Montana (Gaps) Solitaire



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 Virtuoso deal 71830), the gap in row 1 may be filled with the king of hearts near the end of row 2, leaving a new gap behind the six of hearts.  Other cards which could be moved are the three of hearts, jack of clubs, and jack of spades.  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 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.

The deal above is the first time I have ever won on the initial deal, though it took two tries.  On my first attempt, I made several series of undos, ending with two cards (queen and king of diamonds) blocked at the end.   I started over, and found that by making several changes of suit in the top three rows, I was able to finally make the deal come out on the first pass, by moving the following cards in sequence:

KH  7H  7S  KD  8D  6D  8H  6H  JS  3H 
4H  7C  JC  TS  QC  QS  9D  TD  7H  JD 
9S  8S  2D3 3D  8H  QD  9C  2H4 3H  4H 
5H  2S2 3S  QH  KH  7D  8D  8C  9C  5C 
6C  9H  6H  4S  9D  TC  6S  TS  9S  4D 
2D1 3D  4D  2S3 3S  4S  2C2 JH  TD  JC 
7S  3C  4C  TS  5S  6S  7H  8H  5C  JS 
KC  6C  QS  5D  6D  7D  8D  9D  TD  JD 
QC  QD  KC  8S  9S  TS  7S  9H  7C  8C 
9C  TC  JC  QC  8S  9S  TH  JS  TS  JH 
QS  QH  KD  JS  QS  KC  KH 

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).   The deal above was my first win with no redeals; I had previously won once on the first redeal, and occasionally within four deals (three redeals).

Mark Masten has been investigating the game with his own computer solver.  It appears that Gaps is overwhelmingly harder for human players than computers: his computer solver wins over 85% of deals on the first pass under the rules we describe here, where a two of any suit can be played to any row with an empty space at the left.  Over 62% of deals can be won without any suit changes (i.e. without moving a two from the left end of one row to another).   The variant with the suit of each row preassigned has a much lower win rate, just under 25%.    Mark's article has much more detail, including technical details on his solver.

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.

Reduced deck variants



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 83002, with five ranks.   I won by the sequence of moves below: note that the twos of spades and diamonds were moved twice.

2S1 5C  3H  2D2 5S  4S  5S  4H  5H  3C 
2S4 2D1 2C2 3C  4C  5C  3D  4D 

A historical version with a reduced deck is House of Commons, found in Tarbart's 1905 second edition of Patience Games, and Mary Whitmore Jones' 1911 New Games of Patience (Parlett includes it in his 1979 Penguin Book of Patience).  It is played with the 40 cards from ace to ten of each suit, exactly as the standard Montana, with one redeal allowed (the cards are not shuffled, but picked up in order and redealt.)    Surprisingly, Mark Masten's solver says this is just slightly the hardest game to win on the first pass, with a win rate a little under 82%.

Other variants

Boleslaw Cynk devoted an entire book to a set of original variants to Gaps.   All of his versions, rather than using the aces to create random gaps, leave a space at the left end of each row, which can be filled at any time with a two (or an ace in The Blue Moon variant using the full deck).   He also eliminated the shuffling of cards between redeals: the out-of-order cards are picked up from left to right (in each row from top to bottom), and redealt in the same order, leaving a space after the correctly ordered sequence (if any) in each row.   He also created two new allowable strategic moves: the row-exchange rule allows an equal number of cards of two rows (in correct sequence starting from the left end) to be exchanged; the ace-double role rule allows an ace to be placed in a space behind the king of the same suit.   Using these rules, Cynk was able to successfully solve every tableau sent to him by readers of Leisure magazine with no more than two redeals (and most of them with only one).   His reduced-deck version Trains (twos through eights) can usually be won without any redeals, using only the row-exchange rule.   Double Gaps is a full-deck version dealt into a 4x15 array, with two spaces at the left end of each row (aces and the corresponding twos can be played to those spaces immediately).   Cynk estimates the win rate at 80% with no redeals, sometimes without any row-exchanges.

Another notable variant is Robert Harbin's Spaces and Aces (Waddington's Family Card Games, 1972), in which a gap may be filled with any higher card of the same suit.  In his version the aces are removed from the 4x13 array and each placed at the left end of one of the rows (forming a 4x14 array), fixing the suit of that row for the rest of the deal.   There is no redeal; this is a completely open variation.



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.

Cynk, Boleslaw -- Competitive Patience Games, 1966, John Baker, 103pp., hardback

Robert Harbin, Waddington's Family Card Games, 1972, Elm Tree Books (reprinted 1974 by Pan Books, ISBN 0-330-23892-2


Copyright ©2024 by Michael Keller. All rights reserved.  This file was revised on August 14, 2024