On Onwards And Upwards

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In the Baron's latest game, Sir R----- was to place a queen upon the first rank or file of a chessboard and the Baron was to then move it horizontally, vertically or diagonally in its regular manner to a square of higher rank and/or file. Sir R----- was then to do the same in his turn and, alternating thereafter, he who moved the queen to the top-rightmost square was to be declared the winner.

Now I immediately recognised this as strategically equivalent to the ancient Chinese game of picking stones, studied in some detail by one Doctor Whytoff[1]. In this game two piles of stones are put in play with players alternating in taking any number of stones from either pile or an equal number from both, the goal being to empty the piles.
If we equate the distance from the last rank with the number of stones in one pile and that from the last file the number in the other then it becomes apparent that legal moves of the queen are equivalent to picking stones from the piles according to those rules. Doctor Whytoff's analysis revealed that for any specific number of stones in each pile it was possible to deduce which player could force a win.

In the Baron's game we have a comparitively limited choice for the placing of the queen which makes the analysis a great deal simpler. Marking the outcome of Sir R-----'s moves with the white queen and the Baron's with the black, Sir R-----'s goal is


Now clearly, Sir R----- should desire that the Baron be forced to move to a square from which this goal could be reached.


Next, marking out for Sir R----- those squares from which the Baron can be so forced, we have


Again, Sir R----- can identify those squares from which he can move to these goals.


Those unmarked squares from which the Baron must move to a square advantageous for Sir R----- are consequently


and those unassigned squares disadvantageous to the Baron therefore being


meaning that Sir R----- can consequently force a win if he places the queen on either of the remaining squares.


If Sir R----- were willing to think carefully upon his strategy I should most certainly have advised him to take the Baron's wager!
\(\Box\)

References

[1] Wythoff, W. A. A modification of the game of nim, Nieuw Archief voor wiskunde 2, Pages 199-202, 1905-1907.

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