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Number Pyramid 14
Even Moses would like working on this Number Pyramid.
November 21, 2008
By clue 2, the middle number in row 3 minus the rightmost number in row 3 equals 4, so that the highest number possible in the rightmost position of row 3 is 5. By clue 3, the four numbers in row four sum to 8, so that the highest number in the rightmost position of row 4 is also 5. By clue 1, the number at the apex of the pyramid minus the leftmost number in row 4 equals 7, so that the possible numbers at the apex and leftmost in the bottom row are 7-0, 8-1, or 9-2, respectively. If the numbers were 7 at the apex and 0 leftmost in row 4, since the apex and rightmost numbers in rows 2, 3, and 4 sum to 25 (clue 5), the rightmost numbers in rows 2, 3, and 4 would sum to 18. This would give four possible number combinations for the rightmost numbers in rows 2, 3, and 4: 9-8-1, 9-6-3, 9-5-4, and 8-6-4. Given the clues 2 and 3 caps on the rightmost numbers in rows 3 and 4, combinations 9-8-1, 9-6-3, and 8-6-4 would be impossible. Trying 9-5-4, by clues 2 and 3, 9 would be rightmost in row 2. Then 5 could not be rightmost in row 3 (2), so 4 would be, with 5 rightmost in row 4. The middle number in row 3 would be 8 (2). However, there is no way for clue 4 to work given this arrangement. So, 7-0 aren't at the apex and leftmost in row 4. If the numbers were 8 at the apex and 1 leftmost in row 4, since the apex and rightmost numbers in rows 2, 3, and 4 sum to 25 (5), the rightmost numbers in rows 2, 3, and 4 would sum to 17. This would give four possible number combinations for the rightmost numbers in rows 2, 3, and 4: 9-7-1, 9-6-2, 9-5-3, and 7-6-4. Given the clues 2 and 3 caps on the rightmost numbers in rows 3 and 4 being at most 5, any combination with 9 in it would put 9 rightmost in row 2 and 8 then to its left (4)--impossible since 8 would be at the apex. The clues 2 and 3 caps of 5 on the rightmost numbers in rows 3 and 4 would also eliminate 7-6-4 as a possibility. So, 8-1 aren't at the apex and leftmost in row 4. 9 is at the apex and 2 leftmost in row 4 of Number Pyramid 14. Since the apex and rightmost numbers in rows 2, 3, and 4 sum to 25 (5), the rightmost numbers in rows 2, 3, and 4 would sum to 16. This would give five possible number combinations for the rightmost numbers in rows 2, 3, and 4: 8-7-1, 8-6-2, 8-5-3, 7-6-3, and 7-5-4. Given the clues 2 and 3 caps on the rightmost numbers in rows 3 and 4 being at most 5, combinations 8-7-1, 8-6-2, and 7-6-3 are impossible. Trying 8-5-3, by clues 2 and 3, 8 would be rightmost in row 2 with 7 to its left (4). Since 9 is at the apex, by clue 2, 5 couldn't be rightmost in row 3 and would be rightmost in row 4, with 3 then rightmost in row 3. However, the middle number in row 3 would be 7 (2)--no. So, the rightmost numbers are 7-5-4 in some order. By clues 2 and 3, 7 is rightmost in row 2--with 6 to its left (4). By clue 2, 5 is rightmost in row 4 rather than row 3, with 4 rightmost in row 3. 8 is in the middle of row 3 (2). Since row 4 sums to 8, 0 and 1 complete it--by clue 6, 0 is second and 1 third from the left in row 4. Finally, 3 is the leftmost number in row 3. In sum, Number Pyramid 14 is filled as follows:

        9
       6 7
      3 8 4
     2 0 1 5

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