# Prime Pyramid

Prime Pyramid: Compete the pyramid by filling in the missing numbers on each row. Each row must contain the consecutive whole numbers from 1 to the row number but not necessarily in that order. In each row, the numbers from 1 to the row number are arranged such that the sum of any two adjacent numbers (neighboring numbers) is a prime number.

A prime number is a whole number that has exactly two distinct (different) factors--itself and 1. Each row begins with 1 and ends with the number of that row. So, row 2 begins with a 1 and ends with a 2, row 3 begins with a 1 and ends with a 3, and so on.

For example, let’s look at row 4:

• It must contain the numbers 1,2,3, and 4.

• It must begin with 1 and end with 4.

• The sum of each adjacent pair must be a prime number:

• 1 + 2 = 3; 2 + 3 = 5, 3 + 4 = 7

For example, let’s look at row 5:

• It must contain the numbers 1,2,3, 4, and 5.

• It must begin with 1 and end with 5.

• The sum of each adjacent pair must be a prime number:

• However, the solution cannot be 1,2,3,4,5 because the sum of 4 and 5 is 9 (a composite number because 3x3=9)

• Switching the 2 and 4 gives us the solution: 1.4.3.2,5 where each adjacent pair of numbers adds to a prime.

There is only one solution for rows 2 through 6. There are 2 solutions for row 7. Then there are four solutions for row 8 and many more beyond. This week, Mathletes found 12 solutions for row 11. I am guessing there are more solutions. The hardest part of this challenge is finding multiple solutions for a row. It would be terrific if we could find an algorithm for the number of solutions per row.

I like the strategy thinking of a chess game. Look at the row number and then find a number that cannot be paired with that number. It is that number that gets placed first in the row, 2, 4, or 6 down from that spot. Remember, an odd can never be placed next to an odd and an even never next to an even since the sum is always a composite number. Therefore, you only need to analyze the opposite group of numbers. For example, when working on row 7, the only even number that cannot be paired with 7 is 2 so place 2 in the 4th position and then see that you have two groups of two numbers to place. You know that 4 and 5 cannot be paired and 6 and 3 cannot be paired since they sum to 9. So separate them appropriately. Follow this process with every row. Here are some pairs that can not go together.

4 + 5

6 + 3

7 + 2

8 + 1

8 + 7

9 + 6

10 + 5

11 + 4

11 + 10

13 + 2

13 + 8

13 + 12

The K-2nd graders were given a fully solved prime pyramid and were asked only to add each adjacent pair to reinforce adding these numbers mentally and check Mr. Kramer’s work; also, they were able to reinforce their ability to identify prime numbers. If I felt they were sufficiently comfortable with the sums, I challenged them to find solutions on a blank pyramid to row 14.

Third grade and up were challenged to find their own solutions through row 25 and the higher grades were challenged to find as many multiple solutions as possible. This is quite a challenging puzzle teaching so many strategies about problem solving.

The Mathletes and I found mistakes in row 23, 24, and 25 as I had 16 and 17 adjacent to each other, a sum of 33, a composite number of 3 x 11. Luca Ianniello found a beautiful solution to row 23:

1 4 3 2 5 6 7 10 9 8 11 12 17 14 15 16 13 18 19 22 21 20 23

and I suggested to him that row 24 could copy row 23 because 23 + 24 = 47, a prime so:

1 4 3 2 5 6 7 10 9 8 11 12 17 14 15 16 13 18 19 22 21 20 23 24

However, row 25, could not copy row 24 since 24 + 25 = 49, a composite 7 x 7.

Then I saw a simple solution for row 25 by replacing 6 with 24:

1 4 3 2 5 24 7 10 9 8 11 12 17 14 15 16 13 18 19 22 21 20 23 6 25

If your child finds a pattern in the maximum number of solutions in a row, please email me right away.

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