Mr. Markov Tours Candy Land

Introduction

Candy Land TM is a classic children's board game marketed in 1945 by Milton Bradley, now owned by Hasbro. Each player follows a multi-colored path through Candy Land TM to reach the Candy Castle, with each move determined by the draw of one card from a deck of 64 cards. Figure 1 shows a classic game board, copyrighted in 1984. Each player's journey is completely chance-based; there are no decisions made, no skills needed, and no interactions among the players. These properties are precisely what allow a complete mathematical analysis by using the tools of absorbing Markov chains and general probability.

Our analysis follows that of Chutes and Ladders TM (another Milton Bradley game) done by Gadbois [1993], who taught our course on Markov chains in the spring of 2014. We investigate two basic issues:

• the average number of moves to finish the game from any square for one player, and
• the average length of the game for any number of players.

We do not use simulation.

In the actual game, cards are drawn without replacement and without reshuffling. To model this, a dynamic (constantly changing) transition matrix would be necessary. So for simplicity in modeling we assume that for each move of each player, each of the 64 cards is equally likely to be drawn. This assumption is equivalent to assuming that after each draw, the card is replaced and the deck is reshuffled.

Of the 64 cards in the deck, 58 are red, purple, yellow, blue, green, or orange, indicating that the player moves ahead to either the first square or the second square of that color. Specifically, there are exactly two cards of each of the six colors allowing a move ahead two squares, while there are seven cards of two colors (green and orange) allowing a move ahead one square, and eight cards of each of the four other colors. The other six cards are pink and different; drawing one sends the player directly to a unique corresponding pink square on the board, regardless of the player's current location.

Most of the squares along the path are "ordinary," colored red, purple, yellow, blue, green, or orange. But along the way, there are two shortcuts (the "Rainbow Trail" to square 57 and the "Gumdrop Pass" to square 45); in each case, for purposes of this analysis, the square at the beginning of the trail is identified with the square at the end. There are also three "sticky squares" ("Gooey Gumdrops" at square 46, "Lollipop Woods" at square 84, and "Stuck inMolasses Swamp" at square 119); in each case, the player must remain there until a card of the specified color is drawn. Finally, there are six different pink squares, one for each corresponding pink card in the deck.

Each player begins the game off the board, on what will henceforth be called "square 0." Thereafter, squares are numbered from 1 to 132, skipping the square at the beginning of each of the two shortcuts. Unlike some similar games (e.g., Chutes and LaddersTM), the Candy LandTM player need not land on the final square 132 to win but may also go past it.