What is your best shot to win this game of Kayles? Both
players can hit any pin or pair of adjacent pins.
This bowling game, invented by Henry Dudeney, is played with thirteen pins placed in
a row. Only one or two pins could be knocked down by any
single shot. The bowlers stood so close to the pins that
it did not call for much skill to hit any single pin desired,
or any two adjacent ones. Players bowl alternately, one
ball at a time, and the point of the game was to see who
could knock down the last pin.
Mr. P has just rolled a ball and knocked out
pin No. 2. You have a choice of twenty-two different plays:
any of the twelve single pins, or any of the ten open spots
that will bring down two adjacent pins. What is your best
shot to win the game? It is assumed that both players can
hit any pin or pair of pins they wish, and that there is
the best possible play on both sides.
Extra Credit: To have won the game at the start, what pin should Mr. P have knocked out instead of pin No. 2?
You can play this game on paper by drawing 13 evenly spaced lines and taking turns erasing them (or crossing them out).
Some Solutions to the Problem:
You should knock down pin No. 6. This divides the row of pins into groups of one, three, and seven. Then, no matter what play Mr. P makes, he will surely be beaten if you continue to make the best plays. You can also win by knocking down pin No. 10, for this also leaves groups of one, three, and seven.To have won the game at the start, Mr. P should have knocked out pin No. 7 so as to divide the row into two groups of six each. Then whatever you knock out of one group he would duplicate on the other until he won the game.
Correctly solved by:
| 1. James Alarie | Flint, Michigan |