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Number Pyramid 9
With a little work, Number Pyramid 9 comes tumbling down.
July 16, 2007
By clue 3, the three numbers in the third row of Number Pyramid 9 sum to 24; they must be 7, 8, and 9 in some order. By clue 4, the four leftmost numbers also add up to 24. If in row 3 either 7 or 8 were the leftmost number, the most the four leftmost numbers in each row could sum to would be 22 (7, 6, 5, 4) or 23 (8, 6, 5, 4). So, 9 is leftmost in row 3, with 4, 5, and 6 then leftmost in the other rows in some order. By clue 1, the number on top the pyramid minus the sum of the two numbers in the second row equals 1. 4 cannot be at the apex of the pyramid, since row 2 would have at least a 5 or 6 in it. If 5 were atop the pyramid, then row 2 would have 4 and 0 across it (clue 1), with 6 leftmost in row 4. However, given the remaining numbers of 1, 2, and 3 in row 4, there would be no way for clue 5 to work. Therefore, the number atop the pyramid is 6. By clue 1, row 2 has either 5, 0 or 4, 1 across it. If 5, 0 were in row 2 with 4 leftmost in row 4, however, given the remaining numbers of 1, 2, and 3 in row 4, there would be no way for clue 5 to work. So, row 2 has 4, 1 across it, with 5 leftmost in row 4. Given the remaining numbers in row 4 of 0, 2, and 3 and given clue 5, the rightmost number in row 4 is 3 and the number second from the left is 0. 2 is between them to complete the row. Finally, by clue 2, 8 is in the middle and 7 rightmost in row 3. In sum, Number Pyramid 9 is filled as follows:

        6
       4 1
      9 8 7
     5 0 2 3

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