A MATCH MYSTERY

Here is a little game that is childishly simple in its conditions. But it is well worth investigation.

Mr. Stubbs pulled a small table between himself and his friend, Mr. Wilson, and took a box of matches, from which he counted out thirty.

"Here are thirty matches," he said. "I divide them into three unequal heaps. Let me see. We have `14, 11`, and `5`, as it happens. Now, the two players draw alternately any number from any one heap, and he who draws the last match loses the game. That's all! I will play with you, Wilson. I have formed the heaps, so you have the first draw."

"As I can draw any number," Mr. Wilson said, "suppose I exhibit my usual moderation and take all the `14` heap."

"That is the worst you could do, for it loses right away. I take `6` from the `11`, leaving two equal heaps of `5`, and to leave two equal heaps is a certain win (with the single exception of `1, 1`), because whatever you do in one heap I can repeat in the other. If you leave `4` in one heap, I leave `4` in the other. If you then leave `2` in one heap, I leave `2` in the other. If you leave only `1` in one heap, then I take all the other heap. If you take all one heap, I take all but one in the other. No, you must never leave two heaps, unless they are equal heaps and more than `1, 1`. Let's begin again."

"Very well, then," said Mr. Wilson. "I will take `6` from the `14`, and leave you `8, 11, 5`."

Mr. Stubbs then left `8, 11, 3`; Mr. Wilson, `8, 5, 3`; Mr. Stubbs, `6, 5, 3`; Mr. Wilson,`4, 5, 3`; Mr. Stubbs, `4, 5, 1`; Mr. Wilson, `4, 3, 1`; Mr. Stubbs, `2, 3, 1`; Mr. Wilson, `2, 1, 1`; which Mr. Stubbs reduced to `1, 1, 1`.

"It is now quite clear that I must win," said Mr. Stubbs, because you must take `1`, and then I take `1`, leaving you the last match. You never had a chance. There are just thirteen different ways in which the matches may be grouped at the start for a certain win. In fact, the groups selected, `14, 11, 5`, are a certain win, because for whatever your opponent may play there is another winning group you can secure, and so on and on down to the last match."