place these numbers in three horizontal lines so that the units are in the same vertical column. If then the sum of each column is 2 or (i. e. congruent to 0, mod. 2), the set of numbers forms a safe combination. For example,
1 0 0 1,
1 0 1,
1 1 0 0,
or 9,5,12 is a safe combination. It is seen at once that if any two numbers be given, a third is always uniquely determined which forms a safe combination with the two given numbers. Moreover, it is obvious that if a,b,c form a safe combination any two of the numbers determine the remaining one, that is, the system is closed. A particular safe combination which is used later is that in which two piles are equal and the third is zero. In the proofs which follow, the binary scale of notation is used throughout.
Theorem I. If A leaves a safe combination on the table, B cannot leave a safe combination on the table at his next move. B can change only one pile, and he must change one. Since when the numbers in two of the piles are given the third is uniquely determined, and since A left the number so determined in the third pile (i. e., the pile from which B draws) B cannot leave that number. Hence B cannot leave a safe combination.
Theorem II. If A leaves a safe combination on the table, and B diminishes one of the piles, A can always diminish one of the two remaining piles, and have a safe combination. Consider first an example. Suppose A leaves the safe combination nine, five, twelve, and that B draws two from the first pile, leaving the numbers seven, five, twelve, or
1 1 1,
1 0 1,
1 1 0 0.
If A is to leave a safe combination by diminishing one of the piles, it is clear that he must select the third pile, that containing twelve. The number which is safe with 111 and 101 is 10, or two. Hence A must leave two in the pile which contains twelve, or draw ten from that pile, and by doing so he leaves a safe combination.
To prove the general theorem, let the numbers, expressed in the binary scale, be written with the units in a vertical column, and suppose that A left a safe combination. B selects one of the piles and diminishes it. When a number of the binary scale is diminished it is essential to notice that in going
