# KQNC30 – Mathematical Chess

## Preface

Mathematical chess is a two-player educational and entertaining game played on ten digit pieces ( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and six operator pieces (addition, subtraction, multiplication, division, power, and root). The default chessboard is a grid made up of nine vertical and nine horizontal lines equally separated.

A chess piece can move freely on any horizontal or vertical line provided it is not obstructed by other chess pieces. A move of a chess piece can result in the removal of the opponent’s digit and/or operator pieces.

Each mathematical chess game has three phases:

‘ Select now!’ – selecting the game conditions (level of difficulty)

‘ Prepare now!’ – setting up the initial display of the chess pieces

‘ Play now!’ – playing the game based on the mathematical chess rules.

Mathematical chess is a very flexible game with ten or more levels of difficulty, suitable for all students at primary and high schools and even for mature people. Depending on their ages and/or their years at school, the players can select a level among the existing ones or create new levels suitable for their needs.

Mathematical chess offers many benefits that exist in chess playing (European or Chinese chess) and in mathematical learning.

Like chess, mathematical chess teaches many things, including strategic thinking, problem solving, memory improvement, decision-making, concentration, perseverance, logic, observation, analysis, and organisational skills.

Mathematical chess itself also promotes numerical abilities and mental and speedy calculation.

In mathematical chess, strategies are set up and exercised not only in the ‘Play now!’ phase but also in the ‘Prepare now!’ phase.

Like chess, a mathematical chess game is a competition of two players. The competition fosters interest, promotes mental alertness, challenges all students, and elicits the highest levels of achievement (Stephan 1988).

Mathematical chess is an educational game. A learning environment organised around games has a positive effect on students’ attitudes towards learning. This affective dimension acts as a facilitator of cognitive achievement (Allen and Main 1976).

At the 40th World Chess Congress in 1969, Dr Hans Klaus, dean of the School of Philosophy at Humboldt University in Berlin, commented upon the chess studies completed in Germany:

Chess helps any human being to elaborate exact methods of thinking. It would be particularly useful to start playing Chess from the early days … Everybody prefers to learn something while playing rather than to learn it formally … it produces in our children an improvement in their school achievements. Those children who received systematic instructions in Chess improved their school efficiency in different subjects, in contrast with those who did not receive that kind of instruction.

I think that the statements of Dr Hans Klaus can also apply to mathematical chess.

Dr George Ho

May 2017

## Contents

### Chapter 1: Chess Components

Objectives

Chessboard

Chess Pieces

Digit Pieces

Operator Pieces

### Chapter 2: Initial Chess Display

**Phase 1: ‘Select now!’ Phase **

Step 1: The Four Conditions

Step 2: Levels of Difficulty

Remarks

**Phase 2: ‘Prepare now!’ Phase**

Step 1: Placing the Operator Pieces

Step 2: Placing the Digit Pieces

Remarks

**Phase 3: ‘Play now!’ Phase**

Samples of Initial Chess Displays

Sample 1

Sample 2

Sample 3

### Chapter 3: Chess Definitions

Operator Pair

Digit Pair

Attached Digits

Partial Sums

Partial Differences

Partial Products

Partial Quotients

Partial Powers

Partial Roots

Table of Squares, Cubes, and Roots

Single Attached Digit

### Chapter 4: Mathematical Chess Rules

R01: One Move in a Turn

R02: Power of the Operator Piece

R03– Rule of Two Alone Digit Pieces on a Line

R04 – Rule of the Operator Pair

R05 – Rule of the Digit Pair

R06 – Rule of the Partial Values

R07 – Multiple Removals from the Operator Pieces

R08 – Extended Effect of the Operator Pieces

R09 – Removals from the Digit Pieces

R10 – Multiple Removals from the Digit Pieces

R11 – Extended Effect of the Digit Pieces

R12 – Removals without Moving

The Importance of the Digit Piece 0

a) For the Addition Operator

b) For the Subtraction Operator

c) For the Multiplication Operator

d) For the Division Operator

e) For the Power Operator

f) For the Root Operator

How Does a Mathematical Chess Game Finish?

Flexibility of the Mathematical Chess

### Conclusion

My Dream

### Chapter 1

## Chess Components

### Objectives

Mathematical chess is a two-player entertaining and educational game which can help players develop their ability of thinking ahead, their reasoning and most importantly, their numerical skill.

### Chessboard

The chessboard is a square grid made up of an equal number of vertical lines and horizontal lines equally separated. The default grid of nine vertical lines and nine horizontal lines, as shown in the following figure, will be used throughout this document

### Chess Pieces

There are two types of chess pieces: digit pieces and operator pieces .

### Digit Pieces

There are 20 digit pieces: *0, 1, 2, 3, 4, 5, 6, 7, 8, 9* (10 digit pieces for each player)

### Operator Pieces

There are 48 operator pieces for 6 operators: addition, subtraction, multiplication, division, power, root (24 operator pieces for each player)

The *addition and multiplication* *operators* can deal with a variable number of terms.

The *subtraction* *operator* can deal with only two terms. If the first term is less than the second one, increase it by 10. Only positive differences are acceptable by convention.

The *division operator* can deal with only two terms. If a dividend is not divisible by the divisor, increase it by a smallest multiple of 10 and less than 100 so that an integral quotient can be obtained. Only integral quotients are acceptable by convention.

The *power operator* accepts only square (power of 2) and cube (power of 3).

The *root operator* accepts only square root and cube root . A smallest multiple of 10 can be added to a digit so that an integral square root or cube root less than 10 can be obtained.

### Chapter 2

## Initial Chess Display

At the beginning, each player has a box with two compartments, one containing 24 operator pieces (4 additions, 4 subtractions, 4 multiplications, 4 divisions, 4 powers, and 4 Roots) and the other containing 10 digit pieces (from 0 to 9).

A mathematical chess game consists of three phases.

### Phase 1: ‘Select now!’ Phase

Phase 1 has two steps.

### Step 1: The Four Conditions

Both players must agree on the following four conditions:

**Condition 1:** Digit piece 0 is included or excluded.

**Condition 2:** It’s the number of digit pieces applied to both players (usually six or more).

Note that the values of the digit pieces must be the same in both sides.

**Condition 3:** It’s the type and number of operator pieces applied to both players (usually four or less).

**Condition 4:** There are two ways to compare the values generated by an operator. Select one.

**a) Comparison by last digits.** If two values have the same last digit, they are considered the same.

(modulo 10).

e.g. Two sums 4 + 5 + 8 = 17 and 1 + 6 = 7 are considered the same.

This is the default selection for condition 4 and will be used throughout this document.

**b) Comparison by adding all digits until one-digit sum is ****obtained.**

e.g. The two product values 7 x 5 = 35 and 2 x 13 = 26 are considered the same as 3 + 5 = 8 and 2 + 6 = 8.

The two power values 93 =729 and 63 = 216 are considered the same as

7 + 2 + 9 = 18 => 1 + 8 = 9 and 2 + 1 + 6 = 9

### Step 2: Levels of Difficulty

The selection of the types and number of operators determines the level of difficulty of the game.

The ten basic levels of difficulty are:

**Level 1** – addition only, up to four addition operators (+)

**Level 2** – subtraction only, up to four subtraction operators (–)

**Level 3 –** addition and subtraction, up to two addition operators (+)

and up to two subtraction operators (–)

**Level 4 –** multiplication only, up to four multiplication operators (x)

**Level 5 –** division only, up to four division operators (÷)

**Level 6 –** multiplication and division only, up to two multiplication operators (x)

and up to two division operators (÷)

**Level 7 –** power only, up to four power operators (P)

**Level 8 –** root only, up to four root operators (R)

**Level 9 –** one addition operator (+), one subtraction operator (–)

one multiplication operator (x), one division operator (÷)

**Level 10 –** one multiplication operator (x)

one division operator (÷), one power operator (P), one root operator (R)

**Remarks**

a) The players are free to create new levels of difficulty for the game.

b) Based on the level of difficulty selected, the dimension of the chessboard can be set accordingly. For example:

levels: 1, 2, 4, 5, 7, 8 chessboard: 6 x 6

levels: 3, 6 chessboard: 8 x 8

levels: 9, 10 chessboard: 9 x 9, 10 x 10

The default dimension of the chessboard is 9 x 9.

### Phase 2: ‘Prepare now!’ Phase

In this phase, the players place their digit and operator pieces on the chessboard.

This activity has two steps.

### Step 1: Placing the Operator Pieces

The players, in turn, place their operator pieces, one each time, on the chessboard.

### Step 2: Placing the Digit Pieces

The players, in turn, place their digit pieces, one each time, on the chessboard.

The player who started step 1 also starts step 2.

#### Remarks

a) The players determine the locations of their chess pieces on the chessboard.

b) The player who started phase 2 will also start the subsequent phase 3: ‘Play now!’.

c) The initial locations of the chess pieces already placed on the chessboard cannot be changed.

d) The chess rules in subsequent sections can help the players locate wisely their chess pieces on the board.

e) Players are free to create new rules for placing operator pieces and digit pieces on the chessboard and have

an agreement on who will start phase 3.

f) A supervisor (e.g. teacher, parent, friend) can help place the chess pieces in phase 2 for the players.

### Phase 3: ‘Play now!’ Phase

The players, in turn, can move one of their chess pieces (digit or operator piece) in any direction along a horizontal or vertical line.

#### Remark 1

Only one move for each turn is allowed. A move can result in the removal of one or many of the opponent’s chess pieces

(see the rules in subsequent sections).

#### Remark 2

If a favourable display of the chess pieces exists, the player of the turn can remove one or many of the opponent’s chess pieces without moving any of his or her chess pieces.

### Samples of Initial Chess Displays

### Sample 1

Below is a sample of the initial display (after the ‘Prepare now!’ phase) of a chess game with the following selected conditions:

Condition 1: excluding digit piece 0

Condition 2: number of digit pieces for each player = seven

Condition 3: number of operator pieces for each player = three

two addition operators (+), one subtraction operator (–)

Condition 4: comparison by last digits (default)

The initial display of the chess game after the ‘Prepare now!’ phase is as follows:

### Sample 2

Below is a sample of the initial display (after the ‘Prepare now!’ phase) of a chess game with the following selected conditions:

Condition 1: excluding digit piece 0

Condition 2: number of digit pieces for each player = nine

Condition 3: number of operator pieces for each player = four

four multiplication operators (x)

Condition 4: comparison by last digits (default)

The initial display of the chess game after the ‘Prepare now!’ phase is as follows:

### Sample 3

Below is a sample of the initial display (after the ‘Prepare now!’ phase) of a chess game with the following selected conditions:

Condition 1: including digit piece 0

Condition 2: number of digit pieces for each player = ten

Condition 3: number of operator pieces for each player = four

one multiplication operators (x) one division operator (÷)

one power operator (P) one root operator (R)

Condition 4: comparison by last digits (default)

The Initial display of the chess game after the ‘Prepare Now!’ phase is as follows:

### Chapter 3

## Chess Definitions

The following terms will be used to explain the mathematical chess rules.

### Operator Pair

Two operator pieces of the same player form an operator pair when they are on the same line (horizontal or vertical) and not separated by any digit or operator piece of any player.

Examples

### Digit Pair

Two digit pieces of the same player form a digit pair when they are on the same line (horizontal or vertical) and not separated by any digit or operator piece of any player.

Examples

### Attached Digits

A digit piece is attached to an operator piece of any player when it is on the same line of the operator piece and not separated by any other operator piece.

Digits 2, 5, and 4 are attached to the white addition operator.

3 is not attached to the white addition operator.

Digits 3, 2, and 5 are attached to the black subtraction operator.

4 is not attached to the black subtraction operator.

### Partial Sums

A partial sum of an addition operator ( + ) is the sum of any number of digits among the ones attached to the same side of the operator.

A partial sum of an addition operator is also called a partial value of the operator.

Example

The digits attached to the left side of the addition operator (+) are 3, 2 and 9.

The left partial sums generated by 3, 2 and 9 are:

3 + 2 + 9 = 14, 3 + 2 = 5, 3 + 9 = 12, 2 + 9 = 11

The left partial values of the addition operator are 14, 5, 12, and 11.

Notes

a) The attached digits 3, 2, and 9 are not the partial sums of the operator by themselves. .

b) The attached digits 1 and 5 of the addition operator generate the right partial sum: 1 + 5 = 6.

### Partial Differences

A partial difference of a subtraction operator (–) is the difference of any pair of digits among the ones attached to the same side of the operator.

Each pair of digits generates two partial differences obtained by swapping the digits. If the first digit is less than the second one, add ten to it.

A partial difference of a subtraction operator is also called a partial value of the operator.

Example

The left attached digits of the subtraction operator (–) are 3, 5, and 4. The left partial differences generated by 3, 5, and 4 are:

(10 + 3) – 5 = 8, 5 – 3 = 2

(10 + 3) – 4 = 9, 4 – 3 = 1

5 – 4 = 1, (10 + 4) – 5 = 9

The left partial values of the subtraction operator are 8, 2, 9, and 1.

Notes

a) The attached digits 3, 5, 4 are not the partial differences of the subtraction operator by themselves.

b) The right attached digits 6 and 7 of the subtraction operator generate the right partial differences:

(10 + 6) – 7 = 9 and 7 – 6 = 1.

### Partial Products

A partial product of a multiplication operator (x) is the product of any number (≥ 2) of digits among the ones attached to

the same side of the operator.

A partial product of a multiplication operator is also called a partial value of the operator.

Example

The left attached digits of the multiplication operator are 2, 3, and 7.

The left partial products generated by the digits 2, 3, and 7 are:

2 x 3 x 7 = 42, 2 x 3 = 6, 2 x 7 = 14, 3 x 7 = 21

The left partial values of the multiplication operator are 42, 6, 14, and 21.

Notes

a) The attached digits 2, 3, and 7 of the multiplication operator are not the partial products of the operator by themselves.

b) The right attached digits 6 and 8 of the multiplication operator generate the right partial product: 6 x 8 = 48.

### Partial Quotients

A partial quotient of a division operator (÷) is the integral quotient of any pair of digits among the ones attached to the same side of the operator.

Each pair of digits can generate 0, 1, or 2 partial quotients by swapping the digits. A smallest multiple of 10 and less than 100 can be added to the first digit so that an integral quotient can be obtained.

e.g. The pair of attached digits 4 and 8 generates two partial quotients, 3 and 2.

(20 + 4) ÷ 8 = 3 8 ÷ 4 = 2

The pair of attached digits 5 and 7 generates one partial ouotient 5.

(30 + 5) ÷ 7 = 5

The pair of attached digits 9 and 7 generates two partial quotients 7 and 3.

(40 + 9) ÷ 7 = 7 (20 + 7) ÷ 9 = 3

The pair of attached digits 2 and 5 generates zero partial quotient.

A partial quotient of a division operator is also called a partial value of the operator.

Example

The left attached digits of the division operator are 2, 4, and 7.

The left partial quotients generated by 2, 4, and 7 are:

(10 + 2) ÷ 4 = 3 4 ÷ 2 = 2

(40 + 2) ÷ 7 = 6 (10 + 4) ÷ 7 = 2

The left partial values of the division operator are 3, 2, and 6.

Notes

a) The attached digits 2, 4, and 7 of the division operator are not the left partial quotients of the operator by themselves.

b) The right attached digits 9 and 3 of the division operator generate the two right partial quotients:

9 ÷ 3 = 3 (60 + 3) ÷ 9 = 7

### Partial Powers

A partial power of a power operator (P) is the square or cube of any digit among the ones attached to the same side of the operator.

A partial power of a power operator is also called a partial value of the operator.

Example:

The left attached digits of the power operator are 4, 6, and 7.

The left partial powers generated by 4, 6, and 7 are:

4^{2} = 16 6^{2} = 36 7^{2} = 49

4^{3} = 64 6^{3} = 216 7^{3} = 343

The left partial values of the power operator are 16, 36, 49, 64, 216, and 343.

Notes

a) The attached digits 4, 6, and 7 of the power operator are not the partial powers of the operator by themselves.

b) The right attached digits 9 and 8 of the power operator generate the right partial powers:

9^{2} = 81 8^{2} = 64 9^{3} = 729 8^{3} = 512

### Partial Roots

A partial root of a Root Operator (R) is the integral square root or cube root of any digit among the ones attached to the same side of the operator. A smallest multiple of

ten can be added to the digit so that an integral root less than ten can be obtained.

A partial root of a root operator is also called a partial value of the operator.

An attached digit to a root operator can generate one or two partial roots.

Example

The left attached digits of the root operator are 2, 4, and 7.

The left partial roots generated by 2, 4, and 7 are:

Notes

a) The attached digits 2, 4, and 7 of the root operator are not the partial roots of the operator by themselves.

b) The right attached digits 9 and 3 of the root operator generate the right partial roots:

### Table of Squares, Cubes, and Roots

### Single Attached Digit

A single attached digit of the operators addition, subtraction, multiplication and division is considered as a partial value of these operators.

A single attached digit of an addition/subtraction/multiplication/division operator is a partial sum/difference/product/division of this operator.

A single attached digit of a power /root operator is not a partial power/root of this operator.

Examples

### Chapter 4

## Mathematical Chess Rules

The mathematical chess games are played based on the following rules:

### R01: One Move in a Turn

The players, in turn, can move their chess pieces on the chessboard. There is only one move in a turn. A chess piece can move freely along a vertical or horizontal line as long as the move is not obstructed by other chess pieces.

The move is not mandatory. The player can remove the opponent’s chess pieces without moving any of his or her chess piece.

### R02: Power of the Operator Piece

An operator piece, when moving along a line, can remove (or replace) the first opponent’s digit piece or operator piece it encounters provided they are different.

Examples

a) On the first horizontal line, the white addition operator can replace the black subtraction operator. After this move, the black addition operator cannot remove the white addition operator, but it can replace the white digit 2.

b) On the eighth vertical line, the black multiplication operator can replace the white addition operator. Note that it cannot continue removing the white subtraction operator (Rule 1). On the contrary, the white subtraction operator can replace the black multiplication operator.

c) On the second vertical line, the black addition operator cannot remove the white addition operator, but it can remove the white digit 4.

d) On the fourth horizontal line, the white multiplication operator cannot remove the black multiplication operator, but it can remove the black digit 2 on the vertical line 6.

e) On the fourth vertical line, the black subtraction operator can replace the white digit 5. After this move, the white multiplication operator can remove it.

### R03: Rule of Two Alone Digit Pieces on a Line.

When on a line (horizontal or vertical), there are only two digit pieces of different players. One digit can remove (or replace) the other provided they are different.

Example

The white digit 1 can remove the black digit 8 and vice versa.

The white digit 7 cannot remove the black digit 7 and vice versa.

### R04: Rule of the Operator Pair.

When on a line (horizontal or vertical), there is an operator pair, then one operator of the pair can jump over its partner and remove any opponent’s first digit or operator piece situated on the other side. Note that the operator piece to be removed, if any, must be different from the jumping operator.

Examples

a) On the first line, the white addition and subtraction operators form an operator

pair. The white addition operator can jump over the white subtraction operator

and remove the black subtraction operator

b) On the third line, the black addition and subtraction operators form an operator

pair. The black addition operator can jump over the black subtraction operator

and remove the white subtraction operator.

c) On the fourth line, the white multiplication and addition operators form an operator

pair. The white multiplication operator cannot jump over the white addition operator to remove the same black multiplication operator but the white addition operator can remove the black multiplication operator.

d) On the second line, the black multiplication and addition operators form an operator

pair. If the multiplication operator jumps over the addition operator and remove the white digit 6, it will be removed later by the white addition operator on line 4. The best way is removing the white digit 6 by the black addition operator.

### R05: Rule of the Digit Pair

When on a line (horizontal or vertical), there is a digit pair, then one digit of the pair can jump over its partner and remove any opponent’s first digit or operator piece situated on the other side.

Note that the digit piece to be removed, if any, must be different from the jumping digit.

Examples

a) On the first line, the white digit pieces 1 and 9 form a digit pair.

The white digit piece 1 can jump over the white digit pPiece 9 to remove the black subtraction operator.

b) On the second line, the black digit pieces 3 and 7 form a digit pair.

The black digit piece 7 cannot jump over the black digit piece 3 to remove the white digit piece 7 because they have the same value.

On the contrary, the white digit piece 2 can jump over the white digit piece 7 and remove the black digitpPiece 3.

c) On the third line, the black digit pieces 5 and 8 form a digit pair.

The black digit piece 5 can jump over the black digit piece 8 to remove the white subtraction operator..

d) On the fourth line, the white digit pieces 3 and 6 form a digit pair.

The white digit piece 6 can jump over the white digit piece 3 to remove the black multiplication operator or the white digit piece 3 can jump over the white digit

piece 6 to remove the black digit piece 4.

### R06: Rule of the Partial Values

If both sides of an operator piece have partial values with the same last digit , then the operator can remove the opponent’s digits in these two partial values.

Note 1: A single attached digit of an operator other than power and root is also its partial value.

Note 2: Partial value is a common name of all partial sum/difference/product/quotient/power/root.

Example 1

The arrow indicates the move direction of the chess piece.

The left partial sums of the white addition operator (+) are:

1 + 6 + 9 = 16, 1 + 6 = 7 , 1 + 9 = 10, and 6 + 9 = 15

The right partial sum of the white addition operator is 5.

The left partial value, 6 + 9 = 15, and the right partial value, 5, of the white addition operator have the same last digit, 5. Note that partial sum of an addition operator is also called partial value.

=> The white addition operator can remove the opponent’s black digits 9 and 5 in

these two partial values.

Example 2

The left partial sums of the black addition operator (+) are:

2 + 6 + 1 = 9 2 + 6 = 8 2 + 1 = 3 and 6 + 1 = 7

The right partial sum of the black addition aperator (+) is 5 + 3 = 8

The left partial value, 2 + 6 = 8, and the right partial value, 5 + 3 = 8, of the black addition operator have the same last digit, 8.

=> The black addition operator can remove the opponent’s white digits 2, 5, and 3 in these two partial values.

Example 3

The left partial difference of the black subtraction operator (–) is 6.

The right partial differences of the black subtraction operator (–) are:

(10 + 5) – 9 = 6 9 – 5 = 4

The left partial value, 6, and the right partial value, (10 + 5) – 9 = 6, of the black subtraction operator have the same last digit, 6.

=> The black subtraction operator can remove the opponent’s white digits 5 and 3 in these two partial values.

Example 4

The left partial products of the black multiplication operator (x) are:

2 x 8 x 7 = 112 2 x 8 = 16 2 x 7 = 14 8 x 7 = 56

The right partial product of the black multiplication operator (x) is:

6 x 9 = 54

The left partial value, 2 x 7 = 14, and the right partial value, 6 x 9 = 54, of the black multiplication operator have the same last digit, 4.

=> The black multiplication operator can remove the opponent’s white digits 7 and 9 in |

these two partial values.

Example 5

A move of the black addition operator (+) can remove the white digit 6.

A move of the white multiplication operator (x) can remove the black digits 3 and 9, as the left and right partial products of the operator are 7 and 3 x 9 = 27 respectively.

Example 6

The left partial quotients of the white division operator (÷) are:

8 ÷ 2 = 4, (30 + 2) ÷ 8 = 4, (10 + 8) ÷ 3 = 6, (10 + 2) ÷ 3 = 4.

The right partial quotient of the white division operator (÷) is:

(50 + 4) ÷ 9 = 6

The left partial value, (10 + 8) ÷ 3 = 6, and the right partial value, (50 + 4) ÷ 9 = 6, of the white division operator have the same last digit, 6.

=> The white division operator can remove the opponent’s black digits 3 and 4 in these two partial values.

Example 7

The left partial powers of the white power operator (P) are:

2^{2} = 4, 2^{3} = 8, 3^{2} = 9, 3^{3} = 27

The right partial powers of the white power operator (P) are:

4^{2} = 16, 4^{3} = 64, 6^{2} = 36, 6^{3} = 216

The left partial value, 2^{2} = 4, and the right partial value, 4^{3} = 64, of the white power operator, have the same last digit, 4.

=> The white power operator can remove the opponent’s black digits 2 and 4 in these two partial values.

Example 8

A move of the black power operator (P) can remove the white digit 2, as 2^{2} = 4 and 8^{2} = 64 are the left and right partial values of the operator respectively.

A move of the white power operator (P) can remove the black digit 3, as 9^{3} = 729 and 3^{2} = 9 are the left and right partial values of the operator respectively.

Example 9

Example 10

### R07: Multiple Removals from the Operator Pieces

An operator can have multiple pairs of partial values of same last digist and in different sides. All these pairs can contribute to the removal of the opponent’s digits.

Example 1

The arrow indicates the move direction of the chess piece.

The left partial sums of the white addition operator (+) are:

1 + 2 + 5 = 8, 1 + 2 = 3, 1 + 5 = 6, 2 + 5 = 7

The right partial sums of the addition operator are:

3 + 4 + 6 = 13, 3 + 4 = 7, 3 + 6 = 9, 4 + 6 = 10

=> The pair of partial sums 1 + 2 = 3 and 3 + 4 + 6 = 13 results in the removal of

the black digits 1 and 6.

The pair of Partial Sums 2 + 5 = 7 and 3 + 4 = 7 results in the removal of the

black digit 5.

Example 2

The left partial powers of the white power operator (P) are:

2^{2} = 4, 2^{3} = 8, 3^{2} = 9, 3^{3} = 27

The right partial powers of the white power operator are:

4^{2} = 16, 4^{3} = 64, 9^{2} = 81, 9^{2} = 729

=> The pair of partial powers 2^{2} = 4 and 4^{3} = 64 results in the removal of the black digits 2 and 4.

The pair of partial powers 3^{2} = 9 and 9^{3} = 729 results in the removal of the black digit 3.

## R08: Extended Effect of the Operator Pieces

The effect of an operator piece remains after it removes the opponent’s digits, i.e. the operator piece can continue removing the opponent’s digits if it satisfies other conditions.

An operator piece can deny removing any opponent’s digit if this can help its subsequent removals.

Example 1

The left partial sums of the white addition operator (+) are:

1 + 8 + 7 = 16, 1 + 8 = 9, 1 + 7 = 8, 8 + 7 = 15

The right partial sums of the white addition operator (+) are:

4 + 8 + 2 = 14, 4 + 8 = 12, 4 + 2 = 6, 8 + 2 = 10

=> The pair of partial sums 1 + 8 + 7 = 16 and 4 + 2 = 6 results in the removal of the

black digits 1, 7, 4, and 2.

After these removals, the remaining chess pieces on the line are as follows:

The left and right partial sums of the white addition operator (+), 8 and 8, result in the removal of the black digit 8.

Example 2

The left and right partial sums of the white addition operator (+), 6 + 9 = 15 and 5, can result in the removal of the black digits 9 and 5.

After these removals, the remaining chess pieces on the line are as follows:

The white addition operator cannot remove the black subtraction operator because only one move is allowed in a turn. On the contrary, the black subtraction operator can remove the white addition operator. To avoid this case, the white addition operator should not remove the black digit 5.

Example 3

The left partial differences of the white subtraction operator (–) are:

(10 + 5) – 7 = 8, 7 – 5 = 2

The right partial differences of the white subtraction operator are:

2 – 1 = 1, (10 + 1) – 2 = 9, (10 + 2) – 9 = 3, 9 – 2 = 7,

(10 + 1) – 9 = 2, 9 – 1 = 8

=> The pair of partial differences, (10 + 5) – 7 = 8 and 9 – 1 = 8 (also the pair of partial differences 7 – 5 = 2 and (10 + 1) – 9 = 2), can result in the removal of the black digits 7, 1, and 9. Note that the black digit 2 remains.

To also remove the black digit 2, the removal of the black digit 7 should be denied. The remaining chess pieces on the line are as follows:

The left and right partial differences of the white subtraction operator (–), 7 – 5 = 2 and 2, can result in the removal of the black digits 7, and 2.

Example 4

The left and right partial sums of the white addition operator (+) are:

1 + 3 = 4 and 4

=> The black digits, 1 and 4, can be removed. However, the removal of black digit 4 should be denied; otherwise, the black subtraction operator can jump over its partner (the black addition operator) to remove the white addition operator.

### R09: Removals from the Digit Pieces

A move of a digit piece to a location can generate a pair of partial values with the same last digits on both sides of an operator piece (addition, subtraction, multiplication, division, power and root). If this case happens, the owner of the digit piece can remove, among the digit pieces of the partial values, any piece owned by the opponent, including the operator piece.

Example 1

The move of the white digit 4 to the location indicated by the arrow creates a right attached digit to the black addition operator (+).

The left and right partial sums of the black addition operator (+) are 5 + 9 = 14 and 4.

The move of the white digit 4 generates a pair of partial values with the same last digit, 4, on both sides of the black addition operator.

=> The white digit 4 can remove the black digit 5 and the black addition operator.

Example 2

The move of the black digit 9 to the location indicated by the arrow creates a new left attached digit to the white multiplication operator (x).

The two partial products of the white multiplication operator are 9 x 4 = 36 and 3 x 2 = 6.

The move of the black digit 9 generates a pair of partial values with the same last digit, 6, on both sides of the white multiplication operator.

=> The black digit 9 can remove the white digits 4, and 3, and the white multiplication operator.

Example 3

The move of the white digit 7 to the location indicated by the arrow creates a new right attached digit to the black division operator (÷) .

The left partial quotient of the black division operator is:

(50 + 4) ÷ 9 = 6

The right partial quotients of the black division operator are:

(30 + 5) ÷ 7 = 5, (40 + 2) ÷ 7 = 6

The two partial quotients of the black division operator, (50 + 4) ÷ 9 = 6 and (40 + 2) ÷ 7 = 6, have the same last digit, 6.

The move of the white digit 7 generates a pair of partial values with the same last digit, 6, on both sides of the black division operator.

=> The white digit 7 can remove the black digits 9, and 2, and the black division operator.

Example 4

The move of the white digit 3 to the location indicated by the arrow creates a new right attached digit to the black power operator.

The two partial powers of the black power operator , 9^{3} = 729 and 3^{2} = 9, have the same last digit, 9.

The move of the white digit 3 generates a pair of partial values with the same last digit, 9, on both sides of the black power operator.

=> The white digit 3 can remove the black digit 9 and the black power operator.

Example 5

The move of the black digit 9 to the location indicated by the arrow creates a new left attached digit to the white root operator (R).

The two partial roots of the white root operator,

have the same last digit, 3.

The move of the black digit 9 generates a pair of partial values with the same last digit, 3, on both sides of the white root operator.

=> The black digit 9 can remove the white digit 7 and the white root operator.

### R10: Multiple Removals from the Digit Pieces

A move of a digit piece can result in many pairs of partial values having the same last digits and in different sides of the operator pieces. All these pairs can contribute to the removal of the opponent’s digit and operator pieces.

Example 1

The move of the black digit 3 to the location indicated by the arrow, creates two pairs of partial powers having the same last digits and in different sides of the white power operator (P):

2^{2} = 4 4^{3} = 64 3^{2} = 9 9^{3} = 729

The move of the black digit 3 generates two pairs of partial values with the same last digit, 4 and 9, on both sides of the white power operator.

=> The black digit 3 can remove the white digits 2 and 9, and and the white power

operator.

Example 2

The move of the white digit 6 to the location indicated by the arrow creates a pair of partial sums and a pair of partial differences having the same last digits and in different sides of the black addition and the black subtraction operators successively.

8 2 + 6 = 8 for the black addition operator (+)

6 – 2 = 4 9 – 5 = 4 for the black subtraction operator (–)

The move of the white digit 6 generates one pair of partial values with the same last digit, 8, on both sides of the black addition operator, and one pair of partial values with the same last digit, 4, on both sides of the black subtraction operator.

=> The move of the white digit 6 can remove the black digit 8 and the black addition operator in one side and the black digit 5 and the black subtraction operator in the other side.

### R11: Extended Effect of the Digit Pieces

The effect of a digit piece remains after it removes the opponent’s digit and operator pieces – i.e. the digit piece can continue removing the opponent’s digit and operator pieces if it satisfies other conditions.

A digit piece can deny removing any opponent’s digit or operator piece if this can help

its subsequent removals.

Example 1

The move of the black digit 2 to the location indicated by the arrow creates a new left partial root for the white root operator (R).

Two partial roots of the white root operator,

have the same last digit, 8.

The move of the black digit 2 generates a pair of partial values with the same last digit, 8, on both sides of the white root operator.

=> The black digit 2 can remove the white digit 4 and the white root operator.

After these removals, the chess pieces left on the line are:

Two partial sums of the white addition operator, 5 + 2 = 7 and 7, have the same last digit, 7.

=> The move of the black digit 2 can remove the white digit 5 and the white addition operator.

Example 2

The move of the white digit 7 to the location indicated by the arrow, creates two partial products of the black multiplication operator (x) that have the same last digit, 6.

8 x 7 = 56 and 2 x 3 x 6 = 36

=> Instead of removing the black digits 8, 2, 3, 6, and the black multiplication operator, the white digit 7 removes only the black digits 8, 3, 6, and the black multiplication operator.

After these removals, the chess pieces left on the line are:

Two partial sums of the black addition operator, 7 + 2 = 9 and 9, have the same last digit, 9.

=> The move of the white digit 7 can remove the black digits, 2 and 9, and the black addition operator.

### R12: Removals without Moving

A player can remove the opponent’s chess pieces without moving any of his or her digit or operator pieces.

Example 1

A move of the white addition operator (+) to the location indicated by the arrow can remove the black digits 9, 4, and 1, as the left and right partial sums of the operator are

6 + 9 = 15 and 4 + 1 = 5.

After these removals, the chess pieces that will remain on the line are as follows:

After these removals, without moving any black chess piece, the black digit piece 7 can remove the white digit 7 and the white subtraction operator (–).

Example 2

The move of the white addition operator to the location indicated by the arrow results in the removals of the black digits 8, 4, 6, and 3, as 8 + 7 + 4 = 19 and 6 + 3 = 9 have the same last digit, 9.

After these removals, the chess pieces that will remain on the line are as follows:

Without moving any chess piece, the black digit 7 can remove the white multiplication operator and the white digit 7.

After this move, the white addition operator can finally remove the black digit 7.

### The Importance of the Digit Piece 0

The presence of the digit 0 in the attached digits of an operator entails the following properties:

### a) For the Addition Operator

In multiple attached digits, any attached digit, except 0, is also a partial sum of the addition operator.

Example

The left partial sums of the white addition operator are 3, 12, and 9.

Note that the single right attached digit 0 is the right partial sum of the operator.

### b) For the Subtraction Operator

In multiple attached digits, any attached digit, except 0 and its complement with 10, is a partial difference of the subtraction operator.

Example

The left partial differences of the white subtraction operator are as follows:

3 – 0 = 3 (10 + 0) – 3 = 7

(10 + 3) – 9 = 4 9 – 3 = 6

(10 + 0) – 9 = 1 9 – 0 = 9

Note that the single right attached digit 0 is the right partial difference of the operator.

### c) For the Multiplication Operator

Zero (0) is a partial product, but it also decreases the number of partial products of the operator.

Example

The left partial products of the black multiplication operator are 0 and 27.

3 x 0 = 0 3 x 9 = 27 0 x 9 = 0

The right partial product of the operator is 0 x 5 = 0.

The black multiplication operator can remove the white digit pieces 3, 0, and 5.

### d) For the Division Operator

Zero (0) is a partial quotient, but it also decreases the number of partial quotients of the operator. Note that dividing a number by 0 is not acceptable.

Example

The left partial quotients of the black division operator are:

0 ÷ 6 = 0 0 ÷ 8 = 0 (10 + 6) ÷ 8 = 2 (10 + 8) ÷ 6 = 3

The right partial quotient of the black division operator is 0 ÷ 5 = 0

The black division operator can remove the white digit pieces 0, 6, and 5.

### e) For the Power Operator

Zero (0) is a partial power of the operator because 0^{2} = 0, 0^{3} = 0.

Example

The left partial powers of the white power operator are 0, 3^{2} = 9, and 3^{3} = 27.

The right partial powers of the white power operator are 0, 5^{2} = 25, and 5^{3} = 125.

The white power operator can remove the black digit piece 0.

### f) For the Root Operator

Zero (0) is a partial root of the root operator.

Example

### How Does a Mathematical Chess Game Finish?

A mathematical chess game finishes in one of the following cases:

**Case 1:** When a player loses all his or her operator pieces or digit pieces,

the other player is the winner. If this case happens to both players at the same time, the game finishes in a draw.

**Case 2**: The chess game can also stop in one of the following cases:

a) the time limit of the game (30 minutes by default) has passed.

b) the agreement of both players.

c) the decision of the umpire (e.g. teacher, parent).

If necessary, the ranking can be based on the number of chess pieces the player has taken from his or her opponent.

The scores are 2 points for an operator piece and 1 point for a digit piece. The player who has the higher total score, wins the game.

If both players have the same total score, the game finishes in a draw.

### Flexibility of the Mathematical Chess

Mathematical chess offers many flexible choices to make it compatible with the level of knowledge of the players, such as the following:

1) free to choose the dimension of the chessboard

2) free to select the conditions of the game:

a) inclusion or exclusion of the digit piece 0

b) number of digit pieces

c) number and values of operator pieces

3) free to create new levels of difficulty for the game

4) free to set up a time limit for the game.

### Conclusion

### My Dream

With the invention of mathematical chess, its creator aims at providing a means – a weapon – to separate young people at all levels from their deep passion for electronic games, which has damaged their minds with hatred of mathematics.

To be successful, mathematical chess needs initial support from people of educational organisations at all levels, from universities to middle and high schools, in such activities as:

1) distributing the book and explaining essential rules of the mathematical chess

2) organising competitions between students in a class and between students in different classes, different schools …

3) organising yearly competitions between schools in states.

I hope that, eventually, mathematical chess can fulfil its objectives. My dream is to see mathematical chess become a popular game in all middle and high schools, and to see the love and passion of mathematics flowering in the minds of young people around the world.