Monday, July 21, 2014

Li'l Game From Martin Gardner

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A nice old Martin Gardner puzzle to start the week, which he titled "Fifteen Finesse" in his slim "Aha!" volume (I've adapted it, though still using the same example he used):

There is a new carnival game in town called "Fifteen." It involves a board with the numbers 1 through 9 laid out in order (like above). You play against the carny; doesn't matter who goes first, but you will take turns back-and-forth. You each put coins down one-at-a-time on a single number and then "own" that number ('til someone wins or all 9 numbers are used up). The carny will be putting silver dollars down while you put nickels down. The object is to own any THREE numbers that add up to 15, before your opponent does (for example, 3, 5, and 7). Whoever does this gets all the money played, in cases of draws (no winner) you each take your money back.

So we'll take Gardner's example:
You go first putting a nickel on 7. Carny puts a dollar on 8. You put a nickel on 2. Carny puts a dollar on 6 (and blocks you), realizing that you will win the game if allowed to do so (7+2+6). Now you are forced to put a nickel on 1 to block Carny from winning the game on his next move (8+6+1). Carny puts his next dollar on 4. You again block his chance (6+4+5) at a win by putting a nickel on 5. But next Carny places a dollar on 3 and still wins, since 8 + 4 + 3 is 15. You lose your money. The question is, is there any strategy by which you could be assured a win?

Well, needless to say if Carny is giving you odds of a dollar to a nickel, there's no method for you winning a round… but there is a way for Carny to make sure he never loses a round (wins or draws every game). Can you see it?
What Gardner goes on to explain is that this game is "isomorphic" to classic tic-tac-toe (another game one can always win or draw, and never lose, if played strategically).
There are only eight possible 3-digit triplets that are winners for the Fifteen game:


and these can be arranged in a "magic" square formation:

2 9 4
7 5 3
6 1 8

such that EACH horizontal and vertical line and diagonal is one of the winning triplets.
The carny simply keeps a card illustrating this 'magic square' hidden from (your) sight and makes his number selections referring to it (trying to get 3 in a row, while preventing you from doing so), as if he were playing tic-tac-toe. Then, he can't lose (assuming he doesn't screw up ;-).

Clever Carny, clever Martin.

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