A Clock Conundrum

One of the simplest forms of solitaire is a children's game called Clock (it also has other names, and there are other games which go by the same name, but they are more complex). It is completely self-working; once the cards are dealt the player follows a fixed procedure. The win rate is known to be exactly 1 in 13. A few computer solitaire collections (Solitaire's Journey and its successor Mega Solitaire, which both include the game under the name Sun Dial, and Hardwood Solitaire III, which calls it Clock) actually allow the game to play itself (an elaborate form of autoplay) once the first card is turned up and sent to its correct pile.

A single standard deck of 52 cards is shuffled and dealt, face down, into 13 piles of four cards each. The piles are traditionally arranged in the form of a clock face: twelve piles in a circle (the 12 or queen pile at the top, the 1 or ace pile at 1:00, etc., around to the 11 or jack pile at 11:00, and a thirteenth pile, the king pile, in the center).

The top card of the king pile is turned face up. The rank of this card determines what pile the card goes to: if it is another king, it goes *face up* underneath the king pile and another card from the top of the king pile is turned face up. If it is a seven, it is placed face up underneath the seven pile and the top of that pile is turned face up.   This procedure is followed in turn for each card: each card turned up is placed face up at the bottom of the pile corresponding to its rank and the top of that pile is turned face up. The game ends when the fourth king is turned up, as it is placed under the king pile and there are no face down cards left in that pile to  turn up. If all 52 cards are face up at that point, the game is won; otherwise it is lost.

Screen shot from Hardwood Solitaire III, showing the tableau and an autoplayed deal in progress.   The five of diamonds has just been turned up on top of the three pile; it will next go under the five of spades, and the top of that pile will be turned up.   Three kings are already visible; this deal will end when the fourth king appears.

Here is the key point: whenever a card is placed face up under a pile other than the king pile, there will, for a moment, be *five* cards in that pile, until the top card is turned up and moved elsewhere. When the fourth card of a particular rank turns up, it is placed underneath its pile and the fifth card in the pile is turned face up. This card *must* be of a different rank, since there are only four cards of each rank. Looked at from another point of view, while the card most recently turned face up is moving to a pile (even if the same pile it came from), the king pile has three cards and all of the other piles have four each.

So where is the conundrum? Albert Morehead and Geoffrey Mott-Smith, in their usually reliable "The Complete Book of Solitaire and Patience Games", published in 1949, say the following: "If the last face-down card of a pile belongs to that pile, turn next the face-down card of the next pile clockwise around the circle." Similar statements occur in many later books, including Alphonse Moyse Jr.'s "150 Ways to Play Solitaire", which appeared a year later. Nothing similar to this appears in any earlier description of the game I have seen. And for good reason: the situation they describe cannot occur, barring a misdeal (less than four cards dealt initially to a pile) or a defective deck of cards (with five of one rank). Nowhere in any solitaire book have I seen any rule designed to handle a misdeal: the presumption is that a player who misdeals in a solitaire game loses automatically.

Somewhere I read that Morehead and Mott-Smith were rivals of Moyse (they were all noted bridge writers), and that Morehead and Mott-Smith felt that Moyse had heavily copied from their book on solitaire. Could they have put in a deliberate mistake to trip up Moyse? Careful later writers like David Parlett and Peter Arnold who specialize in card games do not repeat Morehead and Mott-Smith's mistake, but it is propagated in books by authors like Sheila Barry, Pierre Crepeau, Franscesca Parodi, and Sloane and Lee.

This article is copyright © 2007 by Michael Keller.  All rights reserved.