[Illustration: 

( 15 Gals )

          (31 Gals) (19 Gals)

    (20 Gals) (16 Gals) (18 Gals)

]

77.—­DIGITS AND SQUARES.

[Illustration: 

+—–­+—–­+—–­+
| 1 | 9 | 2 |
+—–­+—–­+—–­+
| 3 | 8 | 4 |
+—–­+—–­+—–­+
| 5 | 7 | 6 |
+—–­+—–­+—–­+

]

It will be seen in the diagram that we have so arranged the nine digits in a square that the number in the second row is twice that in the first row, and the number in the bottom row three times that in the top row.  There are three other ways of arranging the digits so as to produce the same result.  Can you find them?

78.—­ODD AND EVEN DIGITS.

The odd digits, 1, 3, 5, 7, and 9, add up 25, while the even figures, 2, 4, 6, and 8, only add up 20.  Arrange these figures so that the odd ones and the even ones add up alike.  Complex and improper fractions and recurring decimals are not allowed.

79.—­THE LOCKERS PUZZLE.

[Illustration: 

            A B C
    ================== ================== ==================
    | +--+ +--+ +--+ | | +--+ +--+ +--+ | | +--+ +--+ +--+ |
    | | | | | | | | | | | | | | | | | | | | | | | |
    | +--+ +--+ +--+ | | +--+ +--+ +--+ | | +--+ +--+ +--+ |
    | | | | | |
    | +--+ +--+ +--+ | | +--+ +--+ +--+ | | +--+ +--+ +--+ |
    | | | | | | | | | | | | | | | | | | | | | | | |
    | +--+ +--+ +--+ | | +--+ +--+ +--+ | | +--+ +--+ +--+ |
    | | | | | |
    ================== ================== ==================
    | +--+ +--+ +--+ | | +--+ +--+ +--+ | | +--+ +--+ +--+ |
    | | | | | | | | | | | | | | | | | | | | | | | |
    | +--+ +--+ +--+ | | +--+ +--+ +--+ | | +--+ +--+ +--+ |
    ------------------ ------------------ ------------------

]

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.