The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game...

History The Fibonacci Quilt Sequence The Game Game Length Future Work

The Fibonacci Quilt Game

Alexandra NewlonColgate University

[email protected]

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

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

1


Outline

1 History

2 The Fibonacci Quilt Sequence

3 The Game

4 Game Length

5 Future Work

2


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.

3



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

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

Fn = Fn−1 + Fn−2


Fn = Fn−1 + Fn−2

and F1 = 1, F2 = 2.

4



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

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

Fn = Fn−1 + Fn−2


Fn = Fn−1 + Fn−2

and F1 = 1, F2 = 2.

5


The Fibonacci Quilt Sequence

12

3

4

5

7

9

12

16

21

28

37

49

65

86

114

151

200

265

6



1

23

4

5

7

9

12

16

21

28

37

49

65

86

114

151

200

265

7



12

3

4

5

7

9

12

16

21

28

37

49

65

86

114

151

200

265

8



12

3

4

57

9

12

16

21

28

37

49

65

86

114

151

200

265

9



12

3

4

57

9

12

16

21

28

37

49

65

86

114

151

200

265

10


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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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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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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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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Rule 1

For n ≥ 2,qn + qn+1 → qn+3

15


General Rules

Rule 2

For n ≥ 2,qn + qn+4 → qn+5

16


General Rules

Rule 3

For n ≥ 7, 2qn → qn+2 + qn−5

17


General Rules

Rule 4

For n ≥ 7, qn + qn+3 → qn−5 + qn+4

18


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.

19


General Rules

Rule 5

q1 + q3 → q4


preserves∑


Special Rule

1 + 5 → 2 + 4


20


General Rules

Rule 5

q1 + q3 → q4


preserves∑


Special Rule

1 + 5 → 2 + 4


21


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

22


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



23


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


24


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


25


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


26


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


27


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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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

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

29


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 → 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

30


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



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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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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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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

34





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





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





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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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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Other Results




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Other Results




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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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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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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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Proof Sketch: Strong induction on n.

If n is in the Fibonacci Quilt Sequence, denoted qi

qi−3 + qi−2 = qi


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





qi−3 + qi−2 = qi


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





qi−3 + qi−2 = qi


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


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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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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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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Thank You

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

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