of the next one (but not all the way there).

So finally...

Allan or Brian will win, depending on where Charlie pushes the knight once it gets to spot 53. If he pushes it to somewhere in the 54-58 range, Brian will win. If he pushes it somewhere in the 59-62 range, Allan will win.