Here each column contains four consecutive numbers cyclically arranged, four running in one direction and four in the other. The numbers in the 2nd, 5th, 3rd, and 8th columns of the square may be similarly grouped. The great difficulty lies in discovering the conditions governing these groups of numbers, the pairing of the complementaries in the squares of four and the formation of the diagonals. But when a correct solution is shown, as above, it discloses all the more important keys to the mystery. I am inclined to think this square of two degrees the most elegant thing that exists in magics. I believe such a magic square cannot be constructed in the case of any order lower than 8. 409.--THE BASKETS OF PLUMS. As the merchant told his man to distribute the contents of one of the baskets of plums "among some children," it would not be permissible to give the complete basketful to one child; and as it was also directed that the man was to give "plums to every child, so that each should receive an equal number," it would also not be allowed to select just as many children as there were plums in a basket and give each child a single plum. Consequently, if the number of plums in every basket was a prime number, then the man would be correct in saying that the proposed distribution was quite impossible. Our puzzle, therefore, resolves itself into forming a magic square with nine different prime numbers. [Illustration] A B +-----+-----+-----+ +-----+-----+-----+ | | | | | | | | | 7 | 61 | 43 | | 83 | 29 | 101 | |_____|_____|_____| |_____|_____|_____| | | | | | | | | | 73 | 37 | 1 | | 89 | 71 | 53 | |_____|_____|_____| |_____|_____|_____| | | | | | | | | | 31 | 13 | 67 | | 41 | 113 | 59 | | | | | | | | | +-----+-----+-----+ +-----+-----+-----+ C D +-----+-----+-----+ +-----+-----+-----+ | | | | | | | | | 103 | 79 | 37 | |1669 | 199 |1249 | |_____|_____|_____| |_____|_____|_____| | | | | | | | | | 7 | 73 | 139 | | 619 |1039 |1459 | |_____|_____|_____| |_____|_____|_____| | | | | | | | | | 109 | 67 | 43 | | 829 |1879 | 409 | | | | | | | | | +-----+-----+-----+ +-----+-----+-----+