Montana (Gaps) Solitaire

Montana 2437

A popular solitaire with a number of variants is usually called Montana or Gaps.  In the standard form of this solitaire, the entire deck is dealt into four rows, nonoverlapped, or thirteen cards each.   The four aces are removed, and take no further part in play.  The empty spaces remain (hence the common name Gaps). The object is to arrange the cards into four rows, one in each suit, running from twos at the far left to kings at the right, with gaps at the far right end.   Only one kind of move is permitted: a card may be moved into an empty space if it is the same suit and one higher in rank than the card to its left.  In the diagram above (showing the start of game 2437), the first gap in row 2 may be filled with the king of hearts at the end of row 2, leaving a new gap behind the nine of diamonds. Other cards which could be moved are the five of clubs and four of hearts.  A space at the far left of a row may be filled with any two, even a two already at the start of a row (which may have cards of the same suit in sequence behind it).  This means that the suit being built in each row is not permanently established, and may be changed until all four twos are placed (this is critically important in playing miniature versions, as we will see below).   Nothing can be moved behind a king, so a gap of one or more spaces behind a king are blocked unless we can move that king later.   Gradually you want to build continuous sequences in each suit as far as possible.   It is usually very difficult (probably impossible) to finish all four suits before being blocked by having all four gaps behind kings.  Most versions allow the game to be continued after the player is blocked, by using a redeal in which all of the cards not in sequence from the left end of each row are taken up and shuffled, and dealt row-by row, leaving a gap behind the last card in sequence in each row.

In the deal above I was able to make substantial progress on the initial deal, by moving the following cards in sequence:
4H  9C  7S  TC  4C  JC  QC  5H  2S3 3S 
8D  6H  KH  TD  2H2 8C  9D  QH  3H  2C4
9S  TH  QD  5S  3D  5D  TD  KH  4D  7C 
6S  JD  4H  5D  5H  6D  8C  7S  QD  KD 
8S  9S  TS  JS  2D1 8H  6C  7C  8C  9C 
6H  7D  KS  3D  9H

Note that moves of twos are suffixed by the row to which the two is being moved.   The following position was reached by the time I was blocked:

Montana 2437 blocked

The program then removed all of the cards from the queen of spades rightward in row 1, and similarly from the ten of clubs, king of clubs, and five of clubs.   These were then shuffled by the computer and placed, leaving gaps behind the three of diamonds, six of hearts, jack of spades, and two of clubs, resulting in a new position:

Montana 2437 second deal

From this position, I was able to finish the game with only one redeal:
7H  TC  8D  9D  QD  TH  8C  4D  5D  JC 
QS  KC  QC  3C  QH  5C  7D  4C  9H  9C 
JD  KS  7C  8C  9C  TC  JC  QD  KD  KC 
5C  TH  6C  KH  JH  QH  6D  7D  8H  9H 
TH  JH  QH  KH  8D  7C  9D  8C  TD  9C 
JD  TC  QD  JC  KD  QC  KC 

Montana is a popular game found very often in both books and computer implementations.   Some versions restrict the player to two or three redeals; others allow an unlimited number of redeals (of course, a deal will always come out eventually if redeals are unlimited).   I am only successful on occasion within four deals (three redeals).

The version in Solitaire Virtuoso allows unlimited redeals, and also allows larger and smaller versions to be played by changing the number of suits and ranks.   Select either of these under the Parameters menu. Clicking on a playable card sends it automatically to the spot where it is allowed to go.   Clicking on a two normally sends it to the first available open spot at the left end of a row; if more than one such spot is open, and you want to a send a two to a spot other than the first, you may send it to the second available spot by holding down the Shift key while clicking, or the third available spot with Control + click.   You can also send it to the last available spot with a right-mouse click.   Clicking on the blue stockpile at any time reshuffles.  Clicking on the stock once the game is won automatically writes the full solution to a text file with the prefix MON, followed by the deal number.   This can be edited by hand if you wish to add notes on your solution.   You also write a partially played game to a file using Options/Write Solution, though as yet there is no playback capability.

Montana 50908

Smaller versions are suitable to be played as open solitaires by not allowing any redeals.   I have played quite a few deals with five ranks and four suits, winning about 70 percent of the time.  Professor N. G. de Bruijn wrote an article (see Bibliography) on the four-rank version, which he calls Pretzel Solitaire, but he presets the suit of each row by placing the aces to the left of the rows in the fixed order S-H-D-C.  He claims a win rate of about 45%, but in my opinion much of the richness of strategy lies in choosing which suit to build in each row, and frequently switching suits: playing a two to one row and later to a different row.   Above is a sample deal, number 50908, with five ranks.   I won by the sequence of moves below: note that both red twos were moved twice.

4H  2H2 5D  2D3 3S  4S  2H4 2C2 2D1 3D 
5C  2S3 3S  4S  5S  4D  5H  5D  3C  4C 
5C 

Bibliography
de Bruijn, N. G. -- Pretzel Solitaire as a Pastime for the Lonely Mathematician, pp. 16-24 in The Mathematical Gardner (edited by David A. Klarner), 1981, Wadsworth International/Van Nostrand Reinhold, ISBN 0-442-25336-2
Article on 4x4 Montana, with mathematical theory and many sample deals.  The author suggests playing deals out in your head before you ever move a card.


Copyright 2015 by Michael Keller. All rights reserved.  This file was revised on January 30, 2015.