A "Number Pyramid" is composed of ten different numbers ( usually 0 - 9 ) with four rows. ONE EXAMPLE of a Number Pyramid is as follows:

      0

    1   2

  3  4   5

6  7   8   9

Given the clues below, determine the composition of a new Number Pyramid.

1) The sum of the two numbers in the second row is 11.

2) The sum of the numbers in the bottom row minus the sum of the numbers in the third row equals 10.

3) The rightmost numbers in the four rows must sum to 18.

4) The middle number in the third row minus the leftmost number in the second row should equal 4.

5) Subtracting the top number in the Pyramid from the rightmost number in the second row leaves 4.

6) In the bottom row, the leftmost number is greater than the second number from the left, while the rightmost number is greater than the second number from the right.

For bonus marks, which of the above clues (if any), are not necessary in order to construct the pyramid?

Preliminary assumption: using numbers 0-9

I.
Using clue 1.
Possible sets in row 2:
2/9, 3/8, 4/7, 5/6

II.
Using clue 4.
Possible middle numbers in the third row respectively to point I:
6, 7, 8, 9
Second row has to have smaller number on left side.

III.
Using clue 5.
Possible top numbers respectively to point I and II:
5, 4, 3, 2

IV
Using clue 2.
Possible sets of numbers to use in the third and the fourth row (respectively to point I and II and III):
(0,1,3,4,6,7,8), (0,1,2,5,6,7,9), (0,1,2,5,6,8,9),(0,1,3,4,7,8,9)
Sum of all numbers in the 3rd and the 4th row (respectively to sets above):
29, 30, 31, 32
Possible sums of numbers in the 4th (respectively to sets above):
(29+10)/2 = 19.5, (30+10)/2=20, (31+10)/2 = 20.5, (32+10)/2=21
So the first and the third set are impossible, because you can't sum integers to get not integer.

Considering the 2nd set:
The sum of numbers in the 3rd row is 10 and there is 7 in that row. The only way of getting 10 in that row is to put 1 and 2 there. So the numbers left for the row number 4 are 0,5,6,9 which give 20 in sum. So this set is OK for us.

Considering the 4th se:
The sum of numbers in the 3rd row is 11 and there is 9 in that row. There is no possibility of getting 11 from the rest of numbers from that set.

Conclusion after points I-IV:
1st row: (4),
2nd row: (3,8) (in that order),
3rd row: (2,7,1) or (1,7,2)
4th row: (0,5,6,9) (in any order),

V.
Using clue 3.
The sum of rightmost numbers from the 1st and the 2nd row is 12. The only possible way to get 6 (to have 18 all together) from summing:
(1 or 2 [7 must be in the middle]) + (0 or 5 or 6 or 9) is to sum 1 and 5. so the rightmost numbers in the 3rd and the 4th row are 1 and 5, respectively.

VI
Using clue 6.
The only order in the 4th row meeting requirement of having 5 on the rightmost position is 9605, becouse there is only one smaller number than 5 (0) so it has to stand next to 5. We know that leftmost number must be bigger than the second from left so 9 is the leftmost and 6 is the second from left.

Pyramid has to look like that assumming that we are using numbers from 0 to 9:

4
3 8
2 7 1
9 6 0 5

Posted by Rafal on 2004-09-04 13:14:29