Solitaires : Playing TTT


Playing a Solitaire as experiment 

We propose a game experiment to demonstrate that specialized mental skills are generated during the process of repeated solution of problem, in which players become gradually expert and familiar with one of the game’ strategy. From previous experiments (Luchins and Luchins) we know that the more a payer becomes expert, and the more his reaction becomes fast and automatic. Experimental data available in repeated puzzle solving, may therefore be useful to discriminate between automatic and controlled processes; moreover most individuals, once they have been able to identify one strategy and use it repetitively until it becomes familiar, do not abandon it even in new contexts where better strategies are available.

The Solitaire


The deck consists of 6 cards: 2§, 3§, 4§ and    2© , 3©, 4©.  All six cards are used in each game.  The player may move alternatively from position “Color” and “Number” and may exchange the card with the cards in the four positions named Target, Up, DownColor and DownNumber.In the Target and Up areas the cards are face-up, while in the DownColor and DownNumber areas they are face-down.  Therefore, as soon as the cards are dealt, the player can see the cards in positions Color, Number, Target and Up.  The game ends when the player puts 2© in Target position.  In order to do this, he alternately exchange the cards for one of the cards placed on the board.  There are no restrictions on exchange with cards occupying the Up, DownColor and DownNumber areas.  Exchange with the card in Target area is constrained and the rules are different for the two position, Number and Color.   From Number position the player may exchange his card with the card placed in Target position only if the two cards are of the same color (e.g. exchange 2§, with 4§, or 2© with 3©  ).  From Color position the player may exchange his own card with the card placed in Target position only if they have the same number (e.g. exchange 3§ with 3© or 2§ with 2©). The player moves alternatively from Color and fron Number position and exchange his card with one of the cards on the board until is able to put 2© in the Target area.  There is an additional move, called “Pass”.  When the player decides to “Pass” he skips the move from Color to Number position or viceversa. To simplify, henceforth the following symbols are used:

U  –  exchange the card with the card Up

C  –  exchange the card with the face-down card below Color’ s card  (DownC)

N  –  exchange the card with the face-down card below Number’ s card  (DownN)

T  –  exchange the card with Target

P  –  pass


As it happens for most card games , in our experiment the game starts from a  random card distribution.


Download TTT  Solitaire

 Download programs to evaluate TTT experiments



There are a large number of possible starting configurations, which depend upon the possible card combinations. The starting configurations may be partitioned in two different domains – that we will name respectively Blue and Red – for each of which it exists an optimal strategy. The strategy that is optimal in a Blue configuration is sub-optimal if applied to a Red one and viceversa. It is shown that individuals who are exposed to many runs all starting by a Blue configuration, discover the optimal strategy of this particular sub-domain, and tend to use it also for Red domain, and viceversa. The “modules” in this contexts are specialized mental skills generated in a specific context , the Red (or Blue) configurations, that players use beyond the right domain.  Therefore the deviations from optimal behavior, for this class of problems, can be explained as “locally rational” behaviors generated by an incomplete representation due to the specificity of mental skills and the incomplete categorization of the problem .

Data of previous experiments as far as programs may be found at


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