This is my Math IA that got a raw mark of 17/20, but later got moderated down to a 12/20. I got an overall score of 6 in math, one mark away from a 7.
I wanted to explore an engaging and interesting topic in my Math IA, and chose to analyze Solitaire, which is a card game I really like. If you'r...
International Baccalaureate
Mathematics: Applications and Interpretation Higher Level
Mathematical Exploration
Developing and testing probability
models for Klondike solitaire
Candidate code:
Page count: 15
,1.1 Introduction
Klondike solitaire has remained a behemoth in online gaming in the 30 years since its
launch in 1990, as a free and accessible pastime 1. Yet, the strategic framework behind it
is anything but simple: successfully arranging cards requires close attention and critical
thinking2. This has drawn mathematicians to attempt to determine the probability of
winning a game of solitaire, but the task has proven to be perplexing due to the
immensely complex nature of the standard 52-card deck. To demonstrate, the possible
ways to arrange a deck of cards is expressed as: 52 possibilities for the first card, 51
possibilities for the second, 50 possibilities for the third and so on. In mathematical
terms, this is expressed through the factorial of 52, the value of which is astronomically
large: 52! ≈ 8.0658175 ∙ 1067. Even considering the initial arrangement of Klondike
Solitaire, in which 7 cards are revealed to the player, there are (52 − 7)! = 45! ≈
1.1962222 ∙ 1056 possible arrangements of the undisclosed cards. Hence, the complex
strategy of the game and great number of possible arrangements has restrained the
degree to which existing literature in applied mathematics has successfully developed
probability models for the game.
1.2 Rationale
The extremely large sample space of Klondike solitaire and its connections to game
theory prompted me to conduct my own exploration. It is fascinating how large pools of
possible human behaviors are still analyzed with accuracy through statistics and game
theory in economics, political science, sociology etc. On this premise, to observe
whether it is possible to detect greater associations between the distribution of the
seven revealed cards in Klondike solitaire and different parameters that measure a
player’s success, I decided to examine this game through both theoretical and
experimental approaches to probability and statistical modeling.
1 (Team)
2 Ibid
2
, 1.3 Rules of Solitaire
The aim of Klondike solitaire is to arrange cards in increasing order of their values34. As
shown in Figure 1, the initial layout includes a draw pile of 24 cards from which cards
are drawn, 7 build piles with 28 cards (first with one card, second with two, third with
three and so on) and four foundation piles on which the four suits of cards (hearts,
diamonds, spades, and clubs) are stacked in order of increasing value, beginning with
aces.5
Figure 1: Initial layout of cards6
In each game, a player may either put a card from the build piles to the foundation piles,
and vice versa, place a card or sequence of cards from one build pile to the other or
draw a new card from the draw piles7. Cards are placed in foundation piles on top of a
card with the same color and suit that is directly below them in the order of value8. Also,
cards are ordered in the sequence of value in alternating colors in the build piles9. It is
generally accepted that the fewer moves it takes to stack all cards, the more successful
the player.
3 (Solitaire)
4 In a classic deck, the values of cards in increasing order are A (Ace) (lowest), 2, 3, 4, 5, 6, 7, 8, 9, 10, J
(Jack), Q (Queen), K (King) (highest). Ibid
5 (Klondike Solitaire Rules – How to Play and Deal the Cards)
6 Screenshot taken from the website “Solitaired” https://solitaired.com/?deal=winnable [11.10.2022]
7 (Klondike Solitaire Rules – How to Play and Deal the Cards)
8 Ibid
9 Ibid
3
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