Prime Squares Game

Dennis Shasha

Omniheurist Course

Computer Science


 

Description

Prime numbers hold a strange fascination for mathematicians of all ages. Children grasp the idea quickly and soon recognize that primes are much more dense among small numbers than among big numbers. On the other hand, professional mathematicians do not know but cannot disprove that every even number greater than 4 is the sum of two primes, a conjecture made by Goldbach and Euler in 1742. Here's a page containing fun facts about primes:

This puzzle asks you to find prime squares: a square grid whose rows and columns comprise primes, but where no two row primes or column primes are the same. An ambidextrous prime square is one which also contains primes when the rows are read from right to left. An omnidextrous prime square is an ambidextrous prime square whose columns are primes when read bottom to top and whose diagonals are also primes. One other constraint on prime squares: they cannot begin with a 0.

Here is an example of an ambidextrous prime 3-square:

7 6 9
9 5 3
7 9 7

Because 769, 953, and 797 are all primes as are their columns 797, 659, and 937, this is a prime square. Because the row reversals 967, 359, and 797 are also primes, this is an ambidextrous prime square. One reason it is not omnidextrous is that 956 is even.

Consider the game tictacprime 3: a move consists of placing a single digit in an empty square of a three by three grid. The two players alternate moves except the second player gets the ninth move. When a player's digit completes a segment of three digits in a row, then the number of three digit primes that include the square where the digit was placed (going in any direction) yields a point to that player. The object is to get the most points possible. The prohibition against repeated rows or columns does not apply here. However, a row or column may not begin with a 0. Is there a winning strategy for either player?

Here is a list of primes between 100 and 999:



101 103 
107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 
223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 
337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 
457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 
593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 
719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 
857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 
997

Your Task

To win this game.

Architecture Team

Draw a tic-tac-toe board and then put in the numbers as they appear. Keep track of the score. Look here for a discussion of the problem of constructing prime squares followed by an example of the game.