| From the introduction, a Number Pyramid is composed of the 10 numbers
0-9. By clue 6, the 2nd number from the right in the bottom row is 2.
By clue 1, the topmost number in the pyramid minus the leftmost number in
the bottom row equals 6. The combinations that will then work for these
two numbers are 6-0, 7-1, and 9-3 (the 2 is already placed so 8-2 isn't a
possibility). Trying 6-0, then since the four numbers going down the left
side of the pyramid sum to 21 (clue 2), the leftmost numbers of the middle
two rows would sum to 15; since the 6 would already be in use, the lefthand
numbers of rows 2 and 3 would have to be 8 and 7 in some order. However,
since the four righthand numbers in each row sum to 25 (5), even with the 9
as one of them there is no way for any combination of unused numbers (5, 4, 3, 1)
to make up the needed sum of 10. So, 6-0 does not work for the top and
bottom left numbers. If 9-3 were the number pairing in clue 1, since the
lefthand numbers of the middle two rows would then have to equal 9 (2), the
lefthand numbers of rows 2 and 3 would have to be 8-1 or 5-4 in some order.
If they were 8 and 1, the 8 couldn't be on the left number of the two in row 2,
since by clue 4, the rightmost number in the bottom row minus the sum of the
two numbers in row 2 equals 1, and 9 would have to be the righthand
number of the bottom row for 8 to work at the left of the 2nd row. So,
1 would be the number at the left side of the second and 8 the number at the
left side of the third row. The righthand number of the second row couldn't
be 0, or 2 would have to be the rightmost number of the bottom row (4).
If 4 were the righthand number of the second row, 6 would be the rightmost
number in the bottom row (4)--with another 6 then the rightmost number in the
third row (5), a contradiction. If 5 were the righthand number in the
second row, 7 would be the rightmost number in the bottom row (4)--with 4 then
the rightmost number in the third row (5); however, by clue 3, the middle
number in the third row would be 2, contradicting clue 6. If 6 were the righthand
number in the second row, 8 would be the rightmost number of the bottom
row (4)--with 2 then the rightmost number of the third row (5), contradicting
clue 6. Finally, if 7 were the righthand number in the second row, 9 would
have to be the rightmost number of the bottom row--no, since 9 would already
be the topmost number. So, the 8-1 combination for the leftmost numbers in
rows 2 and 3 doesn't work. Trying the 5-4 possibility, we test 5 in the second
row and 4 in the third. If 0 were the righthand number of the second row, 6
would be the rightmost number in the bottom row (4)--but there would be no
way for the rightmost numbers to sum to 25 (5). If 1 were the righthand
number in the second row, 7 would be the rightmost number in the bottom row
(4)--with 8 then the rightmost number in the third row (5); however, by clue
3, the middle number in the third row would be 2, contradicting clue 6. So,
5 can't work on the left of the second row given the 9-3 pairing. Trying
4 as the leftmost number in row 2 and 5 as the leftmost number in row 3, if
0 were the righthand number of the second row, 5 would be the rightmost
number in the bottom row (4)--but there would be no way for the rightmost
numbers to sum to 25 (5). If 1 were the righthand number in the second row,
6 would be the rightmost number in the bottom row (4)--with another 9 then
the rightmost number in the third row (5)--no, 9 would already be in the
top position. Therefore, the 9-3 combination for the clue 1 numbers
fails. The top number is 7, and the leftmost number in the bottom row is 1.
By clue 2, the lefthand numbers of the second and third rows must sum to
13--either 4-9 or 5-8 in some order. Trying 4-9, the 9 couldn't be the
lefthand number of row 2 (4) so the 4 would, with the 9 the leftmost number
in the third row. 0 couldn't be the righthand number of the second row,
since 5 would then be the rightmost number in row 4 (4) and there would be no
way to sum to 25 as required by clue 5. The only number possible as the
righthand number of the second row would be 3, with 8 then the rightmost
number in the bottom row. However, the rightmost number in the third row
would have to have 7 (5), no. So, 8 and 5 are the lefthand numbers of the
second and third rows. If the 8 were in the second row, by clue 4, 0 would
have to be the other number in row 2 and 9 the rightmost number in the bottom
row. However, by clue 5, a 9 would also have to be the rightmost number in
the third row. Therefore, the lefthand number of the second row is 5 with
the leftmost number in the third row being 8. If the righthand number in
row 2 were 0, the rightmost number in the bottom row would be 6--but there
would be no way for clue 5 to work. The righthand number in row 2 can't be
4 or clue 4 will not work; 3 is the righthand number in the second row.
Then 9 is the rightmost number in the bottom row (4) and 6 is the rightmost
number in the third row (5). By clue 3, the center number in the third row
has is 0. The 4 is the remaining number (second from left) in the bottom
row. In sum, Number Pyramid 1 is as follows:
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