THE CHECKERBOARD CHALLENGE

Akritee Shrestha

Hamilton College ‘13

HRUMC 2013

THE CHECKERBOARD

THE CHECKERBOARD

Ø  The main diagonal of the grid contains no coins

THE CHECKERBOARD

Ø  The main diagonal of the grid contains no coins Ø  The arrangement of the coins is diagonally symmetric

THE CHECKERBOARD

Cover n columns, n < 9 such that an even number of coins remains visible in each row.

THE CHALLENGE

Cover n columns, n < 9 such that an even number of coins remains visible in each row.

THE CHALLENGE

THE SOLUTION?

THE SOLUTION?

THE SOLUTION?

THE SOLUTION?

THE SOLUTION?

THE SOLUTION?

THE SOLUTION?

THE SOLUTION?

THE SOLUTION?

THE SOLUTION!!!

EXISTENCE: DOES A SOLUTION EXIST FOR EVERY

‘CHECKERBOARD’?

CHECKERBOARDS AND MATRICES

CHECKERBOARDS AND MATRICES

0 1 1 0 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 1 0 0

≡

CHECKERBOARDS AND MATRICES

0 1 1 0 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 1 0 0

= A

CHECKERBOARDS AND MATRICES

Aj - jth column of the matrix A

0 1 1 0 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 1 0 0

A =

CHECKERBOARDS AND MATRICES

Aj - jth column of the matrix A

A2 + A4 + A6 + A7 + A8 + A9 =

0 1 1 0 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 1 0 0

A =

244424422

CHECKERBOARDS AND MATRICES

Aj - jth column of the matrix A

A2 + A4 + A6 + A7 + A8 + A9 = or

0 1 1 0 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 1 0 0

A =

A . =

244424422

010101111

244424422

CHECKERBOARDS AND MATRICES

Aj - jth column of the matrix A

A2 + A4 + A6 + A7 + A8 + A9 = or

0 1 1 0 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 1 0 0

A =

A . =

244424422

010101111

244424422

solution matrix

CHECKERBOARDS AND MATRICES

Aj - jth column of the matrix A

A2 + A4 + A6 + A7 + A8 + A9 = or

244424422

A . =

244424422

010101111

CHECKERBOARDS AND MATRICES

A . =

244424422

010101111

WORKING OVER FIELD

is the smallest finite field consisting of two elements 0 and 1. By modular arithmetic, for all integers z z ≡ 0 (mod 2), if z is even z ≡ 1 (mod 2), if z is odd

[ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

[ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

WORKING OVER FIELD

010101111

244424422

A . =

[ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

010101111

244424422

A . =

010101111

000000000

A . = ≡

WORKING OVER FIELD [ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

010101111

244424422

A . =

010101111

000000000

A . = ≡

AX = 0

WORKING OVER FIELD [ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

AX = 0

0 1 1 0 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 1 0 0

A =

WORKING OVER FIELD [ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

AX = 0

0 1 1 0 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 1 0 0

A =

1 1 0 1 0 1 0 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 1 1 1 0 0 0 1 1 1 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0

RREF

(over )

WORKING OVER FIELD [ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

[ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

AX = 0

1 1 0 1 0 1 0 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 1 1 1 0 0 0 1 1 1 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0

x1 = 0, x2 = 1, x3 = 0, x4 = 1, x5 = 0, x6 = 1, x7 = 1, x8 = 1, x9 = 1

x1 x2 x3 x4 x5 x6 x7 x8 x9

000000000

. =

WORKING OVER FIELD [ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

AX = 0

x1 = 0, x2 = 1, x3 = 0, x4 = 1, x5 = 0, x6 = 1, x7 = 1, x8 = 1, x9 = 1

1 1 0 1 0 1 0 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 1 1 1 0 0 0 1 1 1 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0

x1 x2 x3 x4 x5 x6 x7 x8 x9

000000000

. =

WORKING OVER FIELD [ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

AX = 0

x1 = 0, x2 = 1, x3 = 0, x4 = 1, x5 = 0, x6 = 1, x7 = 1, x8 = 1, x9 = 1

1 1 0 1 0 1 0 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 1 1 1 0 0 0 1 1 1 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0

x1 x2 x3 x4 x5 x6 x7 x8 x9

000000000

. =

010101111

X =

WORKING OVER FIELD [ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

EXISTENCE THEOREM

Definition: Let m be an odd number. Over the field , a

checkerboard matrix is an m × m symmetric matrix with diagonal elements equal to 0.

[ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

EXISTENCE THEOREM

Definition: Let m be an odd number. Over the field , a

checkerboard matrix is an m × m symmetric matrix with diagonal elements equal to 0.

Terminology: An elementary product of an m × m matrix

is a product of m elements of the matrix such that the each element in the product is located on a unique row i and a unique column j, where 0 ≤ i, j ≤ m. The set of ordered pairs (i, j) is the corresponding transversal.

[ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

EXISTENCE THEOREM

Definition: Let m be an odd number. Over the field , a

checkerboard matrix is an m × m symmetric matrix with diagonal elements equal to 0.

Terminology: An elementary product of an m × m matrix

is a product of m elements of the matrix such that the each element in the product is located on a unique row i and a unique column j, where 0 ≤ i, j ≤ m. The set of ordered pairs (i, j) is the corresponding transversal.

Theorem: A checkerboard matrix has a non-trivial nullspace.

[ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

PROOF

Let A be an m × m checkerboard matrix. We want to show: det(A) = 0 over .

[ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

PROOF

Let A be an m × m checkerboard matrix. We want to show: det(A) = 0 over .

Det(A) is an alternating sum of elementary products of A.

Since we are working over , +1 = -1.

Det(A) is just an ordinary sum of the elementary products of A.

An m × m matrix has m! elementary products.

[ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

[ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

PROOF

Let A be an m × m checkerboard matrix. We want to show: det(A) = 0 over .

Det(A) is an alternating sum of elementary products of A.

Since we are working over , +1 = -1.

Det(A) is just an ordinary sum of the elementary products of A.

An m × m matrix has m! elementary products. We want to show that the sum of these m! elementary

products is 0.

[ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

[ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

PROOF

We want to show that the sum of these m! elementary products is 0.

PROOF

We want to show that the sum of these m! elementary products is 0.

Case I: If a transversal contains an ordered pair (i, i), i.e it represents a diagonal element, the corresponding elementary product is equal to 0.

PROOF

We want to show that the sum of these m! elementary products is 0.

Case I: If a transversal contains an ordered pair (i, i), i.e it represents a diagonal element, the corresponding elementary product is equal to 0.

Case II: If a transversal represents no diagonal element, it will have a ‘mirror image’ obtained by reflection across the diagonal.

PROOF

We want to show that the sum of these m! elementary products is 0.

Case I: If a transversal contains an ordered pair (i, i), i.e it represents a diagonal element, the corresponding elementary product is equal to 0.

Case II: If a transversal represents no diagonal element, it will have a ‘mirror image’ obtained by reflection across the diagonal.

Since each elementary product consists of an odd number of elements, each transversal is distinct from its mirror image.

PROOF

We want to show that the sum of these m! elementary products is 0.

Case I: If a transversal contains an ordered pair (i, i), i.e it represents a diagonal element, the corresponding elementary product is equal to 0.

Case II: If a transversal represents no diagonal element, it will have a ‘mirror image’ obtained by reflection across the diagonal.

Since each elementary product consists of an odd number of elements, each transversal is distinct from its mirror image.

Since A is symmetric, the elementary products corresponding to the the transversal and its mirror image are equal. Since we are working over , their sum is equal to 0.

[ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

PROOF

We want to show that the sum of these m! elementary products is 0.

Case I: If a transversal contains an ordered pair (i, i), i.e it represents a diagonal element, the corresponding elementary product is equal to 0.

Case II: If a transversal represents no diagonal element, it will have a ‘mirror image’ obtained by reflection across the diagonal.

Since each elementary product consists of an odd number of elements, each transversal is distinct from its mirror image.

Since A is symmetric, the elementary products corresponding to the the transversal and its mirror image are equal. Since we are working over , their sum is equal to 0.

So, the sum of the elementary products is 0.

[ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

PROOF

Let A be an m × m checkerboard matrix. We want to show: det(A) = 0 over .

Det(A) is an alternating sum of elementary products of A.

Since we are working over , +1 = -1.

Det(A) is just an ordinary sum of the elementary products of A.

An m × m matrix has m! elementary products. The sum of these m! elementary products is 0.

QED

[ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

[ ]2

THE CHECKERBOARD CHALLENGEAkritee ShresthaHamilton College

Advisor: Professor Richard Bedient

A standard checkerboard contains a 9 ⇥ 9 grid formed by its vertices. Supposea number of coins are placed on the vertices so that the following two conditionsare satisfied:

1. The diagonal of the grid, which goes from the upper-left to the lower-rightcorner, contains no coins.

2. The arrangement of the coins is diagonally symmetric.

Is it possible to have an even number of coins on each row by covering 8 or fewercolumns?This talk will consider the folllowing checkerboard that satisfies condition (1)and (2) and show that a solution exists.

It will then generalize the problem and prove Theorem 1.

Theorem 1. Let n be an odd number. Let A be a symmetric, n ⇥ n matrix

over the field Z/2Z such that every element on the diagonal is 0. Then there

exists a non-zero vector in the nullspace of A.

Reference

Zulli, Louis. The Incredibly Knotty Checkerboard Challenge. MathematicsMagazine, 1998. 71(5):378-385

1

REFERENCE

L. Zulli, The Incredibly Knotty Checkerboard Challenge, Mathematics Magazine 71(1998), 378-385

ACKNOWLEDGEMENTS

Professor Richard Bedient Hamilton College Mathematics Department Williams College

0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0

0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0

0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0

0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0