(related to Problem: A Match Mystery)
If you form the three heaps (and are therefore the second to draw), any one of the following thirteen groupings will give you a win if you play correctly: * $15,$ $14,$ $1;$ * $15,$ $13,$ $2;$ * $15,$ $12,$ $3;$ * $15,$ $11,$ $4;$ * $15,$ $10,$ $5;$ * $15,$ $9,$ $6;$ * $15,$ $8,$ $7;$ * $14,$ $13,$ $3;$ * $14,$ $11,$ $5;$ * $14,$ $9,$ $7;$ * $13,$ $11,$ $6;$ * $13,$ $10,$ $7;$ * $12,$ $11,$ $7.$
The beautiful general solution to this problem is as follows. Express the number in every heap in powers of $2,$ avoiding repetitions and remembering that $2^0 = 1.$ Then if you so leave the matches to your opponent that there is an even number of every power, you can win. And if at the start you leave the powers even, you can always continue to do so throughout the game. Take, as an example, the last grouping given above — $12, 11, 7.$ Expressed in powers of $2$ we have— $$\begin{array}{rccccc} 12&=&8&4&-&-\\ 11&=&8&-&2&1\\ 7&=&-&4&2&1\\ \hline &&2&2&2&2\\ \end{array}$$
As there are thus two of every power, you must win. Say your opponent takes $7$ from the $12$ heap. He then leaves— $$\begin{array}{rccccc} 5&=&-&4&-&1\\ 11&=&8&-&2&1\\ 7&=&-&4&2&1\\ \hline &&1&2&2&3 \end{array}$$
Here the powers are not all even in number, but by taking $9$ from the $11$ heap you immediately restore your winning position, thus— $$\begin{array}{rccccc} 5&=&-&4&-&1\\ 2&=&-&-&2&-\\ 7&=&-&4&2&1\\ \hline &&-&2&2&2 \end{array}$$
And so on, to the end. This solution is quite general and applies to any number of matches and any number of heaps. A correspondent informs me that this puzzle game was first propounded by Mr. W.M.F. Mellor, but when or where it was published I have not been able to ascertain.
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.
