Pascal Pyramid

Probability Level 3

Pascal's Triangle is the following pattern of numbers

        1
    1   2   1
  1   3   3   1
1   4   6   4   1

Now, consider the Pascal's Pyramid , an extension of Pascal's Triangle. Instead of adding two numbers in the row above, Pascal's Pyramid adds 3 adjacent numbers from the layer above. When viewing from top,

Layer 0 :

Layer 1 :

1
2

1   1
  1

Layer 2 :

1
2
3

1   2   1
  2   2
    1

What is the sum of numbers in layer 8?

The answer is 6561.

1 solution

Marta Reece
Dec 31, 2016

Each number in the next layer is a sum of three of the numbers in the layer before. This includes the numbers on the outside, as these can be thought to be the sums of the appropriately located number(s) below and of zeroes. If each of the numbers is a sum of three numbers, each of the numbers below appears three times in the numbers in the next layer, so the next layer has to have a total value tree times larger than the previous one. Layer 0 has $1=3^{0}$ , layer 1 has $3=3^{1}$ , etc. until layer 8 has $3^{8}=6561$ .

Pascal Pyramid

The answer is 6561.

1 solution

0 pending reports