| By clue 1, the three numbers at the top of Number Pyramid 5 sum to
17, with the largest of the three atop the pyramid. That number must be at
least 7, since if it were 6, the 2nd row could have only at most 5 and 4 in it, a
total of only 15. First, we try 9 at the top of the pyramid. By clue 4,
the four leftmost numbers in the four rows of the pyramid add up to 13.
With 9 at the top, the 2nd row could then have 0-8, 1-7, or 3-5 in
left-to-right order (2-6 would require either 2-0 or 1-1 in the leftmost
positions of the 3rd and 4th rows). Trying the 9/0-8 possibility, by
clue 4, the leftmost values in rows 3 and 4 could be top-to-bottom
3-1 or 1-3. However, trying 3-1, by clue 3 there would be a second 0 as
the second number across row 4. Trying 1-3, by clue 3, the second number
across row 4 would be 2. However, there is no way for clue 2, where the
three numbers in row 3 sum to 8, to work. So, 9/0-8 fails. Trying the
9/1-7 possibility, by clues 4 and 3, the leftmost values in rows 3
and 4 could be top-to-bottom only 0-3. Then the second number across
row 4 would be 2 (clue 3). However, again, there is no way for clue 2,
where the three numbers in row 3 sum to 8, to work. So, 9/1-7 fails.
Trying the 9/3-5 possibility, by clues 4 and 3, the leftmost values in
rows 3 and 4 could be top-to-bottom only 0-1. However, then the second
number across row 4 would be another 0 (3). Therefore, 9/3-5 fails; and
the number atop the pyramid isn't 9. Next, we try 8 at the top of the
pyramid. By clue 4, the four leftmost numbers in the four rows of the
pyramid add up to 13. With 8 at the top, the 2nd row could then have 0-9,
2-7, 3-6, or 4-5 in left-to-right order. Trying the 8/4-5 possibility,
by clue 4, we would have 1-0 or 0-1 as the leftmost values in rows 3 and 4.
However, clue 3 cannot work with either arrangement. Trying the 8/3-6
possibility, by clues 4 and 3, we would have 0-2 as the leftmost values
in rows 3 and 4. Then 1 would be the second number across row 4 (3).
However, there is again no way for clue 2, where the three numbers in row 3
sum to 8, to work. So, 8/3-6 fails. Trying the 8/2-7 possibility, by clues
4 and 3, we would have 0-3 as the leftmost values in rows 3 and 4. Then
the second number across row 4 would be 2 (3) and a repeat. So, 8/2-7 fails.
Trying the 8/0-9 possibility, by clues 4 and 2, we would have 1-4, 3-2, or
4-1 as the leftmost values in rows 3 and 4. Trying 1-4, the second number
across row 4 would be 3 (3). However, since the 2 isn't in row 3 of the
pyramid (2), there is no way for row 3 to sum to 8 (2). Trying 3-2, the
second number across row 4 would be 1 (3). However, there is no way for
the three numbers in row 3 to add up to 8. Trying 4-1, the second number
across row 4 would be 0 (3), a repeat of the leftmost number in row 2.
So, 8/0-9 fails. Therefore, 8 cannot be the number at the apex of the pyramid;
7 is. By clue 4, the four leftmost numbers in the four rows of the pyramid add
up to 13. With 7 at the top, the 2nd row could then have 1-9, 2-8, 4-6,
or 6-4 in left-to-right order. Trying the 7/1-9 possibility, by clues 4
and 2, we would have 5-0, 3-2, or 0-5 as the leftmost values in rows
3 and 4. 5-0 will not work given clue 3. 3-2 will not work given clue 3,
since the 1 in row 4 would be a repeat. Trying 0-5, the second number across
row 3 would be 4 (3). However, since the 2 isn't in row 3 of the pyramid
(2), there is no way for row 3 to sum to 8 (2). So, 7/1-9 fails. Trying
the 7/2-8 possibility, by clue 4, we would have 4-0, 3-1, 1-3 or 0-4 as
the leftmost values in rows 3 and 4. 4-0 will not work given clue 3.
Trying 3-1, the second number across row 3 would be 0 (3); however, there
is no way for row 3 to sum to 8 (2). 1-3 will not work given clue 3,
since the 2 in row 4 would repeat the 2 in row 2. Trying 0-4, the second
number across row 3 would be 3 (3). However, there is no way for row 3
to sum to 8 (2) without a repeat. So, 7/2-8 fails. 7/6-4 cannot work
without 0's being leftmost in rows 3 and 4 (4). So, 4 then 6 fills row 2
of the pyramid. Given clue 4, the leftmost numbers in rows 3 and 4 are
2-0 or 0-2. 2-0 contradicts clue 3, so the leftmost number in row 3 is 0
and in row 4 is 2. Then 1 is the second number across row 4 (3). By
clue 2, the other two numbers in row 3 must be 3 and 5. If 5 were rightmost
in the row, the rightmost number in row 4 would be 6 (5)--a repeat of the
6 in row 2. So, in row 3, 5 is the center and 3 the rightmost number.
By clue 5, the rightmost number in row 4 is 8. 9 to the left of 8 in row
4 completes the pyramid. Number Pyramid 5 is filled as follows:
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