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Number Pyramid 13
Our 13th Number Pyramid doesn't need luck to solve.
July 21, 2008
By clue 1, the numbers in the bottom row of Number Pyramid 13 sum to 28, so that the possible numbers in the bottom row are 9-8-7-4 and 9-8-6-5. By clue 4, the rightmost numbers in the four rows total 11. Therefore, 9 cannot be rightmost in row 4 since there is no combination of numbers with 9 that add to 11. If 8 were in the bottom right location, then the other rightmost numbers would be 0, 1, and 2 with one of them at the apex--not possible given clue 6. If 7 were in the bottom right location, then the other rightmost numbers would be 0, 1, and 3 with one of them at the apex--again not possible given clue 6. If 6 were in the bottom right location, then the other rightmost numbers would be 0, 2, and 3 with one of them at the apex--again not possible given clue 6--or 0, 1, and 4, with the 4 at the apex (clue 6)--but 0 would also have to be leftmost in row 2 (6). If 5 were in the bottom right location, then the other rightmost numbers would be 0, 2, and 4 with the 4 at the apex (clue 6)--but 0 would also have to be leftmost in row 2 (6)--or 1, 2, and 3 and also impossible given clue 4. So, 4 is the rightmost number in row 4 with 9, 8, and 7 being the other bottom row numbers. Given clue 4, the other rightmost numbers are 0, 2, and 5 or 0, 1, and 6. Trying the first possibility, 5 would be at the apex and 1 leftmost in row 2 (6). By clue 7, the four leftmost numbers in the rows sum to 20. Given the 9, 8, and 7 in row 4, the only arrangment possible is 8 leftmost in row 4 and 6 then leftmost in row 3; however, none of the remaining numbers in row 3 can be bigger than 6, contradicting clue 2. So, the rightmost numbers must be 0, 1, and 6, with 6 at the apex and 2 leftmost in row 2 (6). By clue 7, the four leftmost numbers in the rows sum to 20. Given the 9, 8, and 7 in row 4, the only arrangements possible are 7 leftmost in row 4 with 5 leftmost in row 3 or 9 leftmost in row 4 with 3 leftmost in row 3. If 5 were leftmost in row 3, however, clue 2 could not work. So, 3 is leftmost in row 3 and 9 is leftmost in row 4. By clue 2, 5 is in the middle of row 3. By clue 3, 7 is next to 9 in row 4, with 8 to the right of 7. Finally, 0 is rightmost in row 2 and 1 is rightmost in row 3 (5). In sum, Number Pyramid 13 is filled as follows:

        6
       2 0
      3 5 1
     9 7 8 4

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