From the introduction, a Number Pyramid is composed of the 10 numbers
09. 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 20leaving 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, 58; 2) 2, 47; 3) 4, 36;
4) 6, 25; and 5) 8, 14. Trying arrangement 1), 0, 58, 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 6but 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 4but by clue 5 the
alreadyused 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 3but by clue 5 the
alreadyused 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 1but by clue 5 this
arrangement fails. So possibility 1) fails. Trying arrangement 2), 2, 47,
by clue 2, the bottom row contains two of four pairs of numbers: 19, 28,
37, and 46. Given the three numbers already used in possibility 2),
there is no way clue 2 can work. Similarly, arrangement 5), 8, 14, also
conflicts with clue 2. So, neither of arrangements 2) and 5) work. Trying
arrangement 3), 4, 36, 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:


