Here is a little game that is childishly simple in its conditions. But it is 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 anyone 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."

Solutions: 1

**Dudeney, H. E.**: "Amusements in Mathematics", The Authors' Club, 1917

This eBook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this edition or online at http://www.gutenberg.org. If you are not located in the United States, you'll have to check the laws of the country where you are located before using this ebook.