This week the #puzzle is: Can You Pile the Primes? #primes #coding #BruteForce #oeis #subsets

Suppose you want to make two groups with equal sums using the first N2 prime numbers. What is the smallest value of N2 for which you can do this?

The answer is three! (Clearly, that wasn’t actually the puzzle.)

The first three primes are 2, 3, and 5, and you can split them up into two sets: {2, 3} and {5}. Sure enough, 2 + 3 = 5.

Your puzzle involves making three groups with equal sums using the first N3 prime numbers. What is the smallest value of N3 for which you can do this?

And for extra credit:

Now you want to make six groups with equal sums using the first N6 prime numbers. What is the smallest value of N6 for which you can do this?

Can You Pile the Primes?

Intermission

I am very excited, because this is the last fiddler of Q1, and I still have maximum points. Wish me luck.

Solution, possibly incorrect:

Program

Method 1: Write a brute force program. This actually finds a solution rather quickly. N₂ is confirmed, among other things.

Testing N1 =   1. Testing these primes: 2
................. Group 1: 2
................. Sum of groups: 2
................. N1 = 1

Testing N2 =   2. Sum of primes    5 isn't a multiple of 2, rejected.
Testing N2 =   3. Testing these primes: 5,3,2
................. Group 1: 5
................. Group 2: 3,2
................. Sum of groups: 5
................. N2 = 3

Testing N3 =   3. Sum of primes   10 isn't a multiple of 3, rejected.
Testing N3 =   4. Sum of primes   17 isn't a multiple of 3, rejected.
Testing N3 =   5. Sum of primes   28 isn't a multiple of 3, rejected.
Testing N3 =   6. Sum of primes   41 isn't a multiple of 3, rejected.
Testing N3 =   7. Sum of primes   58 isn't a multiple of 3, rejected.
Testing N3 =   8. Sum of primes   77 isn't a multiple of 3, rejected.
Testing N3 =   9. Sum of primes  100 isn't a multiple of 3, rejected.
Testing N3 =  10. Testing these primes: 29,23,19,17,13,11,7,5,3,2
................. Group 1: 29,7,5,2
................. Group 2: 23,17,3
................. Group 3: 19,13,11
................. Sum of groups: 43
................. N3 = 10

Result: N₃ = 10.

And for extra credit:

OEIS

Method 1: The same program.

N₆ has to be at least 57. But then my program gives up. Heat death of the universe kind of thing.

Testing N6 =  57.
Testing these primes:
269,263,257,251,241,239,233,229,227,223,
211,199,197,193,191,181,179,173,167,163,
157,151,149,139,137,131,127,113,109,107,
103,101,97,89,83,79,73,71,67,61,59,53,
47,43,41,37,31,29,23,19,17,13,11,7,5,3,2
Or rather, test left as an exercise for the reader.

Method 2: Try to find a solution with the first 57 primes. And I do!

• 269,199,197,139,137,89,59,37,19
• 263,211,181,163,109,103,53,47,13,2
• 257,223,173,167,107,101,71,41,5
• 251,227,179,157,127,83,61,31,29
• 241,229,193,151,113,97,67,43,11
• 239,233,191,149,131,79,73,23,17,7,3

N₆ = 57.

Method 3: Look for an OEIS sequence. Found. Results confirmed.