Maths project on permutations and combinations class 11

This is a maths project for class 11th made on permutation and combination with reference to the game of chess.

1. de

ae © i
+ + $
DELHI PUBLIC SCHOOL

MATHS

?

(

PROJECT ©

NAME:PRANSHU KUMAR ANAND

2. ACKNOWLEDGEMENT

J, PRANSHU KUMAR ANAND OF STD. XI-A OF
DELHI PUBLIC SCHOOL IS SUBMITTING MY
MATHEMATICS PROJECT; AS:PER CBSE SYLLABUS
„AM GRATEFUL TO MY TEACHERS, FRIENDS AND
PARENTS FOR HELPING AND GUIDING AND CO-
OPERATING ME IN COMPLETING MY MATH'S
PROJECT,

ONCE AGAIN I SINCERELY THANK MY MATH'S:
TEACHER, MRS. LINDA JOSE FOR GUIDING ME TO
COMPLETE MY MATH'S PROJECT.

3. INDE 4

o
o
e
Acknowledgement
] Game of chess

Probability in chess

Moves possible

Probability examples |

P&C examples

4. Ee

INTRODUCTION

Title: Exploring the Strategic Moves: Permutations and Combinations in
Chess

Welcome to the intriguing world of chess. where every move is a
strategic decision that can shape the outcome of the game. In this

project. we delve into the realm of permutations and combinations.
‘applying these mathematical concepts to better understand the
complexity and beauty of chess.

Chess, a timeless game of intellect, requires players to anticipate and
plan their moves meticulously. The application of permutations and
combinations in chess opens up a fascinating avenue to analyze the

various possibilities that unfold on the chessboard.

In this project, we aim to explore how permutations and combinations

play a crucial role in determining the number of possible arrangements
and sequences of moves during a chess game. From the opening to the
endgame. we will unravel the mathematical intricacies that underlie
strategic decisions, leading to a deeper appreciation of the game's
complexity.

Why study permutations and combinations in chess? Because every

move in chess involves a unique set of choices, and understanding the

mathematical principles behind these choices enhances our ability to

analyze, plan, and strategize effectively. As we navigate through this

project. you'll gain insights into how permutations and combinations

influence the decision-making process in chess and contribute to its
rich tapestry of possibilities.

So, fasten your seatbelts as we embark on a journey through the
chessboard, where the synergy of mathematics and strategy creates an
intellectually stimulating experience. Get ready to witness the power of
permutations and combinations in unraveling the mysteries of the game
of chess!

ch

Sa

5. GAME E OR ES >

Title: Understanding Permutations in Chess Openings

Chess, like any other strategic game, begins with a set of initial moves known
as the opening, In this section. we will explore how permutations come into.
play during the opening phase of a chess game.

1. “Piece Arrangements:

- Consider the arrangement of pieces on the board at the start of the game.

The placement of pawns. knights, bishops. rooks. queen. and king involves

various permutations. Each piece has multiple possible squares to occupy.
leading to an exponential number of initial configurations,

2. "Pawn Moves"
‘Analyze the pawn moves during the opening. Pawns can move one or two
squares forward initially, creating different possibilities for the pawn structure.
Understanding these permutations is crucial for developing a strong pawn
structure, which sets the foundation for subsequent moves,

3. “knights and Bishops
- Knights and bishops have specific squares they can move to during the
‘opening. Examining the permutations of these moves allows players to

strategically position their pieces for control and influence over the board.

"Center Control"

The central squares play a vital role in chess, Analyzing the permutations of

‘moves that lead to control over the center helps in formulating a solid opening
strategy,

5. “Development and Coordination:
- Permutations also come into play when considering the development of
leces and thelr coordination. Choosing the order in which pieces are brought

Into the game involves evaluating various possibilities

Understanding the permutations in chess openings is akin to choosing a unique
path at the beginning of a journey. It sets the stage for the middle game, where
combinations become prevalent. In the following pages, we will delve into the
fascinating world of combinations and how they elevate the complexity of
(Chess strategies. Stay tuned as we unravel the interconnected dance of
mathematics and chess on the boardiries of the game of chess!

Sa

6. GAME E (Callbss

Title: Calculating Combinations in Chess Combinations

Now that we have laid the groundwork with permutations in the chess opening.
let's turn our attention to combinations. a concept that takes center stage
during the middle and endgame.

1. "Middle Game complexity"

- As the game progresses into the middle phase. permutations give rise to a
‘multitude of board configurations. Combinations come into play when

assessing the interplay of pieces and their collective impact on the board.

2 “Attacking Combinations:"
Consider combinations that lead to attacks. When multiple pieces.
coordinate to target a specific square or piece, calculating the number of ways
this coordination can occur involves the concept of combinations, This is
crucial for devising effective offensive strategies.

3. “Defensive Combinations “>
- Conversely. defensive combinations are equally important. Analyzing how
pieces can collaborate to defend key squares or pieces requires an
understanding of combinations. The defensive aspect adds an extra layer of
complexity to the game.

4, “Piece Sacrifices
Combinations often involve sacrifices, a strategic maneuver to gain
positional or tactical advantages. Calculating the potential outcomes of
sacrificing a piece requires a keen understanding of combinations and their
Impact on the game dynamics.

5. “Endgame Tactics,

As the game progresses to the endgame, combinations play a pivotal role in

achieving checkmate. Studying combinations in endgames is essential for
‘executing precise and decisive moves that lead to victory.

6. “Calculating Possibilities:
The heart of combinations lies in calculating the number of possible
sequences of moves. This involves evaluating various choices and!

understanding the consequences of each, applying mathematical principles to

the dynamic nature of chess.

‘combinations!

Sa

7. MOVES POSSIBLE

8. LE
PROBABILITY EXAMPLES |

let us consider this configuration of game
phase. Here the best move is to move bishop
to the highlighted square. Or if we consider
total moves dark square bishop can move is 5
out of which 2 leads to losing the bishop.

THE PROBABILITY OF THE MOVES THAT
LEADS TO LOSING BISHOP 15 = 2/5 = 0.4

THE PROBABILITY OF THE MOVES THAT
LEADS TO SURVIVING BISHOP IS = 3/5 = 0.6
THE PROBABILITY OF THE BEST MOVE IS =
Ys = 0.2

let us consider this configuration of endgame
phase. Here there are three ways of doing
forced checkmate to black king. one way is
mate in 3, another is mate is 4, third is mate in 5
THE PROBABILITY OF MATE

THE PROBABILITY OF MATE IN

THE PROBABILITY OF MATE IN 5 = US = 0.2

Now there are 8 white pieces and there is

Bgl ves turn so total number of ways white can

TOTAL KING MOVES = 4

TOTAL PAWN MOVES = 3 ONE STEP MOVE AND 3 TWO STEP MOVE
TOTAL ROOK! (ON E4) MOVES = 7

TOTAL ROOK2 (ON C6) MOVES = 10

TOTAL BISHOP MOVES = 10

TOTAL MOVES = 4+3+3+7+10410 = 37 MOVES
But only one move leads to mate in 3 so probability of that move = 1/37 = 0.027

A professional player will have higher relative probability of finding the

right move.

el I

9. Example 1: Combinations of Capturing Pieces
in Chess

Problem: In a given chess position. a player
has the opportunity to capture 3 out of 5
opponent's pieces. In how many ways can the
player choose which pieces to capture?
Solution: The number of combinations can be
calculated using the formula (n choose 1)

(5 € 3) = SYISNS-3) = (Sx4)/(2%1) = 10

So. there are 10 different ways for the player

to choose 3 out of the 5 opponent's pieces to
© capture

+ You need to decide which opening to
play based on your knowledge of your
‘opponent's likely responses. The order
in which you choose the opening and
your opponent responds adds an
element of permutation and
combination.

Calculation:

+ The number of permutations of
choosing one opening out of three is 31
(factorial). as the order matters.

+ The number of combinations of
choosing one response out of three from
your opponent is (3.C 1)

Example

* Ifyou choose Opening A and your
‘opponent responds with Response Z
that is one specific permutation

+ Ifyou choose Opening B and your
‘opponent responds with Response Y.
that is another specific permutation.

Total Possibilities

+ The total number of possibilities is
Bte(3.C Dese2e 31e

10. P&C EXAMPLES

11. Ee

P&C EXAMPLES

“You are organizing a chess tournament with 12 players (Player 1 to Player 12)
The tournament will have 4 rounds, and each player will play against a unique
‘opponent in each round. Additionally, each player has a repertoire of different
opening strategies they can choose from.
‘Components:
Players:
© There are 12 players participating in the tournament
2, Rounds:
2 The tournament consists of 4 rounds
3.Opening Strategies:
Each player has a repertoire of 3 different opening strategies (Opening A.
Opening 8. Opening C) that they can choose from in each round.
Tasks:
+ You need to create pairings for each round, ensuring that each player faces
a unique opponent in each round
+ Calculate the number of different ways the players can choose their opening
strategies for each round,
+ Determine the probability of a specific player using a specific opening
strategy in a given round.
Calculations:
+ Pairings for Each Round:
2 The number of combinations for choosing 2 players out of 12 is (12 €
2)=121/(21(12-2)1)=66 for each round
© There are 4 rounds, so the total number of pairings is 66x4=264
+ Opening Strategies:
Each player has 3 opening strategies to choose from in each round
2 The number of permutations for choosing 3 strategies out of 3 is 313! for
each player.
© For 12 players, the total number of ways they can choose their
opening strategies is 3!A12=3,110,400
+ Probability:
© Assuming each player is equally likely to choose any opening strategy, the
probability of a specific player using a specific opening strategy is 1/3.
Examples
+ Player 1 chooses Opening A in Round 1. Player 2 chooses Opening B, and so
+ The pairings are unique for each round,
This complex scenario involves permutations and combinations in creating
pairings for a chess tournament, considering different opening strategies for
each player in each round. and calculating the probability of specific choices,

Sa

12. = P&C EXAMPLES E

“You are organizing a chess tournament with 12 players (Player 1 to Player 12)
The tournament will have 4 rounds, and each player will play against a unique
‘opponent in each round. Additionally, each player has a repertoire of different
opening strategies they can choose from.
‘Components:
Players:
© There are 12 players participating in the tournament
2, Rounds:
2 The tournament consists of 4 rounds
3.Opening Strategies:
Each player has a repertoire of 3 different opening strategies (Opening A.
Opening 8. Opening C) that they can choose from in each round.
Tasks:
+ You need to create pairings for each round, ensuring that each player faces
a unique opponent in each round
+ Calculate the number of different ways the players can choose their opening
strategies for each round,
+ Determine the probability of a specific player using a specific opening
strategy in a given round.
Calculations:
+ Pairings for Each Round:
2 The number of combinations for choosing 2 players out of 12 is (12 C
2)=121/21(12-2))*66 for each round.
© There are 4 rounds, so the total number of pairings Is 664-264
+ Opening Strategies:
© Each player has 3 opening strategies to choose from in each round
2 The number of permutations for choosing 3 strategies out of 3 Is 313 for
each player.
© For 12 players, the total number of ways they can choose their opening
strategies is 31112-3,110400
+ Probability:
+ Assuming each player Is equally likely to choose any opening strategy, the
probability of a specific player using a specific opening strategy is 1/3.
Examples:
+ Player 1 chooses Opening A in Round 1, Player 2 chooses Opening B, and so
+ The pairings are unique for each round,
This complex scenario involves permutations and combinations in creating
pairings for a chess tournament. considering different opening strategies for
each player in each round, and calculating the probability of specific choices.

ch

13. Ee

CONCLUSION

In this exploration, we harnessed mathematical
tools such as permutations, combinations, and
probability to unravel the intricate strategies
within chess. From crafting opening moves to
orchestrating tournament pairings, mathematics
became a guiding force, offering strategic insights
and fostering a deeper understanding of the game
The real-life applications showcased the
practicality of these concepts, emphasizing their
role in decision-making and continuous learning.
This journey illuminated the profound intersection
of mathematics and chess, enriching the tapestry
of strategies in this timeless game.