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History The Fibonacci Quilt Sequence The Game Game Length Future Work

The Fibonacci Quilt Game

Alexandra NewlonColgate University

anewlon@colgate.edu

Mentored by Steven J. Miller, Williams College, at theSMALL REU 2019.

Undergraduate Mathematics SymposiumUniversity of Illinois, Chicago, November 2, 2019

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Outline

1 History

2 The Fibonacci Quilt Sequence

3 The Game

4 Game Length

5 Future Work

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

The Fibonacci Sequence

1,1,2,3,5,8,13,21,34,55...

Let F0 = F1 = 1, and for n >= 2

Fn = Fn−1 + Fn−2

Theorem (Zeckendorf)Every positive integer can be written uniquely as the sum ofnon-consecutive Fibonacci numbers where

Fn = Fn−1 + Fn−2

and F1 = 1, F2 = 2.

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

The Fibonacci Sequence

1,1,2,3,5,8,13,21,34,55...

Let F0 = F1 = 1, and for n >= 2

Fn = Fn−1 + Fn−2

Theorem (Zeckendorf)Every positive integer can be written uniquely as the sum ofnon-consecutive Fibonacci numbers where

Fn = Fn−1 + Fn−2

and F1 = 1, F2 = 2.

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

The Fibonacci Sequence

1,1,2,3,5,8,13,21,34,55...

Let F0 = F1 = 1, and for n >= 2

Fn = Fn−1 + Fn−2

Theorem (Zeckendorf)Every positive integer can be written uniquely as the sum ofnon-consecutive Fibonacci numbers where

Fn = Fn−1 + Fn−2

and F1 = 1, F2 = 2.

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

The Fibonacci Quilt Sequence

12

3

4

5

7

9

12

16

21

28

37

49

65

86

114

151

200

265

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

The Fibonacci Quilt Sequence

1

23

4

5

7

9

12

16

21

28

37

49

65

86

114

151

200

265

7

History The Fibonacci Quilt Sequence The Game Game Length Future Work

The Fibonacci Quilt Sequence

12

3

4

5

7

9

12

16

21

28

37

49

65

86

114

151

200

265

8

History The Fibonacci Quilt Sequence The Game Game Length Future Work

The Fibonacci Quilt Sequence

12

3

4

57

9

12

16

21

28

37

49

65

86

114

151

200

265

9

History The Fibonacci Quilt Sequence The Game Game Length Future Work

The Fibonacci Quilt Sequence

12

3

4

57

9

12

16

21

28

37

49

65

86

114

151

200

265

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

FQ-legal Decomposition

Definition (Catral, Ford, Harris, Miller, Nelson)Let an increasing sequence of positive integers qi

∞i=1 be given.

We declare a decomposition of an integer

m = ql1 + ql2 + · · ·+ qlt

(where qli > qli+1) to be an FQ-legal decomposition if for all i , j ,|li − lj | 6= 0,1,3,4 and {1,3} 6⊂ {l1, l2, . . . , lt}.

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

The Fibonacci Quilt Sequence

Definition (Catral, Ford, Harris, Miller, Nelson)The Fibonacci Quilt sequence is an increasing sequence ofpositive integers {qi}∞i=1, where every qi (i ≥ 1) is the smallestpositive integer that does not have an FQ-legal decompositionusing the elements {q1, . . . ,qi−1}.

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Recurrence Relations

Theorem (Catral, Ford, Harris, Miller, Nelson)

Let qn denote the nth term in the Fibonacci Quilt, then

for n ≥ 5,qn+1 = qn−1 + qn−2,

for n ≥ 6,qn+1 = qn + qn−4.

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

General Rules

Inspired by the Two Player Zeckendorf Game

Two players alternate turns, the last person to move wins

Start the game with n 1’s (q1’s)

A turn consists of one of 5 rules, which preserve that∑qi = n by exchanging a pair qi ,qj such that i , j are an

illegal distance apart for a single term or legal pair.

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Rule 1

For n ≥ 2,qn + qn+1 → qn+3

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

General Rules

Rule 2

For n ≥ 2,qn + qn+4 → qn+5

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

General Rules

Rule 3

For n ≥ 7, 2qn → qn+2 + qn−5

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

General Rules

Rule 4

For n ≥ 7, qn + qn+3 → qn−5 + qn+4

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

General Rules

Rule 5

q1 + q3 → q4

To handle base cases, we added additional base rules that

preserves∑

qi = ndoes not produce violation of legality

Special Rule

1 + 5 → 2 + 4

Note: This rule can only be applied when nothing else can bedone.

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

General Rules

Rule 5

q1 + q3 → q4

To handle base cases, we added additional base rules that

preserves∑

qi = ndoes not produce violation of legality

Special Rule

1 + 5 → 2 + 4

Note: This rule can only be applied when nothing else can bedone.

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

General Rules

Rule 5

q1 + q3 → q4

To handle base cases, we added additional base rules that

preserves∑

qi = ndoes not produce violation of legality

Special Rule

1 + 5 → 2 + 4

Note: This rule can only be applied when nothing else can bedone.

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Example Game

1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28...

n = 10 = 9 + 1

1 2 3 4 5 7 910 0 0 0 0 0 0

8 1 0 0 0 0 0 Rule 3: q21 → q2

7 0 1 0 0 0 0 Rule 1: q1 + q2 → q35 1 1 0 0 0 0 Rule 3: q2

1 → q24 0 2 0 0 0 0 Rule 1: q1 + q2 → q33 0 1 1 0 0 0 Rule 5: q1 + q3 → q42 0 1 0 1 0 0 Rule 2: q1 + q4 → q50 1 1 0 1 0 0 Rule 3: q2

1 → q20 0 1 0 0 1 0 Rule 4: q2 + q5 → q61 0 0 0 0 0 1 Rule 4: q3 + q6 → q1 + q7

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Example Game

1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28...

n = 10 = 9 + 1

1 2 3 4 5 7 910 0 0 0 0 0 08 1 0 0 0 0 0 Rule 3: q2

1 → q2

7 0 1 0 0 0 0 Rule 1: q1 + q2 → q35 1 1 0 0 0 0 Rule 3: q2

1 → q24 0 2 0 0 0 0 Rule 1: q1 + q2 → q33 0 1 1 0 0 0 Rule 5: q1 + q3 → q42 0 1 0 1 0 0 Rule 2: q1 + q4 → q50 1 1 0 1 0 0 Rule 3: q2

1 → q20 0 1 0 0 1 0 Rule 4: q2 + q5 → q61 0 0 0 0 0 1 Rule 4: q3 + q6 → q1 + q7

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Example Game

1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28...

n = 10 = 9 + 1

1 2 3 4 5 7 910 0 0 0 0 0 08 1 0 0 0 0 0 Rule 3: q2

1 → q27 0 1 0 0 0 0 Rule 1: q1 + q2 → q3

5 1 1 0 0 0 0 Rule 3: q21 → q2

4 0 2 0 0 0 0 Rule 1: q1 + q2 → q33 0 1 1 0 0 0 Rule 5: q1 + q3 → q42 0 1 0 1 0 0 Rule 2: q1 + q4 → q50 1 1 0 1 0 0 Rule 3: q2

1 → q20 0 1 0 0 1 0 Rule 4: q2 + q5 → q61 0 0 0 0 0 1 Rule 4: q3 + q6 → q1 + q7

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Example Game

1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28...

n = 10 = 9 + 1

1 2 3 4 5 7 910 0 0 0 0 0 08 1 0 0 0 0 0 Rule 3: q2

1 → q27 0 1 0 0 0 0 Rule 1: q1 + q2 → q35 1 1 0 0 0 0 Rule 3: q2

1 → q2

4 0 2 0 0 0 0 Rule 1: q1 + q2 → q33 0 1 1 0 0 0 Rule 5: q1 + q3 → q42 0 1 0 1 0 0 Rule 2: q1 + q4 → q50 1 1 0 1 0 0 Rule 3: q2

1 → q20 0 1 0 0 1 0 Rule 4: q2 + q5 → q61 0 0 0 0 0 1 Rule 4: q3 + q6 → q1 + q7

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Example Game

1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28...

n = 10 = 9 + 1

1 2 3 4 5 7 910 0 0 0 0 0 08 1 0 0 0 0 0 Rule 3: q2

1 → q27 0 1 0 0 0 0 Rule 1: q1 + q2 → q35 1 1 0 0 0 0 Rule 3: q2

1 → q24 0 2 0 0 0 0 Rule 1: q1 + q2 → q3

3 0 1 1 0 0 0 Rule 5: q1 + q3 → q42 0 1 0 1 0 0 Rule 2: q1 + q4 → q50 1 1 0 1 0 0 Rule 3: q2

1 → q20 0 1 0 0 1 0 Rule 4: q2 + q5 → q61 0 0 0 0 0 1 Rule 4: q3 + q6 → q1 + q7

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Example Game

1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28...

n = 10 = 9 + 1

1 2 3 4 5 7 910 0 0 0 0 0 08 1 0 0 0 0 0 Rule 3: q2

1 → q27 0 1 0 0 0 0 Rule 1: q1 + q2 → q35 1 1 0 0 0 0 Rule 3: q2

1 → q24 0 2 0 0 0 0 Rule 1: q1 + q2 → q33 0 1 1 0 0 0 Rule 5: q1 + q3 → q4

2 0 1 0 1 0 0 Rule 2: q1 + q4 → q50 1 1 0 1 0 0 Rule 3: q2

1 → q20 0 1 0 0 1 0 Rule 4: q2 + q5 → q61 0 0 0 0 0 1 Rule 4: q3 + q6 → q1 + q7

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Example Game

1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28...

n = 10 = 9 + 1

1 2 3 4 5 7 910 0 0 0 0 0 08 1 0 0 0 0 0 Rule 3: q2

1 → q27 0 1 0 0 0 0 Rule 1: q1 + q2 → q35 1 1 0 0 0 0 Rule 3: q2

1 → q24 0 2 0 0 0 0 Rule 1: q1 + q2 → q33 0 1 1 0 0 0 Rule 5: q1 + q3 → q42 0 1 0 1 0 0 Rule 2: q1 + q4 → q5

0 1 1 0 1 0 0 Rule 3: q21 → q2

0 0 1 0 0 1 0 Rule 4: q2 + q5 → q61 0 0 0 0 0 1 Rule 4: q3 + q6 → q1 + q7

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Example Game

1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28...

n = 10 = 9 + 1

1 2 3 4 5 7 910 0 0 0 0 0 08 1 0 0 0 0 0 Rule 3: q2

1 → q27 0 1 0 0 0 0 Rule 1: q1 + q2 → q35 1 1 0 0 0 0 Rule 3: q2

1 → q24 0 2 0 0 0 0 Rule 1: q1 + q2 → q33 0 1 1 0 0 0 Rule 5: q1 + q3 → q42 0 1 0 1 0 0 Rule 2: q1 + q4 → q50 1 1 0 1 0 0 Rule 3: q2

1 → q2

0 0 1 0 0 1 0 Rule 4: q2 + q5 → q61 0 0 0 0 0 1 Rule 4: q3 + q6 → q1 + q7

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Example Game

1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28...

n = 10 = 9 + 1

1 2 3 4 5 7 910 0 0 0 0 0 08 1 0 0 0 0 0 Rule 3: q2

1 → q27 0 1 0 0 0 0 Rule 1: q1 + q2 → q35 1 1 0 0 0 0 Rule 3: q2

1 → q24 0 2 0 0 0 0 Rule 1: q1 + q2 → q33 0 1 1 0 0 0 Rule 5: q1 + q3 → q42 0 1 0 1 0 0 Rule 2: q1 + q4 → q50 1 1 0 1 0 0 Rule 3: q2

1 → q20 0 1 0 0 1 0 Rule 4: q2 + q5 → q6

1 0 0 0 0 0 1 Rule 4: q3 + q6 → q1 + q7

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Example Game

1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28...

n = 10 = 9 + 1

1 2 3 4 5 7 910 0 0 0 0 0 08 1 0 0 0 0 0 Rule 3: q2

1 → q27 0 1 0 0 0 0 Rule 1: q1 + q2 → q35 1 1 0 0 0 0 Rule 3: q2

1 → q24 0 2 0 0 0 0 Rule 1: q1 + q2 → q33 0 1 1 0 0 0 Rule 5: q1 + q3 → q42 0 1 0 1 0 0 Rule 2: q1 + q4 → q50 1 1 0 1 0 0 Rule 3: q2

1 → q20 0 1 0 0 1 0 Rule 4: q2 + q5 → q61 0 0 0 0 0 1 Rule 4: q3 + q6 → q1 + q7

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

The Game is Playable

TheoremEvery game terminates in a finite number of moves at anFQ-legal decomposition.

Proof Sketch: The sum of the square roots of the indices ofthe terms is a strict monovarient.

qn ∧ qn+1 −→ qn+3:√

n + 3−√

n −√

n + 1 < 0qn ∧ qn+4 −→ qn+5:

√n + 5−

√n −√

n + 4 < 02qn −→ qn+2 ∧ qn−5:

√n + 2 +

√n − 5− 2

√n < 0

qn ∧ qn+3 −→ qn+4 ∧ qn−5:√n + 4 +

√n − 5−

√n −√

n + 3 < 0

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

The Game is Playable

TheoremEvery game terminates in a finite number of moves at anFQ-legal decomposition.

Proof Sketch: The sum of the square roots of the indices ofthe terms is a strict monovarient.

qn ∧ qn+1 −→ qn+3:√

n + 3−√

n −√

n + 1 < 0qn ∧ qn+4 −→ qn+5:

√n + 5−

√n −√

n + 4 < 02qn −→ qn+2 ∧ qn−5:

√n + 2 +

√n − 5− 2

√n < 0

qn ∧ qn+3 −→ qn+4 ∧ qn−5:√n + 4 +

√n − 5−

√n −√

n + 3 < 0

33

History The Fibonacci Quilt Sequence The Game Game Length Future Work

The Game is Playable

TheoremEvery game terminates in a finite number of moves at anFQ-legal decomposition.

Proof Sketch: The sum of the square roots of the indices ofthe terms is a strict monovarient.

qn ∧ qn+1 −→ qn+3:√

n + 3−√

n −√

n + 1 < 0

qn ∧ qn+4 −→ qn+5:√

n + 5−√

n −√

n + 4 < 02qn −→ qn+2 ∧ qn−5:

√n + 2 +

√n − 5− 2

√n < 0

qn ∧ qn+3 −→ qn+4 ∧ qn−5:√n + 4 +

√n − 5−

√n −√

n + 3 < 0

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

The Game is Playable

TheoremEvery game terminates in a finite number of moves at anFQ-legal decomposition.

Proof Sketch: The sum of the square roots of the indices ofthe terms is a strict monovarient.

qn ∧ qn+1 −→ qn+3:√

n + 3−√

n −√

n + 1 < 0qn ∧ qn+4 −→ qn+5:

√n + 5−

√n −√

n + 4 < 0

2qn −→ qn+2 ∧ qn−5:√

n + 2 +√

n − 5− 2√

n < 0qn ∧ qn+3 −→ qn+4 ∧ qn−5:√

n + 4 +√

n − 5−√

n −√

n + 3 < 0

35

History The Fibonacci Quilt Sequence The Game Game Length Future Work

The Game is Playable

TheoremEvery game terminates in a finite number of moves at anFQ-legal decomposition.

Proof Sketch: The sum of the square roots of the indices ofthe terms is a strict monovarient.

qn ∧ qn+1 −→ qn+3:√

n + 3−√

n −√

n + 1 < 0qn ∧ qn+4 −→ qn+5:

√n + 5−

√n −√

n + 4 < 02qn −→ qn+2 ∧ qn−5:

√n + 2 +

√n − 5− 2

√n < 0

qn ∧ qn+3 −→ qn+4 ∧ qn−5:√n + 4 +

√n − 5−

√n −√

n + 3 < 0

36

History The Fibonacci Quilt Sequence The Game Game Length Future Work

The Game is Playable

TheoremEvery game terminates in a finite number of moves at anFQ-legal decomposition.

Proof Sketch: The sum of the square roots of the indices ofthe terms is a strict monovarient.

qn ∧ qn+1 −→ qn+3:√

n + 3−√

n −√

n + 1 < 0qn ∧ qn+4 −→ qn+5:

√n + 5−

√n −√

n + 4 < 02qn −→ qn+2 ∧ qn−5:

√n + 2 +

√n − 5− 2

√n < 0

qn ∧ qn+3 −→ qn+4 ∧ qn−5:√n + 4 +

√n − 5−

√n −√

n + 3 < 0

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Other Results

TheoremThere is more than one possible game for any n > 3.

TheoremThere are games of even and odd length for any n > 5.

ConjectureThe game is fair.

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Other Results

TheoremThere is more than one possible game for any n > 3.

TheoremThere are games of even and odd length for any n > 5.

ConjectureThe game is fair.

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Other Results

TheoremThere is more than one possible game for any n > 3.

TheoremThere are games of even and odd length for any n > 5.

ConjectureThe game is fair.

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Lower Bound on Game Length

NotationLet L(n) denote the maximum number of terms in an FQ-legaldecomposition of n. Let l(n) denote the minimum number ofterms in an FQ-legal decomposition of n.

Examples:20 = 16 + 4 = 12 + 7 + 1L(20) = 3, l(20) = 2

50 = 49 + 1 = 28 + 16 + 4 + 2L(50) = 4, l(50) = 2

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Lower Bound on Game Length

NotationLet L(n) denote the maximum number of terms in an FQ-legaldecomposition of n. Let l(n) denote the minimum number ofterms in an FQ-legal decomposition of n.

Examples:20 = 16 + 4 = 12 + 7 + 1L(20) = 3, l(20) = 250 = 49 + 1 = 28 + 16 + 4 + 2L(50) = 4, l(50) = 2

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Lower Bound on Game Length

TheoremThe shortest possible game on n is completed in n − L(n)moves.

Proof Sketch: Strong induction on n.If n is in the Fibonacci Quilt Sequence, denoted qi

qi−3 + qi−2 = qi

If n is not in the sequence

n = qi1 + qi2 + · · ·+ qiL(n)

Number of moves:

(qi1 − 1) + (qi2 − 1) + · · ·+ (qiL(n) − 1)

= (qi1 + qi2 + · · ·+ qiL(n))− L(n)

=

n−L(n)

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Lower Bound on Game Length

TheoremThe shortest possible game on n is completed in n − L(n)moves.

Proof Sketch: Strong induction on n.

If n is in the Fibonacci Quilt Sequence, denoted qi

qi−3 + qi−2 = qi

If n is not in the sequence

n = qi1 + qi2 + · · ·+ qiL(n)

Number of moves:

(qi1 − 1) + (qi2 − 1) + · · ·+ (qiL(n) − 1)

= (qi1 + qi2 + · · ·+ qiL(n))− L(n)

=

n−L(n)

44

History The Fibonacci Quilt Sequence The Game Game Length Future Work

Lower Bound on Game Length

TheoremThe shortest possible game on n is completed in n − L(n)moves.

Proof Sketch: Strong induction on n.If n is in the Fibonacci Quilt Sequence, denoted qi

qi−3 + qi−2 = qi

If n is not in the sequence

n = qi1 + qi2 + · · ·+ qiL(n)

Number of moves:

(qi1 − 1) + (qi2 − 1) + · · ·+ (qiL(n) − 1)

= (qi1 + qi2 + · · ·+ qiL(n))− L(n)

=

n−L(n)

45

History The Fibonacci Quilt Sequence The Game Game Length Future Work

Lower Bound on Game Length

TheoremThe shortest possible game on n is completed in n − L(n)moves.

Proof Sketch: Strong induction on n.If n is in the Fibonacci Quilt Sequence, denoted qi

qi−3 + qi−2 = qi

If n is not in the sequence

n = qi1 + qi2 + · · ·+ qiL(n)

Number of moves:

(qi1 − 1) + (qi2 − 1) + · · ·+ (qiL(n) − 1)

= (qi1 + qi2 + · · ·+ qiL(n))− L(n)

=

n−L(n)46

History The Fibonacci Quilt Sequence The Game Game Length Future Work

Distribution of Game Lengths

ConjectureThe distribution of a random game length approachesGaussian as n increases.

Figure: Distribution of 1000 games on n=60

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Future Work

Is there a deterministic game that always results in thelower bound?

What patterns emerge from the winner of certaindeterministic games as n increases?

Does either player have a winning strategy?Analogous result on the Zeckendorf Game shows that forn > 2, player 2 has a winning strategy

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

References

M. Catral, P.L. Ford, P.E. Harris, S.J. Miller, D. Nelson,Legal Decomposition Arising From Non-Positive LinearRecurrences. Fibonacci Quarterly (54 (2016), no. 4,348-365). https://arxiv.org/abs/1606.09312.P. Baird-Smith, A. Epstein, K. Flint, S.J. Miller, TheZeckendorf Game. (2018).https://arxiv.org/abs/1809.04881.The Fibonacci Quilt Game (with S.J. Miller), submittedSeptember 2019 to the Fibonacci Quarterly.https://arxiv.org/abs/1909.01938P. Baird-Smith, A. Epstein, K. Flint, S.J. Miller, TheGeneralized Zeckendorf Game, the Fibonacci Quarterly(Proceedings of the 18th Conference) 57 (2019), no. 55,1–15. https://arxiv.org/abs/1809.04883

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History The Fibonacci Quilt Sequence The Game Game Length Future Work

Thank You

Thank you to Dr. Miller (NSF Grant DMS1561945), the SMALLprogram (NSF Grant DMS1659037) and Williams College.

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