The following solution in fourteen moves was found by Mr. G. Wotherspoon: 8-17, 16-21, 6-16, 14-8, 5-18, 4-14, 3-24, 11-20, 10-19, 2-23, 13-22, 12-6, 1-5, 9-13. As this solution is in what I consider the theoretical minimum number of moves, I am confident that it cannot be improved upon, and on this point Mr. Wotherspoon is of the same opinion. 405.--CARD MAGIC SQUARES. Arrange the cards as follows for the three new squares:-- 3 2 4 6 5 7 9 8 10 4 3 2 7 6 5 10 9 8 2 4 3 5 7 6 8 10 9 Three aces and one ten are not used. The summations of the four squares are thus: 9, 15, 18, and 27--all different, as required. 406.--THE EIGHTEEN DOMINOES. [Illustration] The illustration explains itself. It will be found that the pips in every column, row, and long diagonal add up 18, as required. 407.--TWO NEW MAGIC SQUARES. Here are two solutions that fulfil the conditions:-- [Illustration: SUBTRACTING DIVIDING 11 4 14 13 36 8 54 27 16 7 1 2 216 12 1 2 6 5 3 12 6 3 4 72 9 19 8 15 9 18 24 108 ] The first, by subtracting, has a constant 8, and the associated pairs all have a difference of 4. The second square, by dividing, has a constant 9, and all the associated pairs produce 3 by division. These are two remarkable and instructive squares. 408.--MAGIC SQUARES OF TWO DEGREES. The following is the square that I constructed. As it stands the constant is 260. If for every number you substitute, in its allotted place, its square, then the constant will be 11,180. Readers can write out for themselves the second degree square. [Illustration: 7 53 | 41 27 | 2 52 | 48 30 12 58 | 38 24 | 13 63 | 35 17 ------+-------+-------+------ 51 1 | 29 47 | 54 8 | 28 42 64 14 | 18 36 | 57 11 | 23 37 ------+-------+-------+------ 25 43 | 55 5 | 32 46 | 50 4 22 40 | 60 10 | 19 33 | 61 15 ------+-------+-------+------ 45 31 | 3 49 | 44 26 | 6 56 34 20 | 16 62 | 39 21 | 9 59 ] The main key to the solution is the pretty law that if eight numbers sum to 260 and their squares to 11,180, then the same will happen in the case of the eight numbers that are complementary to 65. Thus 1 + 18 + 23 + 26 + 31 + 48 + 56 + 57 = 260, and the sum of their squares is 11,180. Therefore 64 + 47 + 42 + 39 + 34 + 17 + 9 + 8 (obtained by subtracting each of the above numbers from 65) will sum to 260 and their squares to 11,180. Note that in every one of the sixteen smaller squares the two diagonals sum to 65. There are four columns and four rows with their complementary columns and rows. Let us pick out the numbers found in the 2nd, 1st, 4th, and 3rd rows and arrange them thus :-- [Illustration: 1 8 28 29 42 47 51 54 2 7 27 30 41 48 52 53 3 6 26 31 44 45 49 56 4 5 25 32 43 46 50 55 ]