Before reading this article, if you have not done so already, you might want to read the previous two articles regarding the analysis of Chutes and Ladders and Risk. These will set some context. There is no skill required to play Candyland other than being able to recognise colours. A deck of movement cards it shuffled, and players take turns to move their tokens according to the instructions on the card.
The board consists of a linear track containing spaces, mostly coloured red, green, blue, yellow, orange or purple. In addition there are six pink spaces containing named characters, two bridges, and three of the coloured spaces are sticky more of this later.
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Players alternate in drawing movement cards, most of which show one of six colours, and then moving their token ahead to the next space of that colour. Some cards have two marks of a colour, in which case the player moves his or her marker ahead to the second-next space of that colour. The deck has one pink card for each named location, and drawing such a card moves a player directly to that board location This move can be either forward or backward.
Finally, if a player lands on one of the sticky spaces, he misses his next turn. Landing on the start space to one of the two bridges allows the player to take a short cut.
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My first task was to model the board. There are Visible spaces on the board, which can be stored as a simple array. In addition, there is a space at index zero for the starting location players start off the board. I modeled the bridges as their own state when a player is on a bridge, they are on either space or space First I created a quick stylised map in Excel shown on the left from the game board shown below.
This enabled me to enter the data for the cell colours into an array. I counted the distribution of cards in the game. Assuming my children had not lost any, there were 64 cards supplied in our version of the game.
This is not the case with Candyland. As cards are drawn, they are used and discarded and so the probability of which card will be turned up next is dependent on cards that have already been seen. This is easy to confirm with a thought experiment: For instance, if all four double red cards have already been played, the probability that the next unseen card is also a double red will be zero.
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To work through every possible combination of ways that the cards could be arranged in the deck is computationally infeasible. In longer games, which sometimes happen, all the cards in the deck are used up anyway and need to be reshuffled, so this results in the same situation. The transition matrix for explanation of a transition matrix, review the posting about Chutes and Ladders for this game is x , with each element i , j describing the probability of a player moving from location i to location j on the next move.
By stochastic definition, the sum of all the probabilities in a row adds up to 1. Similarly 20 and 42 and all the other pink squares. Further out, the squares that are accessible with double cards have different probabilities because of their different frequencies in the deck.
With care it is possible to construct a transition matrix for the game which enumerates the probabilities of moving from every game state to every other game state. All that is needed now, is to create a column identity vector with 1. The row vector output will show the probability distribution for where the player token could be after one card draw. Below are the results represented in graphical form, with shading denoting the probabilities.
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Darker red representing high probabilities and Lighter red representing low probabilities. After one draw, the darkest areas are, obviously, the ones that are reached by the cards that have the highest distribution in the deck the single colours. The spaces obtained by the double colours are lighter shaded some lighter than others because there are less of these cards in the deck. If we multiply the above output vector by the transition matrix again, the new output vector represents the superposition of probabilities of where the player token could be after the second drawn card starting from all the weighted starting positions of the first vector.
The result is shown below. The pink squares are the same shade, and there is still no way to reach the finish line with just two card draws.enter
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After the third card draw, the probability distribution looks like the picture below. The clouds are getting wider, and more of the board is shaded. After four draws, and interesting event occurs — it is mathematically possible to win the game! The probability of landing of finishing the game in just four moves is approximately once in every 10, games. There are multiple ways to achieve this victory, but all require that the first card drawn be the pink special card that takes the player to square In our version of the game, this is the Ice Cream.
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By five draws, there are more ways to achieve the victory condition as you can see from the probability increase of the token being on the finish space. Below are thumbnails of the board showing the percentage of finishing increasing with each drawn card. Here is a short video animation showing the probability distribution for the first 50 card draws. Below is a graph of the probability of a single person finishing the game by move- n. The Modal number of card draws to is In the early s, when the dreaded disease polio struck thousands of Americans, Eleanor Abbott, a victim of the disease, sought to invent pastimes for children who were recuperating.
Her most successful idea became Candy Land, a game many people remember fondly as the first board game they ever played. On the advice of her friends, Abbott sent her creation to Milton Bradley, and the company introduced Candy Land in Eager players travel along the rainbow-colored trail, past the Peppermint Stick Forest and the Gumdrop Mountain, navigating impediments like the Molasses Swamp and the Ice Cream Floats. Delightful sweet treats decorate the playing board.
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Wooden playing pieces were used in the past, but today gingerbread men are the game's markers. Drawing simple color cards, players advance to squares of the same color, and the winner is the one who reaches the Candy Castle first. Milton Bradley, now Hasbro, has produced Candy Land for more than 50 years. Today, along with the standard edition, the game is available in CD-ROM, as a hand-held electronic game, and in special character editions.