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
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