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Draft. Do not cite or quote, butdodrink a metric ton of water today.Problem Set 1: Shuf"lingWelcome to PCMI. We know you"ll learn a lot of mathematics

here-maybe some newtricks,maybe some new perspectives onthings with whichyou"re already familiar.Afewthingsyou should know about how the class is organized.

Don"tworryaboutansweringallthequestions.

Ifyou"re

doing that, we haven"t written the problem sets correctly.Some of the problems have yet to be solved. Those are thereallyfun ones.Don"t worry about getting to a certain problem number. Some participants have been knownto spendthe entire session working on one problem(and perhaps afew of its extensions or consequences).

Stop and smell the roses.

Gettingthe correctanswer to a

questionis nota be-alland end-all inthis course.How doesthe questionrelateto othersyou"ve encountered?

How do others think about this question?

Respect everyone"s views.

Remember that you have

somethingtolearnfrom everyone else.Remember that everyone works at a different pace.

Teach only if you have to.

You may feel thetemptation

toteach othersinyourgroup.Fight it!We don"tmean you shouldignore people butgive everyonethe chanceto discover. If youthinkit"s a goodtimetoteach quadratic reciprocity, think again: the problems shouldleadto the appropriate mathematicsrather thanrequiringit.

The same goesfor technology:problems shouldlead

to appropriate uses of technology rather thanrequiringit. Each day has its Stuff.There are problem categories: Im- portantStuff,NeatStuff, Tough Stuff.Check out Important Stuff first.All the keymathematicsinthe course can be found and developedinImportantStuff. It"sImportantDo something really inter- esting, and maybe every- one else will be asked to try it the next day! That also means we want to hear about what you"re doing. Stuff! Everything else is just Neat or Tough. If you didn"t get throughtheImportantStuff,we noticed. . .and we"ll changethe course(yes, literally) to account for it.Every problem set is based on whathappenedinthe previous set, and what happened in the previousclass. Whenyou get to Problem Set3,come back andreadthisWill you remember?

Maybe . . .introduction again.PCMI 20121

Problem Set 1: Shuf"lingDraft. Do not cite or quote, butdodrink a metric ton of water today.Opener Let"s watch a video. Don"t worry, it"s only like 2 minutes long.

1.Doesthe perfectshuffle workforotherdeck sizes? Ifnot,

why not? If so, what stays the same and what changes? 2. Evelynisthinking ofa positiveinteger,and because she"s a mathteachershe callsitx.What information would you know aboutxbased on each statement? a.3xhas last digit 4 b.7xhas last digit 4 c.4xhas last digit 4 d.5xhas last digit 4 3. a. Whatnumber is9101+9100+410-1+410-2?The little 10 here means the number is in base 10. Bases will generally be given when they seem needed, and we"ll try not to be confusing.b. Ben"s favorite number is802.1110. Write it as a sum of powers of 10. 4. a.

Whatnumber is133+032+231+030+13-1?

b.Carol"s favorite base-3 number2110.23. Write it as a sum of powers of 3.Surely you came to PCMI armed with your favorite number in each base.c.Convert 2110.23to base 10. 5. Write each numberas a decimal.Write each numberas a decimal. Write each number as a decimal.These directions either terminate or repeat. a. 12 b. 150
c. 19 d. 29
e. 99
f. 113

2PCMI 2012

Draft. Do not cite or quote, butdodrink a metric ton of water today.Problem Set 1: Shuf"ling6.Hey,wejustmet you,andthisis crazy;buthere"s some

numbers, so make them base three.

We missed you so bad. We

missed you so, so bad. a.9 b.13 c.242d. 19 e. 113
f. 1242

7.Write each number as a base-3 "decimal".

a. 19 b. 12 c. 22
d. 32
e. 113
f. 1242
8. Underwhatcircumstances willa base-10decimal repeat?

9.Under what circumstances will a base-3 decimal repeat?

10.Therepeating decimal.002means.002002002. . . .But

whatnumber isit? Thatdepends onthebase!Acethis problem by finding the base-10 fraction equal to .002 in each given base.

Psst: You did some work

on base 3 already. Ace this base problem, and you"ll see the sine.a.base 3 b.base 4 c.base 5d.base 7 e.basen f.base 2?! 11.

We overheard Sara andJoe debating aboutwhetheror

not the number.99999...was equal to1.WhatdoyouWe didn"t hear who was arguing each side, we mostly just ran away. think?Come up with a convincing argument,andif you already know one, come up with a different one! 12. a. expansion of 1n repeats in 3 digits or less. b. expansion1n repeats in 4 digits or less. 13.

Write13 and 242in basep3instead ofbase 3.Hee hee

hee. Or maybe this turns out to be totally awesome!PCMI 20123

Problem Set 1: Shuf"lingDraft. Do not cite or quote, butdodrink a metric ton of water today.????? ????

14.Aziz has a cube,and he wantsto color itsfaces with

two differentcolors.How manydifferentcolorings are possible?By "different"we meanthat you can"tmake one look like the other through a re-orientation.

15.What about edges?

16. a. Convert 13 to base32 b.Convert 13 to base.4PCMI 2012

Draft. Do not cite or quote, butdocome to class on time tomorrow! Oh, right. Happy 4th!Problem Set 2: Slhi"unfgProblem Set 2: Slhi"unfg

Today"s title comes from the

German word for "shuffling".Opener

Can perfect shuffles restore a deck with 9 cards to its original state? If so, how many perfect shuffles does it take? If not, why not?

Split the cards 5-and-4, and

1.Working with your table, fill in a whole lot ofthistable:The file isinthe computer!

Oh, only one computer per

table, please.http://www.tinyurl.com/perfectshuffle

2.Findthe units digitofeach annoying calculation.PutTheunits digitof 90210 is

0, matching Brenda"s IQ.those calculators away!

a.2314426+573234 b.(46+1)(46+2)(46+3)(46+4)(46+5)You down with ZPP? (Yeah, you know me!)c.714734774794 3.

Find allpossiblevaluesfor the units digitofeach

person"s positive integer. a.

Amy: "Whenyou add5 to mynumber, itendsin

a 2." b.

Brandon: "Whenyou multiplymynumberby3, it

ends in a 7." c.

Carmen: "Whenyou multiplymynumberby6, it

ends in a 4." d.

David: "Whenyou multiplymynumberby 5, itends

in a 3. Yup." 4.

Unlike"base10", inmod10the onlynumbers arethe

remainders when you divide by 10. In mod 10, 6 + 5 = 1This is sometimes called modular arithmetic. Clock arithmetic is mod 12. Four hours from now, it will be four hours later than it is right now.

That last one says box to

the fourth power, by the way.because 1 is the remainder when 6 + 5 is divided by 10.

Answer all these questions in mod 10.

a.2+2=b.34=c.+5=2d.4=2 e.5=3 f.4 =1 5. Repeat the previous problem,except thistime dotheGood news: there are only

7 numbers in mod 7. Bad

news: in mod 7, every Monday is the same.arithmetic inmod 7instead of mod 10.PCMI 20125

Problem Set 2: Slhi"unfgDraft. Do not cite or quote, butdocome to class on time tomorrow! Oh, right. Happy 4th!6.Go backtothe bigtablethatwe all filledintogether.Some examples of bad

patterns: "I noticed that most of the values in the table were numbers." "All the digits in the table could also be found on a computer keyboard." "Some of the numbers in the table were bigger than others, while others were smaller." "The leftmost column increased by 1 each time." "Purple Gingham." "New Mexico got to have the first two columns because they"re in a different time zone."

We"ve provided some blank

space below for you to write your own bad patterns . . .What patterns do you notice?

7.Write each fraction as a base-10 decimal.

a. 15 b. 125
c. 17 d. 37
e. 27
f. 67
g. 113
h. 213
8. Write each base-10 fraction as a base-3 decimal. Some of the answers are alreadygiven, in which case-awesome! a.113 =0.002 3 b. 213
c. 313
d. 913
e. 1013
f. 17 =0.010212 3 g. 37
h. 97
i. 67

9.Write the base-10 decimal expansion of

1142857

10.

Marvin wonders whatkinds ofbehaviorcan happen

withthe base-10decimalexpansion of1n.Be as specific as possible! 11. If1nterminatesin base10,explain howyou couldYou now know the entire plot of the horrible movie

Terminator 1/4: 0.25 Day.

determinethelength of the decimalbased onn,without doing any long division. 12.

Robyn wonders whatkinds ofbehaviorcan happen with

the base-3 decimal expansion of1n .6PCMI 2012

Draft. Do not cite or quote, butdocome to class on time tomorrow! Oh, right. Happy 4th!Problem Set 2: Slhi"unfg13.We overheard Sara andJoe still yelling aboutwhetherA little ditty, bout Sara and

Joe. Two mathematical kids

doin" the best that they know. ornot the number.99999...was equal to1. Isit?Be convincing. 14. a.

Supposeab=0in mod 10. What does this tell you

aboutaandb? b.

Supposecd=0in mod7.Whatdoesthistell you

aboutcandd?It tells you thatathroughd hog the spotlight too much.

No love for the middle of the

alphabet in algebra.15. Investigate shuffling decks ofcardsintothree piles instead of two.Whatarethe options?Doesit "work" like it does with two piles? 16. a.

Investigatethe base-10decimalexpansions ofn41for

different choices ofn. What happens? b. Investigatethebase-3expansions ofn41fordifferentThe fractionn41 is still in base 10 here, so don"t convert 41 to some other number.choices ofn. What happens? 17. a. expansion of 1n repeats in exactly 4 digits. b. expansion of1n repeats in exactly 5 digits.

18.Write 223 and 15.125 in base 2. Then write them in basep2. How cool is that?!

While this problem is

cooler than most math, the

Supreme Court recently

ruled that math cannot actually be cool.????? ???? 19. Aziz has a cube,and he wantsto color itsfaces with two differentcolors.How manydifferentcolorings are possible?By "different"we meanthat you can"tmake one look like the other through a re-orientation. 20. Barbara has an octahedron,and she wantsto color its vertices withtwo differentcolors.How manydifferent colorings are possible?By "different"we meanthat you can"tmake onelooklikethe other through are- orientation.

21.What about edges?Edges? Edges? We don"t

need no stinkin" edges! 22.

Find all solutions tox2-6x+8=0in mod 105 without

use of any technology. There"s probably more.PCMI 20127

Table for Problem Set 3

This table gives the number of shuffles necessary to restore ann-card deck to its original state using the shuffling style from Problem Set 1.# cards# shuffles 42
64
83
106
1210
1412
164
188
2018
226
2411
2620
2818
quotesdbs_dbs47.pdfusesText_47

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