THE LOCKERS PUZZLE.
(
Money Puzzles)
[Illustration:
A B C
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| +--+ +--+ +--+ | | +--+ +--+ +--+ | | +--+ +--+ +--+ |
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| +--+ +--+ +--+ | | +--+ +--+ +--+ | | +--+ +--+ +--+ |
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| +--+ +--+ +--+ | | +--+ +--+ +--+ | | +--+ +--+ +--+ |
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| +--+ +--+ +--+ | | +--+ +--+ +--+ | | +--+ +--+ +--+ |
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| +--+ +--+ +--+ | | +--+ +--+ +--+ | | +--+ +--+ +--+ |
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]
A man had in his office three cupboards, each containing nine lockers,
as shown in the diagram. He told his clerk to place a different
one-figure number on each locker of cupboard A, and to do the same in
the case of B, and of C. As we are here allowed to call nought a digit,
and he was not prohibited from using nought as a number, he clearly had
the option of omitting any one of ten digits from each cupboard.
Now, the employer did not say the lockers were to be numbered in any
numerical order, and he was surprised to find, when the work was done,
that the figures had apparently been mixed up indiscriminately. Calling
upon his clerk for an explanation, the eccentric lad stated that the
notion had occurred to him so to arrange the figures that in each case
they formed a simple addition sum, the two upper rows of figures
producing the sum in the lowest row. But the most surprising point was
this: that he had so arranged them that the addition in A gave the
smallest possible sum, that the addition in C gave the largest possible
sum, and that all the nine digits in the three totals were different.
The puzzle is to show how this could be done. No decimals are allowed
and the nought may not appear in the hundreds place.
Answer:
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.
Informational
