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History The Fibonacci Quilt Sequence The Game Game Length Future Work
The Fibonacci Quilt Game
Alexandra NewlonColgate University
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
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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 < 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
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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
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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)
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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)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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