1 Introduc+on
The report aimed to design a board game by modifying the game of “Snakes and
Ladders” to foster family communicaEon and improve mathemaEcal skills. Using
Markov chains and problem-solving skills, it was ensured that the board game was
Eme-efficient for family play and visually appealing. Technologies including the
graphic calculator and Excel were used for computaEon and simulaEon.
2 Considera+ons
2.1 Observa,ons
Figure 1 Game Board of Classic Snakes & Ladders
1. It was observed that the game modifies the classic game Snakes and Ladders. In tradiEonal Snakes and Ladders:
o Players roll the dice to move around the numbered spaces on the board. Ladders lead upwards, while
landing on a snake’s head sends them downwards.
o The game concludes when the first player rolls the exact number needed to reach the last square on the
10×10 spaced board (Figure 1). Rolling too high results in bouncing back. (Cruz, 2018).
2. It was observed that the number of spaces on the board must range between 20 to 100 and the total number of
“snakes” and “ladders” was in the range between 10 to 25. This maintained the game’s challenging aspect and
short Eme duraEon without overwhelming complexity.
3. It was observed that the duraEon for playing the board game should be short, as evidenced by the ideal Eme
length of 10-30 minutes for a family (The Board Game Family, 2012). In a real-life test, an 18-year-old adult took an
average of 8 seconds to roll a die and move to the desEnaEon space, compleEng one turn.
4. The layout of the board game should be visually appealing. Therefore, the game board design should be clear and
aestheEc.
5. It was observed that technologies such as graphic calculator and Excel were mandatory to use for computaEon
and simulaEon to show problem-solving procedures.
2.2 Assump,ons
1. It was assumed that the players of the game were two adults and two children (4 players) based on the
predominant proporEon of Australian women having 2 children when compleEng ferElity (Australian InsEtute of
Family Studies, 2023).
2. It was assumed that players take turns to roll a singular die to move around the board, with a duraEon of 8
seconds per adult per roll, and 16 seconds per child per roll (observaEon 3), as it was assumed that the Eme taken
for a child was twice as an adult (LibreTexts, 2020).
3. It was assumed that the game would take around 10 to 16 minutes to ensure a joyful experience for players
(observaEon 3).
4. Excel was sufficient to simulate the board game as it is commonly used by board game designers to organise the
data and present the game (Backe, 2022).
5. It was assumed that the simulaEon of 20,000 games could accurately represent real-life situaEons (Quinlan, 2015).
2.3 Game modifica,ons
The adapted game rules remained the same as the tradiEonal game, aside from the eliminaEon of bounce-back rules when
players reach the last square (observaEon 1). The game would finish if one of the players moved to the last square winning
the game. Moreover, the number of spaces on the game board was reduced from 100 to a lower number (observaEon 2).
These modificaEons aim to enhance the game's simplicity, potenEally reducing its duraEon, and making it more family-
friendly.
3 Mathema+cal concepts and procedures
The board game design used Markov chains to describe the board game’s long-term behaviour. The transiEon matrix (T)
represented the probabiliEes for players to move and land on each numbered space by rolling a single die. The long-term
behaviour can be found using the formula !! = !" × $ ! , where !! is the state matrix that could represent each turn,
!" represented the iniEal state matrix, and $ ! was the nth transiEon matrix (Smith, 2019).
1
, c d
e f
Figure 2 Transi;on Matrix (Smith, 2019) Figure 3 State Matrix Sn of the nth turn
If one of the four players won the game, the game would be considered finished. Therefore, the probability for one player
to win the game, &((), was determined. Since the maximum probability is 1, it was assumed that a probability over
0.999999 indicated that the game had finished. The expected value of probability was calculated using the formula
∑(+) = ∑ , ∙ &(,) (Yıldırım, 2020) where ‘x’ represents the number of turns, and ‘P(x)’ represented the probability of the
board game finishing at the xth turn. The product of the expected value represenEng the theoreEcal number of turns to
finish the game and the Eme taken per turn, the theoreEcal duraEon of the game was determined.
The experimental data of the turns to win the game was compared with the theoreEcal value using the percentage error
formula:
|$%&$'"($!)*+,)-$.'$)"/*+|
%/0010 = )-$.'$)"/*+
× 100% (Study.com, 2022).
4 Solu+ons
The iniEal design of a 6 × 6 spaces game board
was shown in Figure 4. The ladders were
represented by green arrows, while the red
arrows indicated the snakes.
Figure 4 Ini;al game board
The Eme taken per turn was 5&$' )1'! = 2 × 8 + 2 × 16 = 48: (assumpEon 2).
4.1 Markov chain matrices
The transiEon matrix shown in Figure 5 was determined by the probabiliEes for the player from a numbered space to the
next space based on rolling a single die. For instance, the probability for the player to move from space 1 to space 3 was
1/6, shown in B3 in Figure 5. Figure 6 demonstrated the iniEal state matrix, consisEng of the number “1” at the top and “0”
at the rest of the 36 spaces which represented the iniEal posiEon of the player at space 1.
The Markov chains shown in Figure 8 are the product of the transiEon matrix and the state matrix represenEng the
resulEng state matrix ajer a certain number of turns. It was calculated by the “MMULTI” shown in Figure 7. For example,
in Figure 7, the state matrix ajer 1 turn, was calculated by “MMULTI(B2:Y25, AA2:AA25)”, where “B2:Y25” is the transiEon
matrix shown in Figure 5 and “AA2:AA25” is the state matrix shown in Figure 6.
2
, From
To
Figure 5 Transi;on Matrix of the Ini;al Design Figure 6 Ini;al State Matrix
Figure 7 Mul;plying matrices in Excel
3
The report aimed to design a board game by modifying the game of “Snakes and
Ladders” to foster family communicaEon and improve mathemaEcal skills. Using
Markov chains and problem-solving skills, it was ensured that the board game was
Eme-efficient for family play and visually appealing. Technologies including the
graphic calculator and Excel were used for computaEon and simulaEon.
2 Considera+ons
2.1 Observa,ons
Figure 1 Game Board of Classic Snakes & Ladders
1. It was observed that the game modifies the classic game Snakes and Ladders. In tradiEonal Snakes and Ladders:
o Players roll the dice to move around the numbered spaces on the board. Ladders lead upwards, while
landing on a snake’s head sends them downwards.
o The game concludes when the first player rolls the exact number needed to reach the last square on the
10×10 spaced board (Figure 1). Rolling too high results in bouncing back. (Cruz, 2018).
2. It was observed that the number of spaces on the board must range between 20 to 100 and the total number of
“snakes” and “ladders” was in the range between 10 to 25. This maintained the game’s challenging aspect and
short Eme duraEon without overwhelming complexity.
3. It was observed that the duraEon for playing the board game should be short, as evidenced by the ideal Eme
length of 10-30 minutes for a family (The Board Game Family, 2012). In a real-life test, an 18-year-old adult took an
average of 8 seconds to roll a die and move to the desEnaEon space, compleEng one turn.
4. The layout of the board game should be visually appealing. Therefore, the game board design should be clear and
aestheEc.
5. It was observed that technologies such as graphic calculator and Excel were mandatory to use for computaEon
and simulaEon to show problem-solving procedures.
2.2 Assump,ons
1. It was assumed that the players of the game were two adults and two children (4 players) based on the
predominant proporEon of Australian women having 2 children when compleEng ferElity (Australian InsEtute of
Family Studies, 2023).
2. It was assumed that players take turns to roll a singular die to move around the board, with a duraEon of 8
seconds per adult per roll, and 16 seconds per child per roll (observaEon 3), as it was assumed that the Eme taken
for a child was twice as an adult (LibreTexts, 2020).
3. It was assumed that the game would take around 10 to 16 minutes to ensure a joyful experience for players
(observaEon 3).
4. Excel was sufficient to simulate the board game as it is commonly used by board game designers to organise the
data and present the game (Backe, 2022).
5. It was assumed that the simulaEon of 20,000 games could accurately represent real-life situaEons (Quinlan, 2015).
2.3 Game modifica,ons
The adapted game rules remained the same as the tradiEonal game, aside from the eliminaEon of bounce-back rules when
players reach the last square (observaEon 1). The game would finish if one of the players moved to the last square winning
the game. Moreover, the number of spaces on the game board was reduced from 100 to a lower number (observaEon 2).
These modificaEons aim to enhance the game's simplicity, potenEally reducing its duraEon, and making it more family-
friendly.
3 Mathema+cal concepts and procedures
The board game design used Markov chains to describe the board game’s long-term behaviour. The transiEon matrix (T)
represented the probabiliEes for players to move and land on each numbered space by rolling a single die. The long-term
behaviour can be found using the formula !! = !" × $ ! , where !! is the state matrix that could represent each turn,
!" represented the iniEal state matrix, and $ ! was the nth transiEon matrix (Smith, 2019).
1
, c d
e f
Figure 2 Transi;on Matrix (Smith, 2019) Figure 3 State Matrix Sn of the nth turn
If one of the four players won the game, the game would be considered finished. Therefore, the probability for one player
to win the game, &((), was determined. Since the maximum probability is 1, it was assumed that a probability over
0.999999 indicated that the game had finished. The expected value of probability was calculated using the formula
∑(+) = ∑ , ∙ &(,) (Yıldırım, 2020) where ‘x’ represents the number of turns, and ‘P(x)’ represented the probability of the
board game finishing at the xth turn. The product of the expected value represenEng the theoreEcal number of turns to
finish the game and the Eme taken per turn, the theoreEcal duraEon of the game was determined.
The experimental data of the turns to win the game was compared with the theoreEcal value using the percentage error
formula:
|$%&$'"($!)*+,)-$.'$)"/*+|
%/0010 = )-$.'$)"/*+
× 100% (Study.com, 2022).
4 Solu+ons
The iniEal design of a 6 × 6 spaces game board
was shown in Figure 4. The ladders were
represented by green arrows, while the red
arrows indicated the snakes.
Figure 4 Ini;al game board
The Eme taken per turn was 5&$' )1'! = 2 × 8 + 2 × 16 = 48: (assumpEon 2).
4.1 Markov chain matrices
The transiEon matrix shown in Figure 5 was determined by the probabiliEes for the player from a numbered space to the
next space based on rolling a single die. For instance, the probability for the player to move from space 1 to space 3 was
1/6, shown in B3 in Figure 5. Figure 6 demonstrated the iniEal state matrix, consisEng of the number “1” at the top and “0”
at the rest of the 36 spaces which represented the iniEal posiEon of the player at space 1.
The Markov chains shown in Figure 8 are the product of the transiEon matrix and the state matrix represenEng the
resulEng state matrix ajer a certain number of turns. It was calculated by the “MMULTI” shown in Figure 7. For example,
in Figure 7, the state matrix ajer 1 turn, was calculated by “MMULTI(B2:Y25, AA2:AA25)”, where “B2:Y25” is the transiEon
matrix shown in Figure 5 and “AA2:AA25” is the state matrix shown in Figure 6.
2
, From
To
Figure 5 Transi;on Matrix of the Ini;al Design Figure 6 Ini;al State Matrix
Figure 7 Mul;plying matrices in Excel
3
