Hampshire College Summer Studies in Mathematics

for mathematically talented high school students

HCSSiM was featured in the Notices of the AMS

Read the article Supporting a National Treasure by Allyn Jackson in PDF form here, you can also read an addendum to Jackson’s piece featured in the February 2004 Notices of the AMS also in PDF form here.

Students say:

“This program is a really interesting experience. First, people over here are different from other mathematicians because not all of them are crazy and all of them are smart. Secondly, there is no competition at all and therefore no hatred.” (Mikhail Shklyar, '02)

“Although math for me didn’t really begin here, it is at Hampshire that I find the true meaning of mathematics.” (Wing Mui, '00)

“In one week at HCSSiM, I have learned more math than I did in 2 years in High School.” (David Levitt, '01)

“The people I’ve met and the experiences I’ve had here have shown me a whole new side to math and it’s really cool!” (Daniel Studenmund, '03)

In the operation of the Summer Studies and in the selection of student and faculty participants, Hampshire College will not discriminate against any person on the grounds of race, creed, color, sex, age, sexual preference, physical disability or national origin. The Hampshire College Summer Studies in Mathematics is an equal opportunity and affirmative action employer.


The Hampshire College Summer Studies in Mathematics is a rigorous math program that is both demanding and expanding. Participants are expected to spend a major portion of each day actively engaged in learning, doing, and sharing mathematics. The daily schedule includes 4 hours of class meetings each morning, Monday-Saturday, the Prime Time Theorem & jeux tables de multiplication in the afternoon, and problem sessions in the evening. Afternoons are devoted to rest, recreation and informal study.

Initially, Summer Studies participants will be partitioned into workshops, each led by a college or university professor and one or two graduate or undergraduate students. Each workshop will actively and intensively investigate problems from many areas of mathematics, with emphasis on unifying themes, recurrent patterns, and fruitful modes of inquiry. The particular topics are not predetermined by syllabus, but will be chosen by the instructors to challenge the interests and abilities of the participants. All students will have access to computer instruction (including TrueBasic and Mathematica) during the workshop, and will be encouraged to use Hampshire’s facilities throughout the summer (but not for games or network cruising).

Midway through our six-week session, students and staff will regroup into “maxi-courses” and “mini-courses” for in-depth studies of particular problems and fields. Students will be able to select topics such as combinatorics, number theory, complex numbers, probability, four-dimensional geometry, fractals and chaos, graph theory, topology, and cellular automata.

Guest lecturers may include recent winners in the Westinghouse/Intel Science Talent Search and other national competitions, authors of MAA and other contest problems, and other mathematicians from colleges, universities, and industry. We’ll have a weekly program of math films to teach, stimulate and to inspire.

Neither grades nor credit are offered. Each student will be asked several times to evaluate his or her own growth; these introspections, together with evaluative comments by instructors, will form the basis for a report which will be made available, at the student’s request, to home schools, college admissions offices, and the like. Participants are also encouraged to make at least one mathematical presentation when they return to their schools.

Your teachers are invited to visit summer studies. Parental visits are discouraged, and other visitors require the approval of the director eight months in advance.

Participants in the Summer Studies will not have time to maintain part-time jobs or similar obligations. There will, however, be a lot of program-related work; we prepare some of our own meals on Sundays, we keep our living and working places clean, we publish a weekly program journal, we show films, we organize trips, picnics, tournaments (in chess, Go, bocce, backgammon, frisbee…) and musical groups. Boredom has never been a complaint at the Summer Studies.

Most of the faculty of the Summer Studies as well as all students will have single rooms in our dormitory. In the past summers, the continual close contact among students and staff has contributed greatly to the creation of a friendly, cooperative, and productive academic and social community.

The Hampshire College dining hall (“SAGA”) provides “all you can eat” at each meal. There is an extensive salad bar. Students will have access to refrigerators and a stove. Special diets (such as kosher or vegetarian) can be arranged.

Hampshire recreation facilities include the Robert Crown Center (which houses a sauna, a gymnasium, a swimming pool, and a climbing wall), as well as the outdoor courts for volleyball, tennis, bocce, and basketball; fields for soccer, Frisbee, and softball, and a pond, as well as many nearby woods, trails, hills and mountains.

Hampshire College, which opened in 1971, is an accredited, independent, experimenting, liberal arts college. Summer Studies students will enjoy the full use of the campus on 550 acres of woods and former farmland. The air-conditioned classrooms of Adele Simmons Hall will be used for our classes, and participants will have access to the College’s libraries, labs, and computer facilities.

Out neighboring institutions in the Pioneer Valley of the Connecticut River – Smith, Mount Holyoke, and Amherst Colleges, and the University of Massachusetts – all have academic, social, cultural, and recreational programs during the summer.

Our Sundays are spent relaxing, biking, hiking, or field-tripping to Boston, to the Berkshire Music Festival at Tanglewood, or to other New England points of interest. These activities are regarded as parts of the program.

The Hampshire College Admissions officers will be available for consultation, as will be representatives from other colleges and universities. Each summer, many alumni of previous HCSSiM’s pay visits and share experiences from their colleges and summer jobs.

Participants are encouraged to bring bikes, musical instruments, athletic equipment, games, puzzles, calculators and computers (either laptop or desktop), and good working habits. Explosives, illegal drugs, firearms, cars, pets, noisy stereos, and sloppy thinking are not permitted.

For many of the 31 previous sessions of the Hampshire College Summer Studies in Mathematics we have enjoyed the support of the National Science Foundation. The NSF is no longer funding programs such as ours. The American Mathematical Society, Hampshire College, alumni of HCSSiM, and other friends have provided support for financial aid funds and some other costs of recent Hampshire Summer Studies. We continue to seek additional sources of assistance (in kind contributions, as well as money) and welcome any suggestions you might have.

The cost for the program is $1989 for room and board, financial aid is also available.

For printed information, please write:

David C. Kelly, Director
Hampshire College Summer Studies in Mathematics
Box NS
Hampshire College
Amherst, MA 01002

A typical Day

Here’s a typical day (Monday - Friday) at HCSSiM:
You’ll get up, and…
go to breakfast at around 8 a.m.
be in class from 8:34 - 12:34 (during the first three weeks, this is all one four-hour class, and during the last three weeks it’s split into a 2.5-hour class and a 1.25-hour class)

eat lunch, and then do stuff during the afternoon. Maybe you’ll play frisbee (even with Kelly!),
or play bocce,
or work on the program journal, or take a nap.
You’re likely to see wildlife about, such as a chipmunk or baby birds.
Then at 5:00 (17:00), you’ll attend Prime Time until 6:00.

(these are Susan Landau of Sun Microsystems and Jim Propp of UW-Madison)
at which point you go to dinner,
then get a break until 7:34 (play frisbee during the break? even in the rain? sure!),
have problem session from 7:34 - 10:34,
hang out until 11:17 (quiet time).

Of course, there are variations on the schedule: on Wednesday evenings, we have math movies from 7:17 - 8ish, so that problem session starts a little bit late. And on Saturdays morning class ends early for special math activities, there’s no prime time, and instead of problem session we watch classic and popular movies.


There are lots of things we do that aren’t mathematics. For example, if you come to HCSSiM, you’ll hike this entire mountain range:
We also take a trip to Boston,

and often a trip to Tanglewood.

Here’s a proof of the Fundamental Theorem of Algebra:


Sometimes we go to Kelly’s house (he’s the director)

Sample Problems   

Problems which seem, at first, to lack sufficient information

  1. Each of three teams, A, B, and C, enters one contestant in each event of a track meet. In each event, p points are awarded to first place, q points for second place and r points for third. p, ,q, and r are positive integers, and p > q > r. Team A won the first event and finished with a total of 9 points; B also finished with a total of 9 points; and C won the meet with 22 points

How many events were there and which team won the second event?

  1. The circle inscribed in a triangle with perimeter 17 cm. has a radius of 2 cm. What is the area of the circle? What is the area of the triangle?

  2. Two-thirds of the way across a narrow railroad bridge, Wiilly Gope hears a train coming towards him. He knows that the trains’s speed is 45 miles per hour and that he has just enough time to reach safety at either end of the bridge. How fast can Willy run?

  3. From the third term on, each term of the sequence of real numbers a(1), a(2), a(3), a(4), a(5), a(6), a(7), a(8), a(9), a(10) is the sum of the preceding two terms; that is, a(n) = a(n-1) + a(n-2), for n = 3, 4, 5, ….10. If a(7) = 17, what is the sum of all the ten terms?

  4. Lewis Carroll posed the following problem: Two travelers spent from 2 o’clock until 9 walking along a level road up a hill and home again; their pace on the level being x miles per hour, up hill y mph, and down hill 2y mph. Find the distance walked. In Carroll’s formulation x and y were given integers. Making use of the additional assumption that the original problem was solvable, find the distance walked.


At time t = 0, 17 electrons are situated at various points on a circular track, 17 miles in circumference. Some of the electrons are moving along the track in a clockwise direction and others are moving in a counterclockwise direction, but all of the electrons are moving at the same speed: 17 miles per second.

Whenever two electrons collide, each instantly reverses direction, losing no time or velocity.

Prove that, at time t = 1 second, each of the positions initially occupied by an electron will again be occupied by an electron traveling in the same direction as the electron initially at that location.

Prove that there will be a time when all 17 electrons will have simultaneously returned to their initial positions and directions.


(this problem was posed by Professor Paul R. Halmos of Indiana University)

"My wife and I were invited to a party recently, a party attended by four other couples. Some of the 10 knew some of the others and some did not, and some were polite and some were not. As a result, a certain amount of handshaking took place in an unpredictable way, subject only to two obvious conditions: No one shook his or her own hand and no husband shook his wife’s hand. When it was all over, I became curious, and I went around the party asking each person: “How many hands did you shake?…And you? …And you?” What answers could I have received? Conceivably, some people could have said None, and others could have given me any number between 1 and 8 inclusive. That’s right isn’t it? Since self-handshakes and spouse-handshakes were ruled out, eight is the maximum number of hands that any one of the party of 10 could have shaken.

I asked nine people (everybody, including my own wife), and each answer could have been any one of the nine numbers 0 to 8 inclusive. I was interested to note, and I hereby report, that the nine different people gave me nine different answers: someone said None, someone said One, and so on and, finally, someone said Eight. Next morning, I told the story to my colleagues, and I challenged them, on the basis of the information just given, to tell me how many hands my wife shook."


A large warehouse contains a number of boxes, each of which is labeled with a natural number (positive integer); there may be boxes with the same number.

Every minute a box is removed from the warehouse. If the box has a label greater than 1, it may be replaced by arbitrarily many boxes with smaller labels. For instance, when a box labeled 17 is removed, it might be replaced by 153 boxes labeled 3, 1017 boxes labeled 8, 289000 boxes labeled 9, 3 boxes labeled 14, and (17!)17! boxes labeled 16.

Show that the warehouse will be emptied within a finite period of time.

(You may want to begin with the special case in which all the boxes initially the warehouse are labeled with 1’s and 2’s.)


Some numbers can be expressed as the sum of 2 or more consecutive positive integers (sometimes in more than 1 way):

                                                17 = 8 + 9
                                                      51 = 25 + ___ = ___ + ___ + ___,
                                                      34 = ________________________________________________________,

Other numbers, including 1, 2, …, 8, … , can not be expressed as the sum of 2 or more consecutive positive integers.

Determine, with proof, the set of numbers which can not be expressed as the sum of 2 or more consecutive positive integers.


Some numbers can be expressed as the sum of 3 or more consecutive positive integers (sometimes in more than 1 way):

                                                30 = __+ __ + __ = __ + __ + __ + __ = __ + __ + __ + __ + __,
                                                      33 = 1 0 + 11 + 12 = 3 + 4 + 5 + 6 + 7 + 8,
                                                      34 = _______________________________________________,
                                                      35 = 5 + 6 + 7 + 8 + 9 = _________________________________ .

Other numbers, including 1, 2, 3,…, 31, 32, … , can not be expressed as the sum of 3 or more consecutive positive integers.

Determine, with proof, the set of numbers which can not be expressed as the sum of 3 or more consecutive positive integers.

Just who are the faculty at HCSSiM?

Well, of course there’s the director, David C. Kelly…

Senior Staff (folks with Ph.D.s, y’know) have included

Ann Trenk, Wellesley College
Rennie Mirollo, Boston College
Rob Hochberg, East Carolina University and Rutgers University/DIMACS (other home page)
sarah-marie belcastro, Xavier University
Tom Hull, Merrimack College
Marty Arkowitz, Dartmouth College
Susan Landau, Sun Microsystems (formerly of UMass Amherst)
Steve Maurer, Swarthmore College
Art Benjamin, Harvey Mudd College
Ed Scheinerman, The Johns Hopkins University
Donal O’Shea, Mt. Holyoke College
Mike Albertson, Smith College
Robert Robson, Oregon State University
Ken Hoffman, Hampshire College
Daniel Loeb, SIG
Larry Carter, UCSD
Don Goldberg, Occidental College
Peter Milletta, last seen in Central America
Mike Spivak, Publish or Perish

Junior Staff (who were undergraduates or graduate students at the time they taught in the program) have included…

Josh Greene, Harvey Mudd College (ug), U Chicago (g)
Wing Mui, Amherst College (ug)
Leathan Graves-Highsmith, Hampshire College (ug)
Ari Turner, Harvard University (g)
Abie Flaxman, Carnegie Mellon University (g)
Lukas Fidkowski, Stanford University (g)
Matt Riddle, Carleton College (ug)
Emily Peters, University of Chicago (ug), UC Berkeley (g)
Tom Wexler, Cornell University (g)
Owen Baker, Harvard University (ug)
Aaron Archer, Cornell University (g)
Cathy O’Neil, MIT (faculty)
Jim Propp, UW-Madison (faculty)
PJ Karafiol, Walter Peyton College prep (faculty)
Steve Strogatz, Cornell (faculty)
Mike Reid, University of Arizona (faculty)
Eric Freeman, UColorado Boulder (faculty)
Pei-Wen Ting, Princeton (ug)
Jean Czerlinski, New College (ug)
Susan Moy, Hampshire College (ug)
Steve Maurer, Swarthmore (faculty)
Matt Cook, Caltech (faculty)
Tom Roby, California State University, Hayward (faculty)
Eric Lander, MIT and Whitehead Institute (faculty)
Leslie Badoian, UCal Berkeley (g)
Marcia Groszek, Dartmouth College (faculty)

(we’re not responsible for broken links on this page.)

Information about, by, and for HCSSiM alumns

coming soon – a perl database of alumn contact info for you to enter your data…

Where have HCSSiM alumns been attending college in recent years?

Here’s a sampling from students who attended HCSSiM in 1998 or later…

Awards won by HCSSiM alumns…

In recent years we’ve had an Intel winner, a Morgan Prize winner, an American Institute of Mathematics Fellow, a Guggenheim Fellow, and a UPA Ultimate Player of the Year. Two HCSSiM alumns are MacArthur Fellows. Alumns often qualify for the USAMO and do well on the Mandelbrot and the Putnam. In particular, 26 HCSSiM-ers have been on the US IMO team in 20 different years, and 16 alumns have placed in the top 5 on the Putnam in 18 different years.

Web pages or sites about HCSSiM, hosted by alumns of the program

Sara Smollett’s YP page
an outdated page by Hank Chien
YP links; click on the yellow pig to see the connection

Here are some not-as-recent alumns of the program, and what they’re doing now. (This list was generated mostly at random. If you want to be added, let us know.)

Daniel Zwillinger is a Senior Principal Engineer at Raytheon, Founder and CTO of Aztec Corporation, and the editor of the CRC Standard Mathematical Tables and Formulae.

Susan Landau is a Senior Staff Engineer at Sun Microsystems, and works primarily with cryptography.

Eric Lander is a Member of the Whitehead Institute, director of the Whitehead’s Center for Genome Research, and professor of biology at MIT. link to a mathematical article about Eric Lander

Carmen Egido is Director of the Applications Research Labs at Intel Corporation.

Danny Glasser works for Total World Domination Industries as a Software Design Engineer.

Eugene Volokh teaches law at UCLA, and is known for his work on free speech and cyberspace law.

Jessica Riskin works on history of science in the Department of History at Stanford University.

Erik Winfree is a faculty member in Computer Science and Computation and Neural Systems at Caltech.

Gregory Sorkin is a Research Staff Member in Mathematics at IBM.

Also at IBM, Martin Wattenberg.

Steven Alexander practices intellectual property law.

Amy Liu is a Physics faculty member at Georgetown University.

Lisa Randall does theoretical high-energy physics at Harvard.

Robert Lipshutz is the Vice President of Affymetrix.

Geoff Davis is a software consultant and works with wavelet image compression.

Timothy Chklovski is a Research Scientist at USC’s Information Sciences Institute.

Aaron Ellison does research in community ecology at Harvard.

Michael Miller is an Epidemiology faculty member at the University of Minnesota.

Julie Ahringer does cancer research at the University of Cambridge.

Scot Kuo is a Bioengineering faculty member at Johns Hopkins.

Bruce K Smith is a Scientific Programmer - Mathematician at Smith-Kettlewell Eye Research Institute.

Donna Crystal Llewellyn is the Director of the Georgia Tech Center for the Enhancement of Teaching and Learning.

PJ Karafiol is in the Math Department at Walter Peyton College Prep (Chicago, IL).

Loren Shure is the director of math and signal processing development at MathWorks.

Mitchell Burman founded Analytics Operations Engineering.

Eve Ostriker is an Astronomy faculty member at the University of Maryland.

Glenn Ellison is an Economics faculty member at MIT.

Susan Feigenbaum is an Economics faculty member at University of Missouri, St. Louis.

Larry Ausubel is an Economics faculty member at the University of Maryland.

Sue Hannaford is a neurobiologist at the University of Puget Sound.

Joe Kilian works at NEC Research Institute, Inc.

Bernard Beard is a Mechanical Engineering faculty member at Christian Brothers University.

Doris Stoffers is an MD/PhD Endocrinology faculty member at U Penn.

Peter Oppenheimer is a Research Engineer at Human Interface Technology Lab, U Washington.

David Steinsaltz is in the Department of Demography at UC Berkeley, and does research in mathematics, aging, music, and more…

…and, of course, we have many alumns who are mathematics faculty, including (but not limited to) Jason Fulman, Elizabeth Wilmer, Japheth Wood, Jennifer Taback, Dave Carlton, Ann Trenk, Bjorn Poonen, Judith Miller, Mike Reid, Dana Randall, Art Duval, John Sullivan, Kathryn Lesh, Susan Staples, Robin Forman, David Gillman, Terry Loring, Benji Fisher, Rennie Mirollo, Paul Feit, Jim Kelliher, Russell Lyons, Victor Wickerhauser, Daniel Ullman, Ed Scheinerman, David Manderscheid, and Larry Riddle.

…and plenty of computer science faculty as well, such as Neil Immerman, Lenore Cowen, Marie desJardins, Calvin Lin, Dina Goldin, David Zuckerman, and Judy Goldsmith.

Finally, a picture…

This was taken in November 2002, in Seventeen, Ohio. Shown are Steve Pav '91, sarah-marie belcastro (senior staff), David Levitt '01, Tom Hull (senior staff), and Abie Flaxman '95 (also junior staff).

Here’s the HCSSiM Online Application Form; we would prefer to receive it by May 15th:


What do you like to be called? 

Street address: 
ZIP code: 

Home phone: 

email address: 


Name of School: 

School address: 

Expected year of graduation: 

Please give us the name of a teacher who will support your application and assist HCSSiM in the selection process. (In other words, make sure you ask the teacher if s/he is willing to sponsor you in view of your interest and ability in mathematics.) The sponsor will be asked to share evaluative comments (email and phone are fine). 

Please check with your parent(s) to be sure it’s okay for you to apply to HCSSiM. Parent’s name: 

Do you want (or really, does your parent want) us to send you a financial aid application form? yes

Now for information about your background and interests:

Please list the math courses you will have completed by July… 

…and those you plan to take next year. 

What other mathematical activities have you enjoyed? 

What are your non-mathematical interests or hobbies? 

If you have previously attended a summer academic program, please name the program and year.

My 17’s and Yellow Pigs

Let’s start with my collection of 17’s; this is an old page (last updated on March 7, 1998), that will be rewritten when I have the time. This page does not contain the mathematical properties of 17. If you want the whole list on the 17’s, get the LaTeX version:

English version: [LaTeX source], [dvi], [gzipped postscript] or [PDF].

French version: [LaTeX source], [dvi], [gzipped postscript] or [PDF].

If you have other interesting 17’s, you can mail them to me.

Note: This site took part in Webs d’Or 96 under the number 1207 = 17 × 71.

My page on yellow pigs.

Historical Facts
You certainly want to know where the 17’s and the yellow pigs come from. Let Daniel Loeb talk…

Ah, there are two mathematicians. David C. Kelly and Mike Spivak. They were graduate students together at Princeton (?) in the 1960’s. They reportedly got the yellow pig 17 idea at a bar.

Since then, Mike Spivak has become a famous author of math text books and has hidden a yellow pig in each book (e.g. one book is dedicated to a chinese (yellow) policeman (pig)) and another has a reference to Steve Neen (?) in the index referring to a page in which he goes whole hog and lets n tend towards infinity or something like that.

David Kelly got involved in summer math programs. Here is some history of that from Larry Carter…

The answer is, it’s probably impossible. In any case, Kelly set up Hampshire by a process of successive improvements. Let me try to recount the History of HCSSiM, although my knowledge of it is incomplete and inaccurate.

Sometime around 1957, possibly in response to the Russian’s launching of Sputnik, a summer program was set up at St. Paul’s School, a prep school in New Hampshire. High school students from all over N.H. came to study their favorite subject. They were also subjected to English classes, daily chapel, mandatory sports, and lights out at 10:30. I suspect that the fact that the program was at a high school, was staffed by high school teachers, and targeted to a state, all made it simpler to start. (Dan: if you want to find out how that program got set up - it still exists - you can write to the Advanced Studies Program, St. Paul’s School, Concord, N.H. 03301. Alan Hall was the original director.)

So we’ve got state funding, decent (but not Hampshire-caliber) students and faculty, and a pretty campus. Sometime in the early 60’s, Kelly found a summer job teaching math there. He set about revising the curriculum, away from freshman math and towards Neat Stuff. Also, whenever he found someone who seemed to love math, he encouraged them to teach there (I joined in '67). I think the NSF funding started coming in under Kelly’s reign (the NSF was bursting with money in those days.)

So now we have a dedicated staff (Dan Heisey and Dave Gay, among others.) I think what happened next is that the NSF declared that they would no longer fund programs that were restricted to a single state, or maybe just St. Paul’s Rules and Regulations got to be too much. In any case, Kelly and Dan Heisey applied to the NSF and got funding to set up a program at UNH, where Dan was a professor. Kelly’s staff of course followed and were joined by Mike Spivak and the Yellow Pig. Students applied since the NSF advertised it along with all the other so-called Summer Studies Training Programs.

This lasted two years (68 and 69), at which point UNH decided they didn’t need Kelly, that they could have a more organized program without him, or something like that. They tried in '70, and were wrong. So even though Hampshire College didn’t exist at the time, Kelly’s proposal to set up a program at Hampshire in '71 was approved by the NSF. From then on, it was mostly easy, since students became staff and word of mouth brings more students. The only hard part was trying to survive during the years the NSF couldn’t support the program. But alumni contributions and Hampshire’s reduced charges for the facilities (plus a high tuition) allowed the program to muddle through.

Kelly gives a talk at each program about seventeen. The program visits his home (in groups) a couple of times. He has a collection of mathematical toys larger than any particular game shop I have seen. In addition he has somewhere between 289 and 4913 yellow pigs lying around his house.

People have tried coming up with rival numbers (e.g. 23); however, Kelly beat them in a competition. The rivals had to name a property of 23, and Kelly (and Don Goldberg) would reply with at least 2 similar properties of 17… In the end, the rivals ran out of interesting properties!


The solution to the following puzzle is unique; in some cases the
knowledge that the solution is unique may actually give you a short-cut
to finding the answer to a particular question, but it’s possible to
find the unique solution even without making use of the fact that the
solution is unique. (Thanks to Andy Latto for bringing this subtlety
to my attention.)

I should mention that if you don’t agree with me about the answer to #20,
you will get a different solution to the puzzle than the one I had in mind.
But I should also mention that if you don’t agree with me about the answer
to #20, you are just plain wrong. :-)

You may now begin work.

  1. The first question whose answer is B is question
    (A) 1
    (B) 2
    © 3
    (D) 4
    (E) 5

  2. The only two consecutive questions with identical answers are questions
    (A) 6 and 7
    (B) 7 and 8
    © 8 and 9
    (D) 9 and 10
    (E) 10 and 11

  3. The number of questions with the answer E is
    (A) 0
    (B) 1
    © 2
    (D) 3
    (E) 4

  4. The number of questions with the answer A is
    (A) 4
    (B) 5
    © 6
    (D) 7
    (E) 8

  5. The answer to this question is the same as the answer to question
    (A) 1
    (B) 2
    © 3
    (D) 4
    (E) 5

  6. The answer to question 17 is
    (A) C
    (B) D
    © E
    (D) none of the above
    (E) all of the above

  7. Alphabetically, the answer to this question and the answer to the
    following question are
    (A) 4 apart
    (B) 3 apart
    © 2 apart
    (D) 1 apart
    (E) the same

  8. The number of questions whose answers are vowels is
    (A) 4
    (B) 5
    © 6
    (D) 7
    (E) 8

  9. The next question with the same answer as this one is question
    (A) 10
    (B) 11
    © 12
    (D) 13
    (E) 14

  10. The answer to question 16 is
    (A) D
    (B) A
    © E
    (D) B
    (E) C

  11. The number of questions preceding this one with the answer B is
    (A) 0
    (B) 1
    © 2
    (D) 3
    (E) 4

  12. The number of questions whose answer is a consonant is
    (A) an even number
    (B) an odd number
    © a perfect square
    (D) a prime
    (E) divisible by 5

  13. The only odd-numbered problem with answer A is
    (A) 9
    (B) 11
    © 13
    (D) 15
    (E) 17

  14. The number of questions with answer D is
    (A) 6
    (B) 7
    © 8
    (D) 9
    (E) 10

  15. The answer to question 12 is
    (A) A
    (B) B
    © C
    (D) D
    (E) E

  16. The answer to question 10 is
    (A) D
    (B) C
    © B
    (D) A
    (E) E

  17. The answer to question 6 is
    (A) C
    (B) D
    © E
    (D) none of the above
    (E) all of the above

  18. The number of questions with answer A equals the number of questions
    with answer
    (A) B
    (B) C
    © D
    (D) E
    (E) none of the above

  19. The answer to this question is:
    (A) A
    (B) B
    © C
    (D) D
    (E) E

  20. Standardized test is to intelligence as barometer is to
    (A) temperature (only)
    (B) wind-velocity (only)
    © latitude (only)
    (D) longitude (only)
    (E) temperature, wind-velocity, latitude, and longitude