On Four Houses

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When last they met, the Baron challenged Sir R----- to a game of cards in which they were to take turns playing the Jacks, Queens, Kings and Aces to four houses, subject to the contraints that no house could contain two cards of the same face value, two cards of the same suit, a Jack and a King of the same colour or a Queen and an Ace of the same colour, with the Baron playing the first card.
If Sir R----- could have contrived to play the last card that did not violate those contraints then he would have had a coin from the Baron's purse. If not, the Baron would have had a coin from his.

Sir R----- should consequently have only entered into this wager if he could have formulated a strategy to force the Baron to play a card that broke those rules or, failing that, to ensure that he himself could play the card that filled the houses without doing so.

If we construct a table in which we record the houses in which we place cards of given values and suits then we find that this wager takes a familiar form. For example, the filled houses


can be represented as

AQJK
1342
4213
2134
3421

Arranged in this fashion, legal plays require that each row, column and quadrant of the table must contain distinct values, showing that this game is equivalent to a four by four, two player variant of the much enjoyed sudoku puzzle, known as dui-doku[1].
Now it is clearly the case that numbers placed in diametrically opposed quadrants, top-left versus bottom-right for example, can impose no constraint upon one another. If, therefore, Sir R----- responds to each card played by the Baron with the one that is represented by placing the same house number in that element of the table 180° removed from it, then his move must be just as legal as the Baron's since such a strategy will ensure that the table will remain rotationally symmetric.
Sir R----- could therefore have ensured that the game could be played until all four houses were filled and, this being a winning condition for him, would have been well advised to have taken the Baron's wager.
\(\Box\)

References

[1] Basu, K. Dui-doku: A 2-player variant of Sudoku, 2008.

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