The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game...
Transcript of The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game...
![Page 1: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/1.jpg)
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
1
![Page 2: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/2.jpg)
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
2
![Page 3: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/3.jpg)
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.
3
![Page 4: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/4.jpg)
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.
4
![Page 5: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/5.jpg)
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.
5
![Page 6: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/6.jpg)
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
6
![Page 7: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/7.jpg)
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
![Page 8: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/8.jpg)
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
![Page 9: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/9.jpg)
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
![Page 10: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/10.jpg)
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
10
![Page 11: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/11.jpg)
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}.
11
![Page 12: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/12.jpg)
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}.
12
![Page 13: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/13.jpg)
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.
13
![Page 14: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/14.jpg)
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.
14
![Page 15: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/15.jpg)
History The Fibonacci Quilt Sequence The Game Game Length Future Work
Rule 1
For n ≥ 2,qn + qn+1 → qn+3
15
![Page 16: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/16.jpg)
History The Fibonacci Quilt Sequence The Game Game Length Future Work
General Rules
Rule 2
For n ≥ 2,qn + qn+4 → qn+5
16
![Page 17: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/17.jpg)
History The Fibonacci Quilt Sequence The Game Game Length Future Work
General Rules
Rule 3
For n ≥ 7, 2qn → qn+2 + qn−5
17
![Page 18: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/18.jpg)
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
18
![Page 19: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/19.jpg)
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.
19
![Page 20: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/20.jpg)
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.
20
![Page 21: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/21.jpg)
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.
21
![Page 22: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/22.jpg)
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
22
![Page 23: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/23.jpg)
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
23
![Page 24: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/24.jpg)
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
24
![Page 25: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/25.jpg)
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
25
![Page 26: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/26.jpg)
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
26
![Page 27: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/27.jpg)
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
27
![Page 28: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/28.jpg)
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
28
![Page 29: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/29.jpg)
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
29
![Page 30: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/30.jpg)
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
30
![Page 31: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/31.jpg)
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
31
![Page 32: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/32.jpg)
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
32
![Page 33: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/33.jpg)
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
![Page 34: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/34.jpg)
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
34
![Page 35: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/35.jpg)
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
![Page 36: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/36.jpg)
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
![Page 37: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/37.jpg)
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
37
![Page 38: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/38.jpg)
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.
38
![Page 39: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/39.jpg)
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.
39
![Page 40: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/40.jpg)
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.
40
![Page 41: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/41.jpg)
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
41
![Page 42: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/42.jpg)
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
42
![Page 43: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/43.jpg)
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)
43
![Page 44: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/44.jpg)
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
![Page 45: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/45.jpg)
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
![Page 46: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/46.jpg)
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
![Page 47: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/47.jpg)
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
47
![Page 48: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/48.jpg)
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
48
![Page 49: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/49.jpg)
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
49
![Page 50: The Fibonacci Quilt Game - web.williams.edu · History The Fibonacci Quilt Sequence The Game Game Length Future Work The Fibonacci Quilt Game Alexandra Newlon Colgate University anewlon@colgate.edu](https://reader034.fdocuments.net/reader034/viewer/2022050610/5fb177658af99a5ec7476e77/html5/thumbnails/50.jpg)
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.
50