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By clue 1, the sum of the two numbers in the second row equals 11, so that the leftmost number must be 2 or greater, since the rightmost number cannot be 10 or 11. By clue 4, the middle number in the third row minus the leftmost number in the second row equals 4. The leftmost number in the second row therefore must be 5 or less, since the middle value in the third row can be no more than 9. So, the leftmost number in the second row is 2, 3, 4, or 5. If the leftmost number in the second row were 2, the other number in the second row would be 9 (clue 1), the middle number in the third row would be 6 (clue 4), and the top number would be 5 (5).
By clue 2, the sum of the numbers in the bottom row minus the sum of the numbers in the third row equals 10. Since the numbers left are 0, 1, 3, 4, 7, and 8, none of 4, 7, and 8 could be in the third row with 6 because no combination of remaining numbers would add to a sum 10 greater than the second row sum. If the 3 were in the second row with 6, then 4, 7, and 8 would sum to 19--but the odd 1 couldn't go in either row, so 3 couldn't be in the second row either. Adding the 0 and 1 to 6 would give 7, but the fourth row would add to 22, contradicting clue 2. Therefore, the 2 cannot work as the leftmost value of the second row. If 4 were the leftmost number in the second row, 7 would be the rightmost (1), 8 would be the number in the middle of the third row (4), and 3 would be the number on top the Number Pyramid (5). Again by clue 2, the sum of the numbers in the bottom row minus the sum of the numbers in the third row equals 10, using 0, 1, 2, 5, 6, and 9. If any of 5, 6, or 9 were in the second row with 8, there would be no way for clue 2 to work; so those numbers would be in the bottom row, giving at least 20. Then the 2 would have to go in the second row with 8 to get 10--but the odd 1 cannot fit. So, the leftmost number in the second row isn't 4. If the leftmost number in the second row were 5, 6 would be the rightmost (1), 9 would be the number in the middle of the third row (4), and 2 would be the number on top the Number Pyramid (5), leaving 0, 1, 3, 4, 7, and 8. Again by clue 2, the sum of the numbers in the bottom row minus the sum of the numbers in the third row equals 10. None of 3, 4, 7, or 8 could be with 9 in the third row because no combination of remaining numbers in the fourth row could satisfy clue 2. With the 0 and 1 being in the second row, however, the sum of 10 would be 12 short of the 3, 4, 7, and 8 in the bottom row, contradicting clue 2. So, the leftmost number in the second row isn't 5 and must be 3. Then 8 would be the rightmost (1), 7 would be the number in the middle of the third row (4), and 4 would be the number on top of the Number Pyramid. Again by clue 2, the sum of the numbers in the bottom row minus the sum of the numbers in the third row equals 10. Of the remaining numbers, neither 5, 6, or 9 could be in the third row or clue 2 couldn't work, so those three numbers add to 20 in the bottom row. In order for clue 2 to be satisfied, the 1 and 2 must go in the third row to sum to 10 and the 0 must go in the bottom row to keep that sum at 20.
By clue 3, the righthand four numbers sum to 18, with 4 in the top and 8 in the second row leaving 6 for the bottom two rows. The only combination that works is 1 as the rightmost number in the third row and 5 as the rightmost number in the fourth row. By elimination, then, 2 is the leftmost figure in the third row. By clue 6, the 0 must be the number next to the 5; while the 9 must be the leftmost number and the 6 next to the 9.
Therefore, the Number Pyramid should look like this:
4
3 8
2 7 1
9 6 0 5
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