Running Head: Trouble 1

Fun Getting into Trouble

Katie Cole, De’Niece Harrison-Hudson, & Lorisha Riley

Olivet Nazarene University

Senior Seminar in Mathematics

Dr. Hathaway & Dr. Brown

December 10, 2013

Trouble 2

In an attempt to unite the realms of linear algebra and statistics in a creative way, we

chose to apply our knowledge in these areas to an ordinary board game. After exploring different

possibilities, we came across the game of Trouble. Providing easy manipulation of rules and only

one factor of chance, Trouble was an ideal choice. We intended to apply the topic of Markov

Chains to this game in order to discover the probabilities of landing on certain spots on the

board, the average number of plays per game, and the effect of multiple players on those

probabilities.

The game of Trouble is a 28-space game board which allows a maximum of four players

with four game pegs per player. There is a designated starting position where a player houses

their game pegs, and finishing point where a player’s peg reaches safety after circling the board

one time. For our purposes, we assumed only one game peg per player in order to concentrate

decision-making of movements to one choice of peg. In addition to this manipulation, we also

limited the number of players to two. Within the game of Trouble, the location of the opposing

players’ peg is critical to your survival because the other player can land on your location,

sending your peg back to the starting position. We designed our experiment with the assumption

that there was only one person six spaces behind your peg at all times, rather than expanding this

to multiple players at varying distances from your location on the board.

In order to apply probabilities to this game, we need a source of chance, in this case the

roll of a di, which determines your next position. You then have the ability to move your peg up

to six spaces on any given turn, with the exception of your first move which requires you to roll a

six in order to leave the starting position and enter the game board. This slight change in starting

requirements made for interesting results within or experiment. One last rule is that you must roll

the exact number of spaces required to exit the game board and enter the finishing zone of safety.

Trouble 3

This means that your probability of reaching the finish depends on the number of spaces you are

from the finish. This rule, along with the requirement of rolling a six to leave the start, are the

only two instances in which a player’s peg might remain in its current location.

A Markov chain is a process consisting of a finite number of states in which moving from

state i to state j would be represented by the probability, pij. In this case, the current state depends

only on the previous state. We can then create a matrix made of these probabilities of moving

states, known as a transition matrix. In terms of playing the game, the probability of staying in

the same position would be represented by pii. Since a player’s peg must be in some location on

the board at any given time, whether still in the starting position or in the game, the sum of

probabilities will equal 1 over any row of the transition matrix.

It is the case of some Markov chains to be considered absorbing in nature. This means

that at least one state within the system cannot be left once it is entered. Within the Trouble

game, this would be the finishing point when your game pegs are safe from other players and can

no longer be moved. In the case of steady state vectors, systems will continue on infinitely,

whereas an absorbing system will eventually reach an end. It is the nature of absorbing systems

to provide information regarding absorption time, or the time it takes a player’s peg to reach the

finish. We can also calculate the probability of absorption based on the peg’s current location.

In the first matrix we analyzed, we used the scenario where there is not a player within

six spaces to the back of your peg. This matrix is shown in Figure 1. The probability of the peg

located in the game space represented by row i moving to the game space represented by column

j is represented by the fraction in the intersection of row i and column j. The first column

represents the Starting home space and the last column represents the Finish home space. The

probability of getting out of the starting spot was 1/6 since the only way to enter the main body

Trouble 4

of the game board is to roll a 6. Then, continuing down the matrix, there is an equal 1/6 chance

of moving from the peg’s current position to one of the next six spaces on the game board. In this

matrix, we did not specify that an exact roll was needed to reach the Finish space, so the

probabilities of reaching Finish when within 6 spaces of the end increase by 1/6 for each

additional game space.

Figure 1

The second matrix we looked at dealt with the scenario in which there is always a peg

within six spaces. This matrix is shown in Figure 2. For this matrix, we added the rule that states

that if your peg is on a game space that another peg lands on, your peg will be sent home.

Therefore, with a peg within six spaces, there is always a 1/6 chance that your peg will be sent

back to the Starting home space. Thus, the probabilities for landing on a playing space within the

main area of play change to 5/36 since there is a 5/6 chance of not being sent home and a 1/6

chance to move forward within six spaces. For this matrix, we also did stipulate that an exact roll

was needed to land in the Finishing home space. Thus, the probabilities in the last column of

Trouble 5

reaching Finish remained 5/36 while the probability of not being able to move increased by 5/36

for each space.

Figure 2

For our third matrix, shown in Figure 3, we looked at the “combined” case of a peg being

within six spaces to send your peg home approximately half of the time. Thus, the probability of

being sent home from any given spot was 1/12 since we multiplied 1/6 by 1/2. Similarly to the

second matrix, the probabilities of moving within the main playing spaces are 11/72, since there

is an 11/12 chance of not being sent home multiplied by a 1/6 chance of moving forward. We

also kept the rule of needing an exact roll to reach the Finish, so the probabilities of not being

able to move to Finish from within six spaces increases by 11/72 for each step closer to the end.

Figure 3

Trouble 6

The next component necessary for our application of Markov Chains is the position

matrix, which can be seen in Figure 4. This is the matrix that tells the probability of a peg being

located on any game space. Once again, the first entry of the position matrix is representative of

the Starting home space while the last entry of the position matrix is representative of the

Finishing home space. For our calculations, we looked at the scenario of beginning solely from

the Start space, so there is a 100% chance, or an entry of 1, as the first and only entry in the

position matrix.

Figure 4

Results are calculated using Markov Chains and their properties. Multiplying the position

matrix by the transition matrix will result in a matrix giving the probabilities of landing on

different game spaces after one move. Then, it is possible to take this new probability matrix and

multiply it by the position matrix to find the probabilities of ending on different spaces after two

moves. This process can be continued step by step, or a shortcut can be used to find the

probability of landing on a game space after an arbitrary number of moves. To do this, multiply

the position matrix by the transition matrix raised to the desired n. The resulting matrix will give

the probabilities of being located on a given spot after n moves.

Trouble 7

Figure 5

In figure 5, there are results of our combined matrix for the probability of landing on

certain spots after the specified number of moves. For example, after 30 moves the probability of

landing on the fifth spot is .679%. Our results indicate that there is not any specific game space

landed on more frequently than others. The exceptions to this result are the spots closet to the

Finish, due to the requirement of exact roll. Also, the Start location carries a larger probability

because of the difficulty of leaving home and the probability of being sent back to Start. For full

results, see tables in the Appendix. We anticipate, on average, it will require 30 or more moves

to complete game.

Trouble 8

Figure 6

Absorption Time is a state which once is entered, it is impossible to leave. Absorbing

matrices are a special category of Markov Chains, counter to steady state matrices. For our

purposes, the absorbing state represents Finish and the completion of game. We start by dividing

our transition matrix into four submatrices, as shown in Figure 6. Begin by locating the identity

matrix, I, and extend to match the size of Q, the submatrix diagonal to I. R is the submatrix

diagonal to the zero submatrix. Now that we have our divisions, begin the process of finding C,

which is the matrix of probabilities of finishing the game from any certain location.

First, subtract Q from I, and then take the inverse of the result; this is N. Next, to find C,

multiply this N by the submatrix R. In our case, C is a column vector of ones due to the fact that

the game will end sometime, regardless of current location. To find the number of moves to

R

I0

Q

Trouble 9

complete the game, we multiplied the column vector C by N. Results can be seen below in

figure 7. Read similarly to figure 5, it is easy to see that the number of moves to complete the

game from the combined matrix in position two is 32.201.

Figure 7

Our experiment using the game of Trouble has provided both challenge and surprise.

Using three different transition matrices: one player, two players with one player always being

six spaces away, and an average of the two scenarios, we have discovered a variety of facts

regarding the game. We have found that it takes more rolls to complete a game on average when

two players are playing rather than one due to the possibility of being sent back to Start. Our

calculations gave us no inclination of a space that is landed on more often than another, other

Trouble 10

than the possibility of spending more time of the spots closest to the finish. This is due to the

requirement of rolling the exact number of spaces to reach the Finish. Our absorption time

calculations indicated a higher number of rolls required to complete a game than first thought;

even after 50 rolls, there is only a 77% chance of finishing the game using the combined matrix.

Trouble was a great choice for this experiment, allowing us to manipulate a number of variables

to reach our specific goals. We have successfully merged the worlds of linear algebra and

statistics in a way that answers many of our questions, while leaving the door open for further

research.