The smallest possible total is 356 = 107 + 249, and the largest sum
possible is 981 = 235 + 746, or 657+324. The middle sum may be either
720 = 134 + 586, or 702 = 134 + 568, or 407 = 138 + 269. The total in
this case must be made up of three of the figures 0, 2, 4, 7, but no
sum other than the three given can possibly be obtained. We have
therefore no choice in the case of the first locker, an alternative in
the case of the third, and any one of three arrangements in the case
of the middle locker. Here is one solution:--
107 134 235
249 586 746
--- --- ---
356 720 981
Of course, in each case figures in the first two lines may be exchanged
vertically without altering the total, and as a result there are just
3,072 different ways in which the figures might be actually placed on
the locker doors. I must content myself with showing one little
principle involved in this puzzle. The sum of the digits in the total is
always governed by the digit omitted. 9/9 - 7/10 - 5/11 -3/12 - 1/13 -
8/14 - 6/15 - 4/16 - 2/17 - 0/18. Whichever digit shown here in the
upper line we omit, the sum of the digits in the total will be found
beneath it. Thus in the case of locker A we omitted 8, and the figures
in the total sum up to 14. If, therefore, we wanted to get 356, we may
know at once to a certainty that it can only be obtained (if at all) by
dropping the 8.