Welcome to the 2014 Hampshire College Summer Studies in Mathematics INTERESTING TEST Try to limit the time you spend on the IT to 5 or 6 hours, indicating as afterthoughts later ideas (which can be sent separately). We donʼt want this application to become a career or to interfere with other studies. One of the reasons youʼve applied to HCSSiM is that you enjoy sharing mathematics—but please do not discuss these questions with others. Instead, encourage those with whom youʼd like to discuss IT problems to apply to the Summer Studies promptly. The IT is not intended to test your Internet search skills; do not use the Internet for IT. Using a calculator is ok and occasionally necessary. We expect much of the material on the IT to be unfamiliar, and you are not expected to complete all of IT's parts. Email us if you suspect that there is an error on the IT or if a problem remains incomprehensible to you. Show your reasoning and computations clearly, using spaces provided, the backs of pages, and additional sheets as needed. We do not need to see your scratch work and we will tire quickly of struggling to find strands of logic or complete sentence. Spend more time doing your work than digitizing it. Feel free to add generalizations, speculations, and questions. Enjoy the IT and return it real soon. Let us know if you canʼt finish within 17 days. Important advice, which you can use at no charge, for the rest of your life in academia: When you do send things electronically, the subject of the email and the name of each and every single attachment should identify you, approximate the date, and give a hint as to the content. Bear in mind that we are getting correspondence from M people; if each person sends K attachments, that's approximately a lot of attachments, some of which are bound to get lost (even if names and dates and content are used lavishly); we don’t feel obligated to open emails titled “my work”; please combine things into as few pdfs (e.g., one) as possible. If sending electronically please email to sgoff@hampshire.edu. Return to David C. Kelly or Susan Goff HCSSiM, Box NS Hampshire College 893 West St. Amherst, MA 01002-3359 When you have completed your work on the test, please use the space below for your comments. Which, if any, of the problems had you considered before? Which did you particularly enjoy? Particularly not enjoy? Thanks. Other comments:

Name: Sponsor: School:

HOW DO YOU DO IT?[You’re permitted, for this IT, to violate the magician’s oath by revealing to non-magicians how you do your tricks.]

Igy Powell: Hi, IT arithmetamagician. This morning, right after breakfast, I picked 17 positive integers less than 289, and I named thema1, a2, ..., a17. Can you guess what my 17 numbers are?

You: Of course not, but I will tell you what your 17 numbers are if you find one number for me.

Igy: No way! I give you one number and you tell me all 17 of my numbers? This I have to see.

You: OK, here... I’ve written down 17 numbers of my own, c1, c2, ..., c17. Please tell me what the sum a1c1 + a2c2 + ...+ a17c17 is.

Igy multiplies, adds, and mutters: That sure is a lot of arithmetic; OK, I’ve found all 17 products and now I just add them up and ... Oh,wow! I get it! Now I see how you know all 17 of my numbers!

How do you do it?

You: Hey, Willy Gope. If it’s not too much trouble, please pick one number from each of these five columns. Don’t tell me which numbersyou choose, but tell me the product of their final digits.WG: I don’t know why you care: but my 5 numbers’ final digits multiply to 490.

You: The sum of the five numbers you chose is 2822.

WG: I’ll check ... hey! that’s right. Wow! Wait a minute.... I know what you did: you justmemorized the sums for each possible product.You: No way! There must be more than 17 sets of numbers you could have picked; that’s toomuch memorization; and I didn’t use notes, a calculator, or even paper and pencil.

17 457 277 187 597215 655 475 385 795413 853 673 583 993512 952 772 682 1092611 1051 871 781 1191

How do you do it?

You: Hey, Steve Neen. I’m glad you’re here with your calculator. How many digits does it display?

SN: Thanks for asking. I was wondering about that last Tuesday, so I investigated and found out: nine.

You: OK, good. So here’s what you do: pick a 1-digit number, then multiply by another 1-digit number, and another, and ... keep goinguntil you get to a product with 9 digits. When you reach a 9-digit display, delete one of the digits and tell me what the other eight digitsare in any order you want. I’ll then tell you what digit you deleted.

SN: Let me get this straight: I get 9 digits, remove one, and tell you the other eight after scrambling them up?

You: That’s ri...on ⇥F?SNi0;` [IT interrupts to bring you alternative endings.] After you’ve predicted the missing digit...

1st scenario: SN (or whoever is speaking in italics): You got it! That’s the digit I left out.

How do you do it?

2nd scenario: You: Ooops. I’m sorry I got it wrong. Let’s try this: think hard about the digit you removed. [pause, while SN orwhoever it is thinks in italics, hard] You: I’m still not getting anything; nothing; are you sure you’re concentrating?

What went wrong?

How will you salvage the trick?

THE GREAT INTERNATIONAL TIC-TAC-TOE COMPETITION

Each of the N Great International Tic-Tac-Toe Competition (GIT 3C) teams played each of the other N-1 teams.

Surprisingly, there were no ties: for each pair of teams Y and P , Y beat P (Y ! P ), or P beat Y (P ! Y ), but not both.

Team ↵ won w↵ games and lost l↵ games.

For each k = 1, 2, ..., N , wk + lk = ..........; and

NPk=1

wk �NP

k=1lk = ............ .

A team T is domineering i↵ for any other team S, T ! S; and

T is quasi� domineering i↵ for any other team S either T ! S

or there exists an intermediary team U , such that T ! U and U ! S.

Can a competition have more than one domineering team? .......

If wT5 = 1 in a 17-team competition could T5 still be quasi-domineering? .......

Which teams in the competition to the right are quasi-domineering? ... ... ... ... ... ...

Could a 289-team competition have exactly one quasi-domineering team? .......

Could all 3 teams in a 3-team competition be quasi-domineering? .......

What do you think about a team that beat a domineering team?

.... .... .... .... .... .... .... ’.... .... .... .... .... ....!

If no team beat more teams than c, prove that c is quasi-domineering.

Please make the win vector

< wA, wB , wC , wD, wE , wF >of the competition above be

< 1, 2, 2, 2, 4, 4 >.

If a competition has no domineering team, prove that it must have at least 3 quasi-domineering teams.

Prove that (NPk=1

(wk)2 =NPk=1

(lk)2.

(*) (nP

k=1ak)2=

nPk=1

a3k SOME SQUARED SUMS ARE SUMS OF SAME CUBES

Especially if you remember that 43=4x4=4+4+4+(2+2), this picture may convince you

that 1 + 2 + 3 + · · ·+ n)2 = (Pn

k=1 k)2 =

Pnk=1 k

3 = 1 + 8 + 27 + · · ·+ n3.

You may use this fact as we discover other ways to create lists < a1, a2, ..., an >

with the property that (nP

k=1ak)2 =

nPk=1

a3k.

Here’s Liouville’s 1857 algorithm: Pick an M (we’ll try M = 153); then find M 0s divisors;finally, count the number of divisors each of M 0s divisors has; those numbers form the list:

153’s divisors 1 3 9 ...... 51 153

their divisors 1 1,3 1,3,9 1,..... 1,3,.....,51 1,3,9,....,51,153

number of ’em 1 2 3 2 4 6It’s not hard to confirm that (1 + 2 + 3 + 2 + 4 + 6)

2

= ....... = 1 + 8 + 27 + 8 + 64 + 216.

It’s hard not to confirm that (1 + 2 + 3 + 2 + 4 + 6)

2

= ....... = 1 + 8 + 27 + 8 + 64 + 216.

What list is generated by M = pn when p is prime?

What list is generated by M = 2014 = 2x13x53?

How about M = 200 = 23x52?

Why does this process work?

We’ll start with L= < 2, 3, 5 > for this 2nd way to find solutions to (*), but any other list would work as well.

We find the quotient

Pa2L

a3

(Pa2L

a)2 =8+27+125(2+3+5)2 = 160

100 =85 . So (!) we create an array of 8 5s, < 5, 5, 5, 5, 5, 5, 5, 5 >, and

multiply each of the 5s by each of the members of L to make a final list that contains 8 10s, 8 15s, and 8 25s.We find that (10+10+10+10+10+10+10+10+15+15+15+15+15+15+15+15+25+25+25+25+25+25+25+25)2 =103 + 103 + 103 + 103 + 103 + 103 + 103 + 103 + 253 + 253 + 253 + 253 + 253 + 253 + 253 + 253.

Test this method with another initial L.

Why does this process work?

HOW DID PECKING CHICKENS MAKE IT TO IT? (*)

For each pair of chickens , and d, , pecks d (,! d), or d pecks , (d ! ,), but not both.

A chicken is a totalbully i↵ it pecks all the other chickens; and it is a quasi� totalbully i↵ for any chicken it doesn’t peck there is

a third chicken such that it pecks that third which then pecks the one it didn’t peck at first–in other words, a quasi-total bully is 1 or 2 pecks

away from every other chicken–but actually it could be 2 peckss away from itself, unless otherwise, or, you know, like that ...

As you’d expect, y and p want to join this flock of chickens, all of which are quasi-total bullies.Can you specify pecking relations among y, p, and the old flock that will maintain the all-quasi-total-bullies-ness of the new and improved flock?

For which values of Q does there exista flock of Q chickens all of which are quasi-total bullies?

POWERFUL TEAMS

On a recent IT page, Igy Powell saw 2 lists of numbers < w1, w2, ..., wN > and < l1, l2, ..., lN > with equal sums and equal sumsof squares. Igy wondered if the lists < 1, 8, 9 > and < 3, 4, 11 > also had that property, but then he accidentally saw that< 1, 5, 9, 17, 18 > and < 2, 3, 11, 15, 19 > satisfied

Pk(ak)

p =Pk(bk)

p for p = 0, 1, 2, 3, and 4.

That amused Igy, so he started to think in base 2, and he decided that 00 would be 1, at least for a few days.

EVil ODious

0 10,3 1,2

0,3,5,6 1,2,4,7

0,3,5,6,9,10,12,15 1,2,4,7,8,11,13,14

0,3,5,6,9,10,12,15,17,18,20,23,24,27,29,30 1,2,4,7,8,11,13,14,16,19,21,22,25,26,28,31

0 1 2

0,5,7 1,3,8 2,4,6

0,5,7,11,13,15,19,21,26 1,3,8,9,14,16,20,22,24, 2,4,6,10,12,17,18,23,25

(*) Why is this problem appropriate for this IT?Well, a competition in whicheveryone plays everyone elseis sometimes called a round-robintournament. Robins are birds, and so are chickens.

When you have completed your workon the 2014 Hampshire College

Summer Studies in MathematicsInteresting Test, please return to the IT’s cover page and

share some of your feelings about it.Thanks.