Eigenvalue Formulas for Matrices: Case Studies | Papers Computer Science

W. McCague 4/16/96 1

INTRODUCTION

The problem to be discussed originally stemmed from an example of a dynamic

programming problem called the nine-tails problem [Heck]. In the nine-tails problem, a

3x3 matrix of coins is given. Each coin has an initial state, heads or tails. When a player

flips a particular coin, each of its neighbors, North, South, East, and West (if they exist),

are also flipped. Heck’s purpose was to provide a simplified example to introduce

dynamic programming. This problem has extended in several directions like considering N

coins in a straight line [Klostermeyer et. al.] or see [Delahan et. al. #2,#3, Goldwasser].

This report deals with yet another variation of coin flipping, 2-dimensional

rectangular arrays and 3-dimensional cubic arrays of coins. In defining a coin’s neighbors,

wrap-around is included. Wrap-around means every coin in the 2-dimensional arrangement

will have exactly 4 neighbors (North, South, East, and West) and every coin in the 3-

dimensional arrangement will have exactly 6 neighbors (includes coins above and below).

Each coin has 2 states (heads and tails) and each coin is given some initial state.

Pointing is defined as selecting a coin causing it and all of its neighbors to be

flipped to the opposite state (heads to tails, or tails to heads). The goal is to determine

which coins to point to in order to have all coins show tails. Pointing to a coin more than

once is not necessary because pointing to a coin a second time resets the states of the coin

and its neighbors. A first question is whether an initial configuration exists for which the

problem is unsolvable. This paper characterizes which sizes of 2-dimensional arrays and

3-dimensional arrays have initial non-solvable configurations. Non-solvable, in this paper,

means that there is not a sequence of pointing that will allow all coins to show tails.

1234

5678

9 101112

13 14 15 16

Figure 1: Illustration of 2-D Coin Layout

Figure 1 illustrates how to number the coins in a 4x4 grid. The neighbors of

horizontal edge coins (5,9,8,12) and the corner coins (1,4,13,16) may not be the expected

neighbors because the visual representation of the coin arrangement is a Klein bottle. For

the Klein bottle, the wrap-around is defined as follows. Coins on the top edge (e.g. 2)

have the normal east (3), west (1), and south neighbors (6), with the north neighbor (14)

being the bottom-most coin in the column of the coin which was pointed to. Coins on the

bottom edge (e.g. 15) have the normal east (16), west (14), and north neighbors (11), with

the south neighbor (3) being the top-most coin of the same column. Coins on the left edge

Partial preview of the text

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INTRODUCTION

The problem to be discussed originally stemmed from an example of a dynamic

programming problem called the nine-tails problem [Heck]. In the nine-tails problem, a

3x3 matrix of coins is given. Each coin has an initial state, heads or tails. When a player

flips a particular coin, each of its neighbors, North, South, East, and West (if they exist),

are also flipped. Heck’s purpose was to provide a simplified example to introduce

dynamic programming. This problem has extended in several directions like considering N

coins in a straight line [Klostermeyer et. al.] or see [Delahan et. al. #2,#3, Goldwasser].

This report deals with yet another variation of coin flipping, 2-dimensional

rectangular arrays and 3-dimensional cubic arrays of coins. In defining a coin’s neighbors,

wrap-around is included. Wrap-around means every coin in the 2-dimensional arrangement

will have exactly 4 neighbors (North, South, East, and West) and every coin in the 3-

dimensional arrangement will have exactly 6 neighbors (includes coins above and below).

Each coin has 2 states (heads and tails) and each coin is given some initial state.

Pointing is defined as selecting a coin causing it and all of its neighbors to be

flipped to the opposite state (heads to tails, or tails to heads). The goal is to determine

which coins to point to in order to have all coins show tails. Pointing to a coin more than

once is not necessary because pointing to a coin a second time resets the states of the coin

and its neighbors. A first question is whether an initial configuration exists for which the

problem is unsolvable. This paper characterizes which sizes of 2-dimensional arrays and

3-dimensional arrays have initial non-solvable configurations. Non-solvable, in this paper,

means that there is not a sequence of pointing that will allow all coins to show tails.

Figure 1: Illustration of 2-D Coin Layout

Figure 1 illustrates how to number the coins in a 4x4 grid. The neighbors of

horizontal edge coins (5,9,8,12) and the corner coins (1,4,13,16) may not be the expected

neighbors because the visual representation of the coin arrangement is a Klein bottle. For

the Klein bottle, the wrap-around is defined as follows. Coins on the top edge (e.g. 2)

have the normal east (3), west (1), and south neighbors (6), with the north neighbor (14)

being the bottom-most coin in the column of the coin which was pointed to. Coins on the

bottom edge (e.g. 15) have the normal east (16), west (14), and north neighbors (11), with

the south neighbor (3) being the top-most coin of the same column. Coins on the left edge

(e.g. 5) have the normal north (1), south (9), and east neighbors (6), with the west

neighbor (4) being on the right edge and in the same row as the north neighbor. Coins on

the right edge (e.g. 8) have the normal north (4), south (12), and west neighbors (7), with

the east neighbor (9) being on the left edge and in the same row as the south neighbor.

The coins in the corners follow these basic rules, except the west neighbor of the top-left

coin (1) is the coin in the bottom-right corner (16), and the east neighbor of the bottom-

right coin (16) is the coin in the top-left corner (1).

BASIC MATHEMATICAL THEORY

Figure 2: Adjacency Matrix of a 4x4 Grid

The 16x16 adjacency matrix corresponding to the 4x4 arrangement of coins

(Figure 1) is shown in Figure 2. Notice that each row is a one-step circular shift from the

previous row. This is known as a circulant matrix [Davis]. The fact that we have a

circulant plays an important role in simplifying the calculations done in this work.

For the first row of the 16x16 circulant matrix of Figure 2 (which arises from a 4x4 coin

arrangement) the eigenvalues are as follows:

λ ω ω ω ω j j j j j

= 1 + 1 + 4 + 12 +^15

where the constants a 0 , a 1 , a 4 , a 12 , a 15 = 1 and the others = 0. In a similar way, the NxN

arrangement of coins leads to eigenvalues that are given by:

λ ω ω ω ω j j j

N

N N

= 1 + 1 + + 2 −^ + N^2 −^1

Combining Equations (1) and (2) gives the following equation:

π π j

Cos

 ∗^ ∗

Now the eigenvalues λ j are set to 0 and the previous equation is simplified to obtain:

 ∗^ ∗

 +^

 ∗^ ∗

Cos (^) 2

Cos

π π

Equation (3) applies to the NxN array of coins. Similar calculations yield:

 +^

 ∗^ ∗

Cos

M N

Cos

π π

for the MxN array. And for the LxMxN array:

 ∗^ ∗

Cos

L M

Cos

L M N

π π π

Equations (3), (4), and (5) will be the starting point for the calculations in

the next 3 sections. Another fact that is needed:

“If θ is rational in degrees, say θ = 2πr for some rational number r, then the

only rational values of the trigonometric functions of θ are as follows: cos

θ = 0, ±1/2, ±1; ...” [Niven] (FACT 1)

For the scope of this paper, it is assumed that each of the Cos terms is rational.

This means each Cos term can take on only the values 0, ±1/2, or ±1. This assumption is

supported by finding determinants of several different sizes of matrices (using

Mathematica 2.1) shown in Figure 4. The 4x3 and 4x9 coin arrangements are the only

arrangements tested that yield at least one 0 eigenvalue. These two arrangements, under

the assumption, should have a 0 eigenvalue (see Conclusion). If any other arrangement

had a 0 eigenvalue, the determinant would be zero, and the assumption would not hold.

This allows for each of Equations (3), (4), and (5) to be solved by cases by exhaustively

considering all combinations in which 0, ±1/2, and ±1 can be used to solve each equation.

It may be the case that Cos terms with irrational numbers can produce accurate results.

This possibility is not explored in this paper. Obviously, the irrational numbers would

have to cancel. For example, if one Cos term evaluated to ( 3 2 +1) and was added to

another Cos term that evaluated to (- 3 2 +1) the result of the addition would be a rational

number.

Size 3x3 3x4 3x5 3x6 3x7 3x8 3x Det 20 405 125 320 849 19845 5780

Size 4x3 4x4 4x5 4x6 4x7 4x8 4x Det 0 45 -375 5120 -2535 2205 0

Figure 4: Determinants

Determining Unsolvable Cases for an NxN Circulant Matrix

This section explores the existence of non-solvable configurations for the NxN

arrangement of coins. Recall Equation (3) from the Basic Mathematical Theory section of

this paper:

Cos

^ ∗^ ∗

 +^

 ∗^ ∗

π π −

Let:

X =

∗ π ∗ j

(X Equation)

Y =

2 ∗ π ∗ j N

(Y Equation)

Giving the equation:

Cos(X) + Cos(Y) = -1/

For this section and from Fact 1 in the Basic Mathematical Theory section of this paper,

there are 4 possible cases:

CASE 1 CASE 2 CASE 3 CASE 4

Cos(X) 0 -1 -0.5 0. Cos(Y) -0.5 0.5 0 -

substituting into Equation (I) for N:

(^4) ( 2 1 )

2 ∗ = ∗ + ∗

j k (^) 

j k

∨ 4 ( 2 1 )^

2 ∗ = ∗ + ∗

j k (^) 

j k

then dividing both sides by 4 ∗ j^2

( ) ( )

1 2

j k k

∨ ( ) ( )

1 2

j k k

taking the inverse of both sides

( )

Case 1 Conclusion: Since both denominators have a factor of 3 and neither numerator

has a factor of 3, j cannot be an integer and still satisfy these equations.

CASE 2: Cos(X)=-1, Cos(Y)=1/

X

π AND X = ( 2 ∗ k + 1 ) ∗( π ) combining both equations

( ) ( )

π π

k multiplying by

N^2

2 ( 2 1 )

∗ j = ∗ k + ∗ N (I)

Y

2 ∗ π ∗ AND Y = ( ∗ k + )∗  

π ∨ Y = ( ∗ k + )∗  

combining both equations

( )

π j π N

k ∨ ( )

π j π N

k multiplying by

3 ∗ N

6 ∗ j = ( 6 ∗ k 1 + 1 )∗ N ∨ 6 ∗ j = ( 6 ∗ k 2 + 5 )∗ N (II)

solving for N in Equation (II):

( )

N

( )

N

substituting into Equation (I) for N:

( ) ( )

∗ = ∗ + ∗

j k 

∨ ( ) ( )

∗ = ∗ + ∗

j k 

multiplying by

2 ∗ j^2

( ) ( )

j^2

k k

∨ ( ) ( )

j^2

k k

inverting both sides

( )

Case 2 Conclusion: Since both denominators are even and both numerators are odd, j

cannot be an integer and still satisfy these equations.

CASE 4 : Cos(X)=1/2, Cos(Y)=-

X

π AND X = ( ∗ k + )∗ 

π ∨ X = ( ∗ k + )∗  

combining both equations

( )

π j π

k ∨ ( )

π j π

k multiplying by 3

∗ N

(^6) ( 6 1 1 )

∗ j = ∗ k + ∗ N^2 ∨

(^6) ( 6 2 5 )

∗ j = ∗ k + ∗ N^2 (I)

Y

j N

2 ∗ π ∗ AND Y = (^) ( 2 ∗ k + (^1) )∗ ( π (^) ) combining both equations

( ) ( )

π π

j N

k multiplying by^

N

2j = (2k+1) * N (II)

solving for N in Equation (II):

N

substituting into Equation (I) for N:

(^6) ( 6 1 1 )

2 ∗ = ∗ + ∗

j k (^) 

j k

∨ (^6) ( 6 2 5 )

2 ∗ = ∗ + ∗

j k (^) 

j k

multiplying by

6 ∗ j^2

( )

j^2

( )

j^2

inverting both sides

( )

Case 4 Conclusion: Since both denominators are even and both numerators are odd, j

cannot be an integer and still satisfy these equations.

Conclusion for NxN: Since none of the four cases could satisfy the equation for the

NxN circulant, any NxN circulant matrix is always solvable.

Determining Unsolvable Cases for an MxN Circulant Matrix

This section explores the existence of non-solvable configurations for the MxN

arrangement of coins. Recall Equation (4) from the Basic Mathematical Theory section of

this paper:

Cos

j M N

Cos

j N

 +^

 ∗^ ∗

π π −

Let:

X =

π j

M N

(X Equation)

Y =

2 ∗ π ∗ j N

(Y Equation)

Giving the equation:

Cos(X) + Cos(Y) = -1/

For this section and from Fact 1 in the Basic Mathematical Theory section of this paper,

there are 4 possible cases:

CASE 1 CASE 2 CASE 3 CASE 4

Cos(X) 0 -1 -0.5 0. Cos(Y) -0.5 0.5 0 -

We know that:

Cos(ß) = 0 => ß = (2k+1) (π/2)

Cos(ß) = -1 => ß = (2k+1) (π) (Cos Equations)

Cos(ß) = -0.5 => ß = (6k1+2) * (π/3) ∨ ß = (6k2+4) * (π/3)

Cos(ß) = 0.5 => ß = (6k1+1) * (π/3) ∨ ß = (6k2+5) * (π/3)

where k, k1, and k2 are integers and ß represents the angle.

In the following cases, equations are formed by combining the X and Y Equations with the

four Cos Equations. Also k, k1, k2, k3, k4 and k5 are integers. The scope of the

variables and equations used within each case are limited to that case.

CASE 2: Cos(X)=-1, Cos(Y)=1/

X

j M N

2 π AND X = (^) ( 2 ∗ k + (^1) ) ∗( π (^) ) combining both equations

X ( ) ( )

M N

= k

π multiplying by

M ∗ N

2 ∗ j = ( 2 ∗ k + 1 )∗ M ∗ N (I)

Y

2 ∗ π ∗ AND Y = ( ∗ k + )∗  

π ∨ Y = ( ∗ k + )∗  

combining both equations

( )

π j π N

k ∨ (^) ( )

π j π N

k multiplying by 3 ∗^ N

6 ∗ j = ( 6 ∗ k 1 + 1 )∗ N ∨ 6 ∗ j = ( 6 ∗ k 2 + 5 )∗ N (II)

solving for N in Equation (II):

( )

N

( )

N

substituting into Equation (I) for N:

( ) ( )

j k M 

∨ ( ) ( )

j k M 

multiplying by

deno ators j

min 2 ∗

( 6 ∗ k 1 + 1 ) = ( 2 ∗ k + 1 ) ∗ M ∗ 3 ∨ ( 6 ∗ k 2 + 5 ) = ( 2 ∗ k + 1 ) ∗ M ∗ 3

Case 2 Conclusion: Since the left hand sides are not divisible by 3 and the right hand

sides are, M cannot be an integer and still satisfy these equations.

CASE 3: Cos(X)=-1/2, Cos(Y)=

X

j M N

2 π AND (^) X = ( ∗ k + )∗ 

X = ( ∗ k + )∗  

combining both equations

( )

π j π

M N

k ∨ (^) ( )

π j π

M N

k multiplying by 3 ∗^ M ∗^ N

6 ∗ j = (^) ( 6 ∗ k 1 + (^2) )∗ M ∗ N ∨ 6 ∗ j = (^) ( 6 ∗ k 2 + (^4) )∗ M ∗ N (I)

Y

j N

2 ∗ π ∗ AND (^) Y = ( ∗ k + )∗  

π combining both equations

( )

π j π

k multiplying by 2 ∗^ N

4j = (2k+1) * N (II)

solving for N in Equation (II):

N

substituting into Equation (I) for N:

6 ( 6 1 2 )

j k M (^) 

j k

∨ 6 ( 6 2 4 )

j k M (^) 

j k

multiplying by

k j

3 ∗ ( 2 ∗ k + 1 ) = ( 6 ∗ k 1 + 2 ) ∗ M ∗ 2 ∨ 3 ∗ ( 2 ∗ k + 1 ) = ( 6 ∗ k 2 + 4 ) ∗ M ∗ 2

Case 3 Conclusion: Since the left hand sides are odd and the right hand sides are even,

M cannot be an integer and still satisfy these equations.

Determining Unsolvable Cases for an LxMxN Circulant Matrix

This section explores the existence of non-solvable configurations for the LxMxN

arrangement of coins. Recall Equation (5) from the Basic Mathematical Theory section of

this paper:

Cos

j L

Cos

j L M

Cos

j L M N

^ ∗^ ∗

 +^

π π π −

Let:

X =

2 ∗ π ∗ j

(X Equation)

Y =

π j L M

(Y Equation)

Z =

π j L M N

(Z Equation)

Giving the equation:

Cos(X) + Cos(Y) + Cos(Z) = -1/

For this section and from Fact 1 in the Basic Mathematical Theory section of this paper,

there are 18 possible cases:

CASE 1 2 3 4 5 6 7 8

Cos(X) -1 -1 -1 -1 -0.5 -0.5 -0.5 -0.5 -0. Cos(Y) 0 0.5 1 -0.5 0 1 0.5 -0.5 - Cos(Z) 0.5 0 -0.5 1 0 -1 -0.5 0.5 1

CASE 10 11 12 13 14 15 16 17 18

Cos(X) 0.5 0.5 0.5 0 0 0 0 1 1 Cos(Y) 0 -0.5 -1 0 0.5 -1 -0.5 -1 -0. Cos(Z) -1 -0.5 0 -0.5 -1 0.5 0 -0.5 -

We know that:

Cos(ß) = 0 => ß = (odd #) * (π/2)

Cos(ß) = -1 => ß = (odd #) * (π)

Cos(ß) = 1 => ß = (odd #) * (π)

Cos(ß) = -0.5 => ß = (even # except multiples of 6) * (π/3)

Cos(ß) = 0.5 => ß = (odd # except multiples of 3) * (π/3)

where ß represents the angle.

Then written mathematically we have:

Cos(ß) = 0 => ß = (2k+1) (π/2)

Cos(ß) = -1 => ß = (2k1+1) (π)

Cos(ß) = 1 => ß = (2k2) (π) (Cos Equations)

Cos(ß) = -0.5 => ß = (6k3+2) * (π/3) ∨ ß = (6k4+4) * (π/3)

Cos(ß) = 0.5 => ß = (6k5+1) * (π/3) ∨ ß = (6k6+5) * (π/3)

where k, k1, k2, k3, k4, k5, and k6 are integers and ß represents the angle.

In the following cases, equations are formed by combining the X, Y, and Z Equations with

the five Cos Equations. Also k, k1, k2, k3, k4, k5, k6, k7, k8, k9, and k10 are integers.

The scope of the variables and equations used within each case are limited to that case.

Due to the large number of cases and the complexity of each, below I will show two

cases: Case 8 (which has a solution) and Case 9 (which does not have a solution). Case 8

is the only case that has a solution, and the remaining 17 do not yield a solution.

CASE 8 : Cos(X) = -1/2, Cos(Y) = -1/2, Cos(Z) = 1/

X

2 ∗ π ∗ AND X = ( ∗ k + )∗  

π ∨ X = ( ∗ k + )∗  

combining both equations

( )

π j π L

k ∨ (^) ( )

π j π L

k solving for L

( )

L

( )

L

(I)

this means L=3*k or simply L mod 3 = 0

Y

L M

2 π AND Y = ( ∗ k + )∗ 

π ∨ Y = ( ∗ k + )∗ 

combining both equations

( )

π j π

L M

k ∨ ( )

π j π

L M

k solving for L*M

( )

L M

( )

L M

(II)

Z

L M N

2 π AND Z = ( ∗ k + )∗  

π ∨ Z = ( ∗ k + )∗ 

combining both equations

CASE 9 : Cos(X) = -1/2, Cos(Y) = -1, Cos(Z) = 1

X

j L

2 ∗ π ∗ AND (^) X = ( ∗ k + )∗  

X = ( ∗ k + )∗  

combining both equations

( )

π j π

k ∨ (^) ( )

π j π

k solving for L

( )

L

( )

L

(I)

Y

j L M

2 π AND Y = (^) ( 2 ∗ k 1 + (^1) )∗ ( π)

combining both equations

( ) ( )

π π

L M

k solving for M

( )

k L^

M (II)

substituting L from Equation (I) into Equation (II):

( )

2 (^ )

2 1 1

M ∨

( )

2 (^ )

2 1 1

M reducing

( )

M ∨

( )

M (III)

Case 9 Conclusion: Since both denominators have a factor of 3 and neither numerator

has a factor of 3, M cannot be an integer and still satisfy Equation (III).

Conclusion for LxMxN: Case 8 satisfied the equation for the LxMxN circulant, thus,

any LxMxN circulant matrix where L mod 3 = 0, M mod 3 ≠ 0, N=2*k8 ∧ N mod 3 ≠ 0

(L is divisible by 3, M is not divisible by 3, N is even and not divisible by 6) is not always

solvable and all others that do not satisfy the aforementioned are always solvable. Also

note that since L cannot equal M, the NxNxN circulant matrix is always solvable.

CONCLUSION

Eigenvalue Formulas for Matrices: Case Studies, Papers of Computer Science

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Partial preview of the text

Download Eigenvalue Formulas for Matrices: Case Studies and more Papers Computer Science in PDF only on Docsity!

INTRODUCTION

The problem to be discussed originally stemmed from an example of a dynamic

programming problem called the nine-tails problem [Heck]. In the nine-tails problem, a

3x3 matrix of coins is given. Each coin has an initial state, heads or tails. When a player

flips a particular coin, each of its neighbors, North, South, East, and West (if they exist),

are also flipped. Heck’s purpose was to provide a simplified example to introduce

dynamic programming. This problem has extended in several directions like considering N

coins in a straight line [Klostermeyer et. al.] or see [Delahan et. al. #2,#3, Goldwasser].

This report deals with yet another variation of coin flipping, 2-dimensional

rectangular arrays and 3-dimensional cubic arrays of coins. In defining a coin’s neighbors,

wrap-around is included. Wrap-around means every coin in the 2-dimensional arrangement

will have exactly 4 neighbors (North, South, East, and West) and every coin in the 3-

dimensional arrangement will have exactly 6 neighbors (includes coins above and below).

Each coin has 2 states (heads and tails) and each coin is given some initial state.

Pointing is defined as selecting a coin causing it and all of its neighbors to be

flipped to the opposite state (heads to tails, or tails to heads). The goal is to determine

which coins to point to in order to have all coins show tails. Pointing to a coin more than

once is not necessary because pointing to a coin a second time resets the states of the coin

and its neighbors. A first question is whether an initial configuration exists for which the

problem is unsolvable. This paper characterizes which sizes of 2-dimensional arrays and

3-dimensional arrays have initial non-solvable configurations. Non-solvable, in this paper,

means that there is not a sequence of pointing that will allow all coins to show tails.

Figure 1: Illustration of 2-D Coin Layout

Figure 1 illustrates how to number the coins in a 4x4 grid. The neighbors of

horizontal edge coins (5,9,8,12) and the corner coins (1,4,13,16) may not be the expected

neighbors because the visual representation of the coin arrangement is a Klein bottle. For

the Klein bottle, the wrap-around is defined as follows. Coins on the top edge (e.g. 2)

have the normal east (3), west (1), and south neighbors (6), with the north neighbor (14)

being the bottom-most coin in the column of the coin which was pointed to. Coins on the

bottom edge (e.g. 15) have the normal east (16), west (14), and north neighbors (11), with

the south neighbor (3) being the top-most coin of the same column. Coins on the left edge

(e.g. 5) have the normal north (1), south (9), and east neighbors (6), with the west

neighbor (4) being on the right edge and in the same row as the north neighbor. Coins on

the right edge (e.g. 8) have the normal north (4), south (12), and west neighbors (7), with

the east neighbor (9) being on the left edge and in the same row as the south neighbor.

The coins in the corners follow these basic rules, except the west neighbor of the top-left

coin (1) is the coin in the bottom-right corner (16), and the east neighbor of the bottom-

right coin (16) is the coin in the top-left corner (1).

Figure 2: Adjacency Matrix of a 4x4 Grid

The 16x16 adjacency matrix corresponding to the 4x4 arrangement of coins

(Figure 1) is shown in Figure 2. Notice that each row is a one-step circular shift from the

previous row. This is known as a circulant matrix [Davis]. The fact that we have a

circulant plays an important role in simplifying the calculations done in this work.

For the first row of the 16x16 circulant matrix of Figure 2 (which arises from a 4x4 coin

arrangement) the eigenvalues are as follows:

= 1 + 1 + 4 + 12 +^15

where the constants a 0 , a 1 , a 4 , a 12 , a 15 = 1 and the others = 0. In a similar way, the NxN

arrangement of coins leads to eigenvalues that are given by:

N

N N

= 1 + 1 + + 2 −^ + N^2 −^1

Combining Equations (1) and (2) gives the following equation:

 ∗^ ∗

Now the eigenvalues λ j are set to 0 and the previous equation is simplified to obtain:

 ∗^ ∗

 +^

 ∗^ ∗

Equation (3) applies to the NxN array of coins. Similar calculations yield:

 +^

 ∗^ ∗

for the MxN array. And for the LxMxN array:

 ∗^ ∗

Equations (3), (4), and (5) will be the starting point for the calculations in

the next 3 sections. Another fact that is needed:

“If θ is rational in degrees, say θ = 2πr for some rational number r, then the

only rational values of the trigonometric functions of θ are as follows: cos

θ = 0, ±1/2, ±1; ...” [Niven] (FACT 1)

For the scope of this paper, it is assumed that each of the Cos terms is rational.

This means each Cos term can take on only the values 0, ±1/2, or ±1. This assumption is

supported by finding determinants of several different sizes of matrices (using

Mathematica 2.1) shown in Figure 4. The 4x3 and 4x9 coin arrangements are the only

arrangements tested that yield at least one 0 eigenvalue. These two arrangements, under

the assumption, should have a 0 eigenvalue (see Conclusion). If any other arrangement

had a 0 eigenvalue, the determinant would be zero, and the assumption would not hold.

This allows for each of Equations (3), (4), and (5) to be solved by cases by exhaustively

considering all combinations in which 0, ±1/2, and ±1 can be used to solve each equation.