| From the introduction, a Number Pyramid is composed of the 10 numbers
0-9. Therefore, the sum of all the numbers in the Pyramid is 45. By clue 1,
numbers in the 3rd row of Number Pyramid 3 sum to 12; while by clue 2,
the four numbers in the 4th row sum to 20--leaving a sum of 13 for the top
and 2nd rows of the pyramid. By clue 3, the rightmost number in row 2 - the
leftmost number in that row equals 3. We then have these possible
arrangements for the 1st and 2nd rows: 1) 0, 5-8; 2) 2, 4-7; 3) 4, 3-6;
4) 6, 2-5; and 5) 8, 1-4. Trying arrangement 1), 0, 5-8, by clue 4 the leftmost
number in the 4th row minus the leftmost number in the 3rd row equals 3; while
by clue 2, the endmost numbers in row 4 sum to 10 as do the middle two
numbers of the row. If the leftmost number in the 4th row were 4 and the
leftmost in the 3rd row were then 1 (4), the rightmost number in row 4
would be 6--but clue 5 would conflict with this arrangement. If the
leftmost number in the 4th row were 6 and the leftmost in the 3rd row were
then 3 (4), the rightmost number in row 4 would be 4--but by clue 5 the
already-used 5 would be needed to complete this arrangement. If the
leftmost number in the 4th row were 7 and the leftmost in the 3rd row were
then 4 (4), the rightmost number in row 4 would be 3--but by clue 5 the
already-used 8 would be needed to complete this arrangement. If the
leftmost number in the 4th row were 9 and the leftmost in the 3rd row were
then 6 (4), the rightmost number in row 4 would be 1--but by clue 5 this
arrangement fails. So possibility 1) fails. Trying arrangement 2), 2, 4-7,
by clue 2, the bottom row contains two of four pairs of numbers: 1-9, 2-8,
3-7, and 4-6. Given the three numbers already used in possibility 2),
there is no way clue 2 can work. Similarly, arrangement 5), 8, 1-4, also
conflicts with clue 2. So, neither of arrangements 2) and 5) work. Trying
arrangement 3), 4, 3-6, by clue 4, either 8 or 9 would be the leftmost
number in row 4. Trying 8, with 5 then the leftmost number in row 3 (4)
and 2 the rightmost number in row 4 (2), there is no way for clue 5 to work.
Trying 9, the 6 needed in row 3 (4) would have already been used. So,
arrangement 3) also fails and we have arrangement 4): 6 at the apex of the
pyramid, 2 to the left and 5 to the right in row 2. Since the 2, 5, and 6
are used, by clue 4, the leftmost number in row 4 can't be 9, 8, 1, or 0;
and by clue 5, the leftmost number in row 4 can't be 4. The leftmost number
in row 4 is 3 or 7. If it were 3, with the 7 then the rightmost number in
the 4th row (2) and the 0 the leftmost number in the 3rd row (4), clue 5
will not work. So, the number leftmost in row 4 is 7 and in row 3 4 (4).
3 is at the far right in row 4 (2). By clue 5, the rightmost number in the
3rd row is 8; 0 then is in the middle of that row (1). Finally by clue 2,
the 1 is left of the 9 in the middle of row 4. In sum, Number Pyramid
3 is as follows:
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