Friday, 9 December 2016

Figured Tour with Legendre Numbers

At the August Bank Holiday Rapid-Play at the Hastings Chess Club I somehow did well enough to win a £5 book voucher. I spent it on "Professor Stewart's Incredible Numbers". On page 46 there is an account of numbers that cannot be expressed as a sum of three squares. A-M. Legendre found the formula (4^k)(8n + 1) for all such numbers. Here is a figured tour that I just constructed this afternoon showing all the Legendre Numbers less than 65 on the diagonals.
They consist of the eight odd numbers that are one less than a multiple of eight (that is 8n - 1) and the two even numbers 28 and 60. They occur in pairs that differ by 32 thus allowing a symmetric arrangement in a symmetric knight tour. The reader may like to try constructing a similar tour with the odd numbers in a different sequence or the even numbers on different cells.


  1. I shared this in Facebook, because it seemed relevant to a discussion. I tried to send you an email about a few matters - maybe I didn't have the right address - please check your junk folder maybe or contact me with a better email address if you have one. All the best.

  2. Just noticed your comment. There is an email address at bottom of this page:

  3. Hi George, thanks for all your work with chess variants and the various chess pieces. Your list of chess pieces is excellent. I was wondering if you would add the huygens (chess piece) to the catalog. It is used in Trappist-1 and can also be played in variations of Chess on an Infinite Plane.

    This piece brings new strategic concepts to variant chess games. The huygens is also a piece which is interesting to
    mathematicians doing work in the field of game theory, including those who study chess games played on an infinite chessboard (such as the widely popular YouTube video which has gone viral among chess players).

    (link here):

    Here are the details:

    name: Huygens

    History: Named after Christiaan Huygens, a prominent Dutch mathematician and astronomer.
    Chess piece invented by vickalan, and was first used in mid-2016.

    Movement: The huygens jumps prime numbers of squares in orthogonal directions (so jumps
    2, 3, 5, 7, 11,... etc. number of squares). It is sometimes played with a different
    minimum jump distance, so that it is not a close-attacking piece.

    I'll also be happy to send you the icon for this piece, and a move diagram.

    Best regards!