At 09:07 PM 5/30/03, David Wilson wrote:
I've never seen this trick before. What properties of the subsquares make this trick work? Would it be possible to perform this trick with a square that has distinct entries? Entries 1 through 64?
The array is an addition table; that is, each entry is the sum of a value that depends on the row and another value for the column: 9 7 6 4 2 0 3 1 ------------------------ 3 | 12 10 9 7 5 3 6 3 | 0 | 9 7 ... | 2 | 11 9 ... | 1 | ... | 7 | | 4 | | 6 | | 5 | The rules require you to pick out one entry in each row and column of your 4x4 subsquare; their sum is thus the sum of four consecutive row values and four consecutive column values. The possible consecutive row-value sums are 6, 10, 14, 18, 22; for the column values, 26, 19, 12, 9, 6. The first set are all congruent mod 4, if the second set were all different mod 4, that would suffice to pick out which number in the second set contributed to the sum. As it is, the largest and smallest are congruent, but as the difference between them is greater than the range of first set, this case can be decided by checking if the sum is more or less than, say, 30. It's easy to create such a table with distinct entries; in fact, it's no trouble at all to obtain entries 1 to 64, as asked: Just write the entries down in order! The column values are 1,2,3,4,5,6,7,8 (with four-term sums ranging from 10 to 26), and row values 0,8,16,24,32,40,48,56 (with four-term sums congruent mod 32). -- Fred W. Helenius <fredh@ix.netcom.com>