Concentration

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Problem Description

Here is a version of the memory-testing  card game, "Concentration".   The objective is to select matching pairs  from an array of face down cards by turning two cards face up, one after the other, on each turn.   When a match is found, the matching cards are removed and the player gets another turn.   When the two cards do not match, they are returned to their original face down positions.  

As a single player game, the objective is to find all matching pairs in the fewest moves.  In the two-player game, the player that finds the most pairs is the winner.    

Background & Techniques

This program was prompted by a request from a high school student looking for a class Delphi programming project.   By the time I had developed  a graphical demo that would flip or remove cards as a starting point, he had moved on to something different.   So I decided to complete the logic part and post it.

The game allows one or two players and the number of cards in the array can be selected by the user.    

Non-programmers are welcome to read on, but may want to skip to the bottom of this page to download executable version of the program.

The U_CardComponent previously introduced, automates the card image handling.   When a new game is started two decks are created -  one with 52 cards and one with twice the number of pairs specified by the user.    After shuffling the temporary 52 card deck, the first "nbrpairs" cards are used to set the suit and card values of the first half of the real deck,.  The lower half of the deck has card values and suits set to match the first half, ensuring pairs.   The play deck is shuffled of course before dealing them into a face down array. 

The checking and scoring logic is embedded in an OnMouseUp exit  for each TCard object.   The first card clicked of a turn is saved and when a second card is clicked, its value and suit are compared to the 1st card.   Based on results, scores are updated, cards made invisible or simply turned back face down.    When all pairs are matched, the winner and score are displayed and the game is over. 

Expected game lengths

What is the expected length of a game if the player has perfect memory?   That is, once a card has been turned  our player will remember exactly where it is located; it is, in effect, left face up  Since a perfectly lucky game would require N turns to match N pairs, and a perfectly unlucky game would require 2N-1 turns.    Simplistic reasoning says that the average game should be halfway between the shortest and longest possible games which  slightly less than 1.5 times the number of pairs .  My analytical skills bogged down in trying to calculate a value, so I resorted to writing a "Concentration Study" program to play several thousand "perfect memory" games and chart the game length statistics.   I looks like the average game is slightly higher than 1.5 times N.   In other words, there seem to be more ways to be be unlucky than there are to be lucky.  I'll wait for a real mathematician to verify this, or tell me that I have a bug in the program.  In any event the Concentration Study program  can  be downloaded from the links below if you want to play with it.  

Addendum April 12, 2009:  If the original version of Concentration was not difficult enough, here is Version 2 which adds the option of requiring 3 cards to be matched before they are removed from the board.  This version also adds the option of matching on card value and suit, as in the original version, or , on card value only. 

Running/Exploring the Program 

Suggestions for Further Explorations

Draw three cards per turn and create triplets?
Implement computer as an opponent - we would have to make him somewhat forgetful of the locations of cards seen.

 

Original Date: May 21, 2004 

Modified: February 18, 2016

 
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