Prime Pyramid

A prime number is a whole number that has exactly two factors--itself and 1. Last week, we introduced a new discovery regarding the lack of randomness of prime numbers. I have developed a challenge called a prime pyramid where 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. In each row, the numbers from 1 to the row number are arranged such that the sum of any two adjacent number is a prime number.

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 are 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.

The kindergarteners and first graders first worked with a fully solved prime pyramid and were asked only to add each adjacent pair to reinforce adding these numbers mentally; 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,

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