| By clue 1, the apex of Number Pyramid 6 minus the leftmost number in
row 2 equals 6. Therefore, there are four possible combinations for the
number at the top of the pyramid and the leftmost number in row 2: 9-3, 8-2,
7-1, and 6-0. If the two numbers were 9 and 3, then the leftmost numbers in
rows 3 and 4 would sum to 5 (clue 2) and could be 5-0, 4-1, 1-4, or 0-5 in
rows 3 and 4 respectively. 5-0 would also give a 5 as the second number from
the right in row 4 (3). 1-4 would make 9 the second number from the right in
row 4 (3). 0-5 would make 10 the second number from the right in row 4 (3).
If the combination were 4-1, by clue 3, the second number from the left in
row 4 would be 6. Since the three numbers in row 3 add to 17 (2), the
only combination available would be 8 and 5 in some order. However, the
remaining two numbers in row 4 would sum to 12 (5)--impossible given the remaining
numbers. So, the 9-3 combination for the number at the top of the pyramid and
the leftmost number in row 2 cannot work. If the two numbers at the top of
the pyramid and leftmost in row 2 were 8-2, then the leftmost numbers in
rows 3 and 4 would sum to 7 (clue 2) and could be 7-0, 6-1, 4-3, 3-4, 1-6, or
0-7 in rows 3 and 4 respectively. Neither 1-6 nor 0-7 could fit because the
second number from the left in row 4 would then be greater than 9 (3). 4-3
couldn't work because clue 3 would give a second 8 in the pyramid. If the pair
were 7-0, by clue 3, the second number from the left in row 4 would be 5.
Then the rightmost two numbers in row 4 would equal 14 (5) and would require
digits already used to make that total. If the pair were 6-1, by clue 3, the
second number from the left in row 4 would be another 6. If the pair
were 3-4, by clue 3, the second number from the left in row 4 would be 9.
Then the rightmost two numbers in row 3 would equal 14 (2) and would require
digits already used to make that total. So, the 8-2 combination for the
number at the top of the pyramid and the leftmost number in row 2 cannot work.
If the two numbers at the top of the pyramid and leftmost in row 2 were 7-1,
then the leftmost numbers in rows 3 and 4 would sum to 9 (clue 2) and could
be 9-0, 6-3, 5-4, 4-5, 3-6, or 0-9 in rows 3 and 4 respectively. None of
4-5, 3-6, or 0-9 could work because the second number from the left in row 4
would then be greater than 9 (3). If the combination were 9-0, by clue 3,
the second number from the left in row 4 would be 5. Since the three numbers
in row 4 add to 19 (5), the only combination available for the rightmost two
digits in row 4 would be 8 and 6. By clue 4, the 6 would be the rightmost
number in row 4, but a second 9 would have to be rightmost in row 3. If the
combination were 6-3, by clue 3, the second number from the left in row 4
would be 8. Since the three numbers in row 3 add to 17 (2), the only
combination available for the rightmost two numbers in row 3 would be 9 and 2.
By clue 4, the 9 would be the rightmost number in row 3, but a second 6 would
have to be rightmost in row 4. If the combination were 5-4, by clue 3, the
second number from the left in row 4 would be 9. However, since the three
numbers in row 3 add to 17 (2), there is no way for the remaining numbers to
add to the required 12. So, the 7-1 combination for the number at the top of
the pyramid and the leftmost number in row 2 cannot work and we are left with
6-0. Then the leftmost numbers in rows 3 and 4 would sum to 11 (clue 2) and
could be 9-2, 8-3, 7-4, 4-7, 3-8, or 2-9 in rows 3 and 4 respectively. None of
4-7, 3-8, or 2-9 can work because the second number from the left in row 4
would then be greater than 9 (3). If the combination were 9-2, by clue 3,
the second number from the left in row 4 would be 7. However, since the four
numbers in the bottom row sum to 19 (5), the rightmost two numbers would add
to 10 and there is no combination of the remaining numbers that can add to 10.
If the combination were 8-3, by clue 3, the second number from the left in
row 4 would be another 8. Therefore, the leftmost numbers in rows 3 and 4 of
the pyramid are 7 and 4 respectively. 9 is to the immediate right of the 4
(3). By clue 2, the rightmost two numbers in row 3 must sum to 10. The only
remaining combination summing to 10 is 8 and 2. By clue 4, the 8 must be
rightmost in row 3, with 5 then rightmost in row 4; the 2 is the center digit
in row 3. By clue 5, the remaining number in row 4 is 1, with a 3 then the
rightmost digit in row 2. In sum, Number Pyramid 6 is as follows:
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