Enigmatic Code

Programming Enigma Puzzles

Enigma 1539: Board with primes?

From New Scientist #2702, 4th April 2009 [link]

As I have mentioned in a previous Enigma puzzle, a “Snakes and Ladders” board consists of a 10-by-10 grid of squares. In row 1 (at the bottom) the squares are numbered from 1 to 10 from left to right. Then in row 2 the squares are numbered 11 to 20 from right to left. In row 3 they are 21 to 30 from left to right again, and so on, ending up at square 100 in column 1 and row 10.

I have been cutting up such a board. I have cut out a rectangle consisting precisely of some of its squares. It runs from a prime-numbered row of the original board to a higher prime-numbered row inclusive, and it runs from a prime-numbered column to a higher prime-numbered column inclusive. Furthermore, the total of all the numbers in my rectangle is a prime.

What is that total?



2 responses to “Enigma 1539: Board with primes?

  1. Jim Randell 17 April 2012 at 8:36 am

    Here’s my original Perl code. It runs in 13ms.

    use strict;
    use Enigma qw/prime/;
    # primes below 10
    my @PRIMES = qw/2 3 5 7/;
    my ($R1, $R2, $C1, $C2, $R, $C, $SUM);
    for $R1 (@PRIMES) {
      for $R2 (@PRIMES) {
        next unless $R1 < $R2;
        for $C1 (@PRIMES) {
          for $C2 (@PRIMES) {
            next unless $C1 < $C2;
            $SUM = 0;
            for $R ($R1..$R2) {
              for $C ($C1..$C2) {
                $SUM += ($R % 2 ? 10 * ($R - 1) + $C : 10 * $R + 1 - $C);
            next unless prime($SUM);
            print "R[$R1,$R2] x C[$C1,$C2] = $SUM\n";

    Solution: The sum of the numbers in the rectangle is 449.

    • Jim Randell 17 April 2012 at 8:37 am

      And here’s my Python solution. It runs in 37ms.

      from itertools import combinations, product
      from enigma import irange, chunk, is_prime, printf
      # primes below 10
      primes = (2, 3, 5, 7)
      # pairs of primes s.t. a < b
      pairs = list(sorted(x) for x in combinations(primes, 2))
      # make a grid for the board
      grid = list(chunk(irange(1, 100), 10))
      for i in irange(1, 9, step=2):
        grid[i] = list(reversed(grid[i]))
      # compute sums
      for (cs, rs) in product(pairs, repeat=2):  
        s = sum(sum(grid[r][cs[0] - 1:cs[1]]) for r in irange(rs[0] - 1, rs[1] - 1))
        if not prime(s): continue
        printf("sum={s} rows={rs} cols={cs}")

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