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 September 4, 2006 By clue 2, the two numbers in the second row of Number Pyramid 7 sum to 6, while by clue 5, the rightmost number in row 2 minus the number at the apex of Number Pyramid 7 equals 1. Reducing these clues to equations, letting the leftmost number in row 2 equal X, the rightmost number in row 2 equal Y, and the pyramid apex equal Z, we have X + Y = 6 for clue 2 and Z = Y - 1 for clue 5. Summing the two equations, we get X + Z = 5, meaning that the leftmost number in row 2 and the apex of the pyramid must sum to 5. By clue 3, the leftmost number in row 4 minus the 3rd number in row 4 equals 6. Therefore, there are four possible combinations for the leftmost number in row 4 and the 3rd letter in row 4: 6-0, 7-1, 8-2, and 9-3. Trying 6-0, since the leftmost number in row 2 and the apex sum to 5, the leftmost number in row 3 would be 9 (clue 1). However, there is no way for clue 4 to work. Trying 7-1, since the leftmost number in row 2 and the apex sum to 5, the leftmost number in row 3 would be 8 (1). However, since the 1 would already be used, again there is no way for clue 4 to work. Trying 8-2, since the leftmost number in row 2 and the apex sum to 5, the leftmost number in row 3 would be 7 (1). However, since the 2 would already be used and since 1 in the center of row 3 would make the rightmost number in row 3 8, there is no way for clue 4 to work. Therefore, the leftmost number in row 4 and the 3rd letter in row 4 are 9 and 3 respectively. Since the leftmost number in row 2 and the apex sum to 5, the leftmost number in row 3 is 6 (1). By clue 4, the middle number in row 3 must be 1 or 2. If the middle number in row 3 were 1, the rightmost number in row 3 would be 7. By clue 2 and the numbers we would already have in the pyramid, the number combination for row 2 would be 4-2 or 2-4. 4-2 cannot work because the apex would then be 1 (5) and 2-4 cannot work because the apex would be 3 (5). So, the middle number in row 3 is 2 and the rightmost number in row 3 is 8. By clue 2 and the numbers we would already have in the pyramid, the number combinations for row 2 could be 5-1 or 1-5. If the combination were 1-5, the apex would be 4 (5). The two remaining numbers would be 0 and 7--but this combination cannot fit clue 6. So, row 2 has 5 and then 1 in it, with 0 at the apex (5). By clue 6, the two remaining numbers of 4 and 7 must be the second and rightmost numbers in row 4 respectively. In sum, Number Pyramid 7 is filled as follows:     0    5 1   6 2 8  9 4 3 7  