+-----+-----+-----+ | | | | | 121 | 114 | 119 | |_____|_____|_____| | | | | | 116 | 118 | 120 | |_____|_____|_____| | | | | | 117 | 122 | 115 | | | | | +-----+-----+-----+ First write down any consecutive numbers, the smallest being greater than 1--say, 2, 3, 4, 5, 6, 7, 8, 9, 10. The only factors in these numbers are 2, 3, 5, and 7. We therefore multiply these four numbers together and add the product, 210, to each of the nine numbers. The result is the nine consecutive composite numbers, 212 to 220 inclusive, with which we can form the required square. Every number will necessarily be divisible by its difference from 210. It will be very obvious that by this method we may find as many consecutive composites as ever we please. Suppose, for example, we wish to form a magic square of sixteen such numbers; then the numbers 2 to 17 contain the factors 2, 3, 5, 7, 11, 13, and 17, which, multiplied together, make 510510 to be added to produce the sixteen numbers 510512 to 510527 inclusive, all of which are composite as before. But, as I have said, these are not the answers in the smallest numbers: for if we add 523 to the numbers 1 to 16, we get sixteen consecutive composites; and if we add 1,327 to the numbers 1 to 25, we get twenty-five consecutive composites, in each case the smallest numbers possible. Yet if we required to form a magic square of a hundred such numbers, we should find it a big task by means of tables, though by the process I have shown it is quite a simple matter. Even to find thirty-six such numbers you will search the tables up to 10,000 without success, and the difficulty increases in an accelerating ratio with each square of a larger order. 412.--THE MAGIC KNIGHT'S TOUR. +----+----+----+----+----+----+----+----+ | 46 | 55 | 44 | 19 | 58 | 9 | 22 | 7 | +----+----+----+----+----+----+----+----+ | 43 | 18 | 47 | 56 | 21 | 6 | 59 | 10 | +----+----+----+----+----+----+----+----+ | 54 | 45 | 20 | 41 | 12 | 57 | 8 | 23 | +----+----+----+----+----+----+----+----+ | 17 | 42 | 53 | 48 | 5 | 24 | 11 | 60 | +----+----+----+----+----+----+----+----+ | 52 | 3 | 32 | 13 | 40 | 61 | 34 | 25 | +----+----+----+----+----+----+----+----+ | 31 | 16 | 49 | 4 | 33 | 28 | 37 | 62 | +----+----+----+----+----+----+----+----+ | 2 | 51 | 14 | 29 | 64 | 39 | 26 | 35 | +----+----+----+----+----+----+----+----+ | 15 | 30 | 1 | 50 | 27 | 36 | 63 | 38 | +----+----+----+----+----+----+----+----+