The Cookie Monster Problem - MIT...
Transcript of The Cookie Monster Problem - MIT...
![Page 1: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/1.jpg)
The Cookie Monster Problem
Leigh Marie Braswell
Under Mentorship of Tanya Khovanova
PRIMES Conference 2013
![Page 2: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/2.jpg)
The Problem
Cookie Monster is always hungry.
We present him with a set of cookie jars, each filled withsome number of cookies.He wants to empty them as quickly as possible. But...On each of his moves, he must choose a subset of the jarsand take the same number of cookies from each.
![Page 3: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/3.jpg)
The Problem
Cookie Monster is always hungry.We present him with a set of cookie jars, each filled withsome number of cookies.
He wants to empty them as quickly as possible. But...On each of his moves, he must choose a subset of the jarsand take the same number of cookies from each.
![Page 4: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/4.jpg)
The Problem
Cookie Monster is always hungry.We present him with a set of cookie jars, each filled withsome number of cookies.He wants to empty them as quickly as possible. But...
On each of his moves, he must choose a subset of the jarsand take the same number of cookies from each.
![Page 5: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/5.jpg)
The Problem
Cookie Monster is always hungry.We present him with a set of cookie jars, each filled withsome number of cookies.He wants to empty them as quickly as possible. But...On each of his moves, he must choose a subset of the jarsand take the same number of cookies from each.
![Page 6: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/6.jpg)
Example
16 10 6 3
−10 −10
6 0 6 3
−6 −6
0 0 0 3
−3
0 0 0 0
![Page 7: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/7.jpg)
Example
16 10 6 3
−10 −10
6 0 6 3
−6 −6
0 0 0 3
−3
0 0 0 0
![Page 8: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/8.jpg)
Example
16 10 6 3
−10 −10
6 0 6 3
−6 −6
0 0 0 3
−3
0 0 0 0
![Page 9: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/9.jpg)
Example
16 10 6 3
−10 −10
6 0 6 3
−6 −6
0 0 0 3
−3
0 0 0 0
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Example
16 10 6 3
−10 −10
6 0 6 3
−6 −6
0 0 0 3
−3
0 0 0 0
![Page 11: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/11.jpg)
Example
16 10 6 3
−10 −10
6 0 6 3
−6 −6
0 0 0 3
−3
0 0 0 0
![Page 12: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/12.jpg)
Example
16 10 6 3
−10 −10
6 0 6 3
−6 −6
0 0 0 3
−3
0 0 0 0
![Page 13: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/13.jpg)
Initial Observations
Cookie Monster may perform his moves in any order.
Jars with equal number of cookies may be treated thesame.Jars may be "emptied" without being emptied.Is there a procedure the monster can follow that will alwayslead to the optimal solution?
![Page 14: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/14.jpg)
Initial Observations
Cookie Monster may perform his moves in any order.Jars with equal number of cookies may be treated thesame.
Jars may be "emptied" without being emptied.Is there a procedure the monster can follow that will alwayslead to the optimal solution?
![Page 15: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/15.jpg)
Initial Observations
Cookie Monster may perform his moves in any order.Jars with equal number of cookies may be treated thesame.Jars may be "emptied" without being emptied.
Is there a procedure the monster can follow that will alwayslead to the optimal solution?
![Page 16: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/16.jpg)
Initial Observations
Cookie Monster may perform his moves in any order.Jars with equal number of cookies may be treated thesame.Jars may be "emptied" without being emptied.Is there a procedure the monster can follow that will alwayslead to the optimal solution?
![Page 17: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/17.jpg)
General Algorithms
Empty the Most Jars Algorithm: the monster reduces thenumber of distinct jars by as many as he can for eachmove.
{7,4,3,1}
−4−−−−−−→Jar 1, Jar 2
{3,0,3,1} = {3,1}
![Page 18: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/18.jpg)
General Algorithms
Empty the Most Jars Algorithm: the monster reduces thenumber of distinct jars by as many as he can for eachmove.{7,4,3,1}
−4−−−−−−→Jar 1, Jar 2
{3,0,3,1} = {3,1}
![Page 19: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/19.jpg)
General Algorithms
Empty the Most Jars Algorithm: the monster reduces thenumber of distinct jars by as many as he can for eachmove.{7,4,3,1}
−4−−−−−−→Jar 1, Jar 2
{3,0,3,1} = {3,1}
![Page 20: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/20.jpg)
General Algorithms
Take the Most Cookies Algorithm: the monster takes asmany cookies as possible for each move.
{20,19,14,7,4}
−14−−−−−−−−−−→Jar 1, Jar 2, Jar 3
{6,5,0,7,4} = {6,5,7,4}
![Page 21: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/21.jpg)
General Algorithms
Take the Most Cookies Algorithm: the monster takes asmany cookies as possible for each move.{20,19,14,7,4}
−14−−−−−−−−−−→Jar 1, Jar 2, Jar 3
{6,5,0,7,4} = {6,5,7,4}
![Page 22: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/22.jpg)
General Algorithms
Take the Most Cookies Algorithm: the monster takes asmany cookies as possible for each move.{20,19,14,7,4}
−14−−−−−−−−−−→Jar 1, Jar 2, Jar 3
{6,5,0,7,4} = {6,5,7,4}
![Page 23: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/23.jpg)
General Algorithms
Binary Algorithm: the monster takes 2k cookies from alljars that contain at least 2k cookies for k as large aspossible.
{18,16,9,8}
−16−−−−−−→Jar 1, Jar 2
{2,0,9,8} = {2,9,8}
None of these algorithms are optimal in all cases.
![Page 24: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/24.jpg)
General Algorithms
Binary Algorithm: the monster takes 2k cookies from alljars that contain at least 2k cookies for k as large aspossible.{18,16,9,8}
−16−−−−−−→Jar 1, Jar 2
{2,0,9,8} = {2,9,8}
None of these algorithms are optimal in all cases.
![Page 25: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/25.jpg)
General Algorithms
Binary Algorithm: the monster takes 2k cookies from alljars that contain at least 2k cookies for k as large aspossible.{18,16,9,8}
−16−−−−−−→Jar 1, Jar 2
{2,0,9,8} = {2,9,8}
None of these algorithms are optimal in all cases.
![Page 26: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/26.jpg)
General Algorithms
Binary Algorithm: the monster takes 2k cookies from alljars that contain at least 2k cookies for k as large aspossible.{18,16,9,8}
−16−−−−−−→Jar 1, Jar 2
{2,0,9,8} = {2,9,8}
None of these algorithms are optimal in all cases.
![Page 27: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/27.jpg)
CM(S)
Suppose S is the set of the numbers of the cookies in thejars.
The Cookie Monster number of S, CM(S), is the minimumnumber of moves Cookie Monster must use to empty all ofthe jars in S.Suppose CM(S) = n and Cookie Monster follows anoptimal procedure.After he performs move n, all jars are empty.Therefore, each jar may be represented as the sum ofsome moves.
![Page 28: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/28.jpg)
CM(S)
Suppose S is the set of the numbers of the cookies in thejars.The Cookie Monster number of S, CM(S), is the minimumnumber of moves Cookie Monster must use to empty all ofthe jars in S.
Suppose CM(S) = n and Cookie Monster follows anoptimal procedure.After he performs move n, all jars are empty.Therefore, each jar may be represented as the sum ofsome moves.
![Page 29: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/29.jpg)
CM(S)
Suppose S is the set of the numbers of the cookies in thejars.The Cookie Monster number of S, CM(S), is the minimumnumber of moves Cookie Monster must use to empty all ofthe jars in S.Suppose CM(S) = n and Cookie Monster follows anoptimal procedure.
After he performs move n, all jars are empty.Therefore, each jar may be represented as the sum ofsome moves.
![Page 30: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/30.jpg)
CM(S)
Suppose S is the set of the numbers of the cookies in thejars.The Cookie Monster number of S, CM(S), is the minimumnumber of moves Cookie Monster must use to empty all ofthe jars in S.Suppose CM(S) = n and Cookie Monster follows anoptimal procedure.After he performs move n, all jars are empty.
Therefore, each jar may be represented as the sum ofsome moves.
![Page 31: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/31.jpg)
CM(S)
Suppose S is the set of the numbers of the cookies in thejars.The Cookie Monster number of S, CM(S), is the minimumnumber of moves Cookie Monster must use to empty all ofthe jars in S.Suppose CM(S) = n and Cookie Monster follows anoptimal procedure.After he performs move n, all jars are empty.Therefore, each jar may be represented as the sum ofsome moves.
![Page 32: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/32.jpg)
General Bounds for CM(S)
TheoremFor all |S| = m, dlog2(m + 1)e ≤ CM(S) ≤ m.
![Page 33: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/33.jpg)
Interesting Sequences
We now present our Cookie Monster with interestingsequences of cookies in his jars.
First, we challenge our monster to empty a set of jarscontaining cookies in the Fibonacci sequence.Define the Fibonacci sequence as F0 = 0, F1 = 1, andFi = Fi−2 + Fi−1 for i ≥ 2.A jar with 0 cookies and 2 jars containing 1 cookie areirrelevant, so our smallest jar will contain F2 cookies.
![Page 34: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/34.jpg)
Interesting Sequences
We now present our Cookie Monster with interestingsequences of cookies in his jars.First, we challenge our monster to empty a set of jarscontaining cookies in the Fibonacci sequence.
Define the Fibonacci sequence as F0 = 0, F1 = 1, andFi = Fi−2 + Fi−1 for i ≥ 2.A jar with 0 cookies and 2 jars containing 1 cookie areirrelevant, so our smallest jar will contain F2 cookies.
![Page 35: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/35.jpg)
Interesting Sequences
We now present our Cookie Monster with interestingsequences of cookies in his jars.First, we challenge our monster to empty a set of jarscontaining cookies in the Fibonacci sequence.Define the Fibonacci sequence as F0 = 0, F1 = 1, andFi = Fi−2 + Fi−1 for i ≥ 2.
A jar with 0 cookies and 2 jars containing 1 cookie areirrelevant, so our smallest jar will contain F2 cookies.
![Page 36: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/36.jpg)
Interesting Sequences
We now present our Cookie Monster with interestingsequences of cookies in his jars.First, we challenge our monster to empty a set of jarscontaining cookies in the Fibonacci sequence.Define the Fibonacci sequence as F0 = 0, F1 = 1, andFi = Fi−2 + Fi−1 for i ≥ 2.A jar with 0 cookies and 2 jars containing 1 cookie areirrelevant, so our smallest jar will contain F2 cookies.
![Page 37: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/37.jpg)
Fibonacci and CM(S)
TheoremWhen S = {F2, . . . ,Fm}, then CM(S) = dm
2 e.
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N-nacci
Each term starting with the n-th is the sum of the previousn terms.
For example, in the 3-nacci sequence, otherwise known asTribonacci, each term after the third is the sum of theprevious three terms.0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149 . . .Tetranacci, Pentanacci similar
![Page 39: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/39.jpg)
N-nacci
Each term starting with the n-th is the sum of the previousn terms.For example, in the 3-nacci sequence, otherwise known asTribonacci, each term after the third is the sum of theprevious three terms.
0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149 . . .Tetranacci, Pentanacci similar
![Page 40: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/40.jpg)
N-nacci
Each term starting with the n-th is the sum of the previousn terms.For example, in the 3-nacci sequence, otherwise known asTribonacci, each term after the third is the sum of theprevious three terms.0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149 . . .
Tetranacci, Pentanacci similar
![Page 41: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/41.jpg)
N-nacci
Each term starting with the n-th is the sum of the previousn terms.For example, in the 3-nacci sequence, otherwise known asTribonacci, each term after the third is the sum of theprevious three terms.0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149 . . .Tetranacci, Pentanacci similar
![Page 42: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/42.jpg)
Tribonacci Example
{1,2,4,7,13,24,44}
−24−−−−−−−→Jar 6 , Jar 7
{1,2,4,7,13,0,20} = {1,2,4,7,13,20}
−13−−−−−−−→Jar 5, Jar 6
{1,2,4,7,0,7} = {1,2,4,7}
"Empty" 3 jars with 2 moves!
![Page 43: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/43.jpg)
Tribonacci Example
{1,2,4,7,13,24,44}
−24−−−−−−−→Jar 6 , Jar 7
{1,2,4,7,13,0,20} = {1,2,4,7,13,20}
−13−−−−−−−→Jar 5, Jar 6
{1,2,4,7,0,7} = {1,2,4,7}
"Empty" 3 jars with 2 moves!
![Page 44: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/44.jpg)
Tribonacci Example
{1,2,4,7,13,24,44}
−24−−−−−−−→Jar 6 , Jar 7
{1,2,4,7,13,0,20} = {1,2,4,7,13,20}
−13−−−−−−−→Jar 5, Jar 6
{1,2,4,7,0,7} = {1,2,4,7}
"Empty" 3 jars with 2 moves!
![Page 45: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/45.jpg)
Tribonacci Example
{1,2,4,7,13,24,44}
−24−−−−−−−→Jar 6 , Jar 7
{1,2,4,7,13,0,20} = {1,2,4,7,13,20}
−13−−−−−−−→Jar 5, Jar 6
{1,2,4,7,0,7} = {1,2,4,7}
"Empty" 3 jars with 2 moves!
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Tribonacci and CM(S)
Theorem
When S = {T3, . . . ,Tm}, then CM(S) = d2m3 e − 1.
![Page 47: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/47.jpg)
General Nacci CM(S)
TheoremWhen Cookie Monster is presented with the first m − (n − 1)distinct n-nacci numbers, CM(S) = dn−1
n me − (n − 2).
![Page 48: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/48.jpg)
Beyond Nacci
The monster wonders if he can extend his knowledge ofnacci sequences to non-nacci ones.
Define a Super-n-nacci sequence as S = {k1, . . . , km}where ki+n ≥ ki+n−1 + · · ·+ ki for i ≥ 1.Cookie Monster suspects that since he already knows howto consume the nacci sequences, he might be able tobound CM(S) for Super naccis.
![Page 49: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/49.jpg)
Beyond Nacci
The monster wonders if he can extend his knowledge ofnacci sequences to non-nacci ones.Define a Super-n-nacci sequence as S = {k1, . . . , km}where ki+n ≥ ki+n−1 + · · ·+ ki for i ≥ 1.
Cookie Monster suspects that since he already knows howto consume the nacci sequences, he might be able tobound CM(S) for Super naccis.
![Page 50: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/50.jpg)
Beyond Nacci
The monster wonders if he can extend his knowledge ofnacci sequences to non-nacci ones.Define a Super-n-nacci sequence as S = {k1, . . . , km}where ki+n ≥ ki+n−1 + · · ·+ ki for i ≥ 1.Cookie Monster suspects that since he already knows howto consume the nacci sequences, he might be able tobound CM(S) for Super naccis.
![Page 51: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/51.jpg)
Super Nacci
Theorem
For Super-n-nacci sequences with m terms, CM(S) ≥ d (n−1)mn e.
![Page 52: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/52.jpg)
Bounds Depending on Growth of Sequence
The n-nacci sequences all give monic recursive equations,xn = xn−1 + xn−2 + · · ·+ x1 + 1.
Therefore, any n-nacci sequence approximates ageometric sequence, specifically αr , αr2, αr3, . . . where r isa real root of the characteristic polynomial and α is somereal number.Cookie Monster wonders if there is any relationshipbetween r and n−1
n , the coefficient for CM(S) of n-nacci.
![Page 53: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/53.jpg)
Bounds Depending on Growth of Sequence
The n-nacci sequences all give monic recursive equations,xn = xn−1 + xn−2 + · · ·+ x1 + 1.Therefore, any n-nacci sequence approximates ageometric sequence, specifically αr , αr2, αr3, . . . where r isa real root of the characteristic polynomial and α is somereal number.
Cookie Monster wonders if there is any relationshipbetween r and n−1
n , the coefficient for CM(S) of n-nacci.
![Page 54: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/54.jpg)
Bounds Depending on Growth of Sequence
The n-nacci sequences all give monic recursive equations,xn = xn−1 + xn−2 + · · ·+ x1 + 1.Therefore, any n-nacci sequence approximates ageometric sequence, specifically αr , αr2, αr3, . . . where r isa real root of the characteristic polynomial and α is somereal number.Cookie Monster wonders if there is any relationshipbetween r and n−1
n , the coefficient for CM(S) of n-nacci.
![Page 55: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/55.jpg)
Bounds Depending on Growth of Sequence
3 4 5 6 7 8
0.5
1.0
1.5
2.0
Figure: Real root (red) approaches 2 as fraction (blue) approaches 1.
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Bounds Depending on Growth of Sequence
Theorem
The real root of xn = xn−1 + xn−2 + · · · x + 1 may beapproximated as 2− ε where ε = 1
2n−1 .
Thus, n ≈ log2(1 + 12−r ), so n−1
n = 1− 1n ≈ 1− 1
log2(1+1
2−r ).
Therefore, for large n, the Cookie Monster coefficientapproximates a function of r .
![Page 57: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/57.jpg)
Bounds Depending on Growth of Sequence
Theorem
The real root of xn = xn−1 + xn−2 + · · · x + 1 may beapproximated as 2− ε where ε = 1
2n−1 .
Thus, n ≈ log2(1 + 12−r ), so n−1
n = 1− 1n ≈ 1− 1
log2(1+1
2−r ).
Therefore, for large n, the Cookie Monster coefficientapproximates a function of r .
![Page 58: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/58.jpg)
Bounds Depending on Growth of Sequence
Theorem
The real root of xn = xn−1 + xn−2 + · · · x + 1 may beapproximated as 2− ε where ε = 1
2n−1 .
Thus, n ≈ log2(1 + 12−r ), so n−1
n = 1− 1n ≈ 1− 1
log2(1+1
2−r ).
Therefore, for large n, the Cookie Monster coefficientapproximates a function of r .
![Page 59: The Cookie Monster Problem - MIT Mathematicsmath.mit.edu/research/highschool/primes/materials/2013/conf/3-1-Braswell.pdfThe Problem Cookie Monster is always hungry. We present him](https://reader034.fdocuments.us/reader034/viewer/2022050509/5f9a69f61a5b816c0d2237db/html5/thumbnails/59.jpg)
Bounds Depending on Growth of Sequence
ConjectureThere exist m jars in a recursive sequence with characteristicequation of the form xn = xn−k1 + xn−k2 + · · · xn−km such thatCM(S) = dqme where q is any rational number between 0 and1.
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Future Research
CM(S) = n characterizationExplore more interesting sequences with regard to CM(S)
Is computing CM(S) an NP-hard problem?Introduce more monsters, more dimensions of cookies, ora game to play with the cookies
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Acknowledgements
Tanya KhovanovaPRIMESMy parents
Thanks for listening!