Onwards And Upwards


Onwards And Upwards

Felicitations of the season Sir R-----! I trust that the cold weather has not chilled your passion for wager?

I should not have doubted it sir! Come, take a mug of mulled cider and a seat by the hearth, the better to chase off Mister Frost.

I propose a game that is something of a devotion amongst the Knights Templar. As an honourary member I have, on those occasions when our paths have chanced to cross, assisted them in the defence of missionaries in foreign lands, swiftly dispatching any particularly suspicious looking gangs of heathens to make safe our charges with their message of peace, love and forgiveness.
Now, after a hard day's bodyguarding we would encamp and ponder matters spiritual whilst playing their game, it being allegorical of the soul's progress through the celestial spheres. As a well travelled man of reason I am quite aware that the celestial bodies are, in fact, worlds no more peopled by angelic hosts than is our own, but when surrounded by heavily armed priests who are quite convinced otherwise I believe that the wise man should keep his own counsel on such matters.

Here, the first rank and file of this chessboard shall represent the fixed Earthly realm and those squares in the second not deemed terrestrial, the Lunar sphere. In likewise fashion we progress through the Mercurial, Venusian, Solar, Martian and Jovian spheres, with the top-rightmost square representing the Saturnian.
You shall place a queen upon one of the Earthly squares. I shall then move it in its fashion towards the Saturnian, forbidden at each turn from remaining still or moving to a lower sphere. My having moved you shall do likewise and we shall continue onwards and upwards in this manner until one of us reaches the highest, whereupon he shall have a coin from the other's purse.

That loathesome student I know, upon hearing these rules, immediately started wittering on about writing off some stones that had been picked from him. Quite what business of mine he thought his loss was I cannot fathom.

Now take another mug and mull over your strategy!


I think this is a win for the first player (it reminds me of nim), but I'm not dead sure what moves are allowed. I thought I'd be able to move down a file while remaining in the same sphere (I've found that I can move up and remain in the same sphere). I also thought I'd be able to move diagonally down, so long as I do not move to a lower sphere, but that would enable infinite move games - a draw. So I guess the moves are up or right or diagonally up and right. That being the case, it's still a win for the first player so long as they start in (row,file) = (1,4) or (4,1), as then the 2nd player cannot get to any of the safe squares, (5,3),(3,5),(7,6),(6,7) or (8,8), but only to squares that can get to a safe square.

Hmm, so which NxN boards are wins for the first player, and what are its safe squares?

I see the general case is Wythoff's Nim, also tsyan-shidzi (choosing stones). Very nice, and new to me. Now, can I get them interested down the pub and win myself a few beers, that's what counts.

Well spotted sir! Correct on both counts!

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