Table Of ContentTen Essays on Fizz Buzz
Meditations on Python, mathematics, science,
engineering, and design
Joel Grus
Thisbookisforsaleathttp://leanpub.com/fizzbuzz
Thisversionwaspublishedon2020-07-06
ISBN978-0-9824818-2-0
ThisisaLeanpubbook.LeanpubempowersauthorsandpublisherswiththeLean
Publishingprocess.LeanPublishingistheactofpublishinganin-progressebook
usinglightweighttoolsandmanyiterationstogetreaderfeedback,pivotuntilyou
havetherightbookandbuildtractiononceyoudo.
©2020JoelGrus
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
1. 100PrintStatements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
AlgorithmandAbstraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
ReusabilityandChangeability . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
TestabilityandFizzBuzz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
TestabilityBeyondFizzBuzz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
GeneratingtheprintStatements . . . . . . . . . . . . . . . . . . . . . . . . . 10
2. if/elif/elif/else . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
MindtheOrder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
DivisibilityandModulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
“We’reAdders,WeNeedLogstoMultiply” . . . . . . . . . . . . . . . . . . . 18
SomeOtherChecks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
SomeWell-KnownTricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Eliminatingelif . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3. TheCycleof15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
EquivalenceClasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
NoneasaSentinel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
TruthinessandLogic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
dict.get . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4. Euclid’sSolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
PrimeNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
PerformanceOptimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Factorization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
WhatPrimeFactorizationHastoDowithFizzBuzz . . . . . . . . . . . . . 42
CONTENTS
GreatestCommonDivisorsandLeastCommonMultiples . . . . . . . . . . 43
gcdandFizzBuzz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Euclid’sAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5. Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
SinandCos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
int . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
PiandTau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
OperatorOverload . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6. ABigNumber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Decimal,Binary,andHexadecimal . . . . . . . . . . . . . . . . . . . . . . . . 58
ASecretEncoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
PrefixCodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
HuffmanCoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
DataModeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
OurDataModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
ConstructingtheCode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Digression:MutableDefaultValues . . . . . . . . . . . . . . . . . . . . . . . . 73
Digression:heapq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
“AccidentallyQuadratic” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
RegularExpressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7. InfiniteIterables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Iterables,Iterators,andLazyInfiniteSequences . . . . . . . . . . . . . . . . 84
itertools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
zipvsmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
PuttingItAllTogether . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Bonus:PowerFizz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
8. RandomGuessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
random . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
PseudorandomnessandtheSeed . . . . . . . . . . . . . . . . . . . . . . . . . 99
CreatingaPseudorandomGenerator . . . . . . . . . . . . . . . . . . . . . . . 100
AdvancesintheSameIterator . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
ValuesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
CONTENTS
ArgumentUnpacking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
TheCycleof15,Redux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Wheredid23_977_775comefrom? . . . . . . . . . . . . . . . . . . . . . . . . 109
Puttingitalltogether . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
9. MatrixMultiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
n-DimensionalArrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Digression:Recursion,Iteration,andTailRecursion . . . . . . . . . . . . . . 115
NumPy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
WorkingWithArrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
FizzBuzzasaMatrixMultiplicationProblem. . . . . . . . . . . . . . . . . . 123
HowDidIChooseThoseWeights? . . . . . . . . . . . . . . . . . . . . . . . . 125
10. FizzBuzzinTensorflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
MachineLearning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
FizzBuzzasaMachineLearningProblem . . . . . . . . . . . . . . . . . . . . 128
LearningandGeneralization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
FeatureEngineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
DeepLearning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
HowDoesItWork? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
AbouttheAuthor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Introduction
FizzBuzzisthefollowing(simple)problem:
Print the numbers from 1 to 100, except that if the number is divisible by
3,insteadprint“fizz”;ifthenumberisdivisibleby5,insteadprint“buzz”;
andifthenumberisdivisibleby15,insteadprint“fizzbuzz”.
It originated as a children’s game, but has since taken on a new life as a lowest-
common-denominatorlitmustestforassessingcomputerprogrammers.
If you are an experienced programmer, it is an extremely easy problem to solve.
Becauseofthis,ithastakenonathirdlifeastheprototypicalbad interviewproblem.
Everyone knows that it’s the question you ask people to make sure that they’re not
completelyincompetentas programmers. Accordingly,if yourinterviewerasks you
to solve it, he’s suggesting he thinks it possible that you’re completely incompetent
asaprogrammer.Youwouldnotbewrongtofeelinsulted!
Myassociationwiththisproblembeganin2016,whenIwroteablogpostcalledFizz
Buzz in Tensorflow, the (possibly fictional) story of one such insulted programmer
whodecidedtoshowuphisinterviewerbyapproachingFizzBuzzasadeeplearning
problem. This post went modestly viral, and ever since then I have been seen as a
thoughtleaderintheFizzBuzzspace.
Accordingly,overtheyearsIhavecomeupwithand/orcollectedvariousotherstupid
and/orcleverwaysofsolvingFizzBuzz.Ihavenotbloggedaboutthem,asIamnot
the sort of person who beats a joke to death, but occasionally I will tweet about
them,andrecentlyinresponsesomeonesuggestedthatIwriteabookon“100Ways
ofWritingFizzBuzzinPython.”
Now,Icouldprobablycomeupwith100waysofsolvingFizzBuzz,butmostofthem
would not be very interesting. Luckily for you, I was able to come up with 10 that
areinterestinginvariousways,eachofwhichturnedouttobeagoodlaunching-off
point for (sometimes meandering) discussions of various aspects of coding, Python,
Introduction ii
Fizz Buzz, mathematics, software design, technical interviewing, and various other
topics.Hence“TenEssaysonFizzBuzz”.¹
In many ways this is a strange book. Its goal is not to teach you a specific field or a
specific technology. I hope you will learn a lot from reading it, but it’s not really a
book that you’d read in order to learn anything in particular. Most technical books
areaboutspecifictechnicaltopics;thisonesortofisn’t.
Nonetheless, it is a technical book. Each essay contains code that implements a
different solution of Fizz Buzz. Each essay uses code to illustrate its ideas. Each
essay represents my current best thinking about how to solve problems using code.
Ifyouhaveacodingjob,youshouldfeelnoreluctancetoexpensethisbooktoyour
employer.
AsIprimarilycodeinPython,allofthesesolutionswillbeinPython.Someofthem
will use features that were only introduced in Python 3.6, and some of them will
demonstrate concepts and techniques that are mostly specific to Python. On some
levelthisisdeeplyaPythonbook.However,mygoalwastowriteabookthatwould
be interesting and enlightening even if you are not a Python programmer. I’ll leave
ittoyoutojudgehowsuccessfulIwas.
IfIhaveexploredmorewaysofsolvingFizzBuzzthanothers,itisbecauseIstoodon
theshouldersofgiants.The“if/elif/elif/else”solutionisthecanonicalone;the“100
printstatements”solutionisobvious;the“randomguessing”solutionIlearnedabout
from a Stack Overflow question; the remaining solutions I’m pretty sure I came up
withmyself,althoughformanyofthemitisanearcertaintythatotherpeoplehave
independentlycomeupwithsimilar(ifnotbasicallyidentical)solutions.That’show
programmingworks.
Thesolutionsfromthisbookareallavailableatgithub.com/joelgrus/fizzbuzz,incase
youdon’tfeelliketypingthemintothecomputeryourself.
¹AfterIstartedwritingthisbookIdiscoveredthereisablogpost“TwentyWaystoFizzBuzz”,veryfewofwhich
overlapwiththese.
1. 100 Print Statements
FizzBuzzoriginatedasagameforchildren.Theideawasthatthechildrenwouldsit
inacircleandgoaroundinsequencecallingoutthenextnumber;substituting“fizz”
or “buzz” or “fizzbuzz” as appropriate; punishing mistakes according to house rules
withvaryinglevelsofcruelty.
Well, if children can come up with the correct outputs without using a computer,
thensocanwe.Thissuggestswhatispossiblytheleastimaginativesolution–figure
outthecorrectoutputsbyhandandexplicitlyprinteachone:
print('1')
print('2')
print('fizz')
print('4')
print('buzz')
print('fizz')
print('7')
print('8')
print('fizz')
print('buzz')
print('11')
print('fizz')
print('13')
print('14')
print('fizzbuzz')
print('16')
print('17')
print('fizz')
print('19')
print('buzz')
print('fizz')
100PrintStatements 2
print('22')
print('23')
print('fizz')
print('buzz')
print('26')
print('fizz')
print('28')
print('29')
print('fizzbuzz')
print('31')
print('32')
print('fizz')
print('34')
print('buzz')
print('fizz')
print('37')
print('38')
print('fizz')
print('buzz')
print('41')
print('fizz')
print('43')
print('44')
print('fizzbuzz')
print('46')
print('47')
print('fizz')
print('49')
print('buzz')
print('fizz')
print('52')
print('53')
print('fizz')
print('buzz')
print('56')
print('fizz')
100PrintStatements 3
print('58')
print('59')
print('fizzbuzz')
print('61')
print('62')
print('fizz')
print('64')
print('buzz')
print('fizz')
print('67')
print('68')
print('fizz')
print('buzz')
print('71')
print('fizz')
print('73')
print('74')
print('fizzbuzz')
print('76')
print('77')
print('fizz')
print('79')
print('buzz')
print('fizz')
print('82')
print('83')
print('fizz')
print('buzz')
print('86')
print('fizz')
print('88')
print('89')
print('fizzbuzz')
print('91')
print('92')
print('fizz')
