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PRIMES STEP students with Dr. Tanya Khovanova and PRIMES Chief Research Advisor Prof. Pavel Etingof, Sixth Annual PRIMES conference, May 2016, MIT.

PRIMES STEP students' research papers

"Who Is Guilty?" (2016)
Alternator Coins (2016), published in Math Horizons 25:1 (2017): 22-25.
On Variations of Nim and Chomp (2017), published in The Minnesota Journal of Undergraduate Mathematics 4:1 (September 2018)
PRIMES STEP Plays Games (2017), published in Math Horizons 26:2 (2018): 10-13.
On Base 3/2 and its Sequences (2018)
Variants of Base 3 over 2 (2018)
Chip-Firing and Fractional Bases (2018)
It's Common Knowledge (2019)


Try these problems   What is PRIMES STEP?

Problem 1. Mrs. Fullhouse has 2 sons, 3 daughters, 2 cats and 1 dog. How many children does she have?


Problem 2. Detective Radstein is investigating a robbery. He apprehends three suspects: Anne, Bill, and Caroline. The detective knows that no one else could have participated in the robbery. During the interrogation the suspects make the following statements:

Anne: I didn't do it. Bill did it alone.
Bill: I didn't do it. Caroline did it.
Caroline: I didn't do it. Bill did it.

Detective Radstein also discovered that all three suspects are members of a clan called the Halfsies. Every time they speak, they make two statements, one of which is a lie and the other of which is true. Who committed the robbery?


Problem 3. One day you meet your friend Alice enjoying a nice walk with her husband Bob and their son Carl. They are excited to see you and they invite you to their party.

Alice: Please, come to our party on Sunday at our place at 632 Elm St. in Watertown.
Bob: My wife likes exaggerating and multiplies every number she mentions by 2.
Carl: My dad compensates for my mom's exaggerations and divides every number he mentions by 4.
Alice: Our son is not like us at all. He doesn't multiply or divide. He just adds 8 to every number he mentions.

Where is the party?


Problem 4. There are n ants placed randomly on a stick that is 1 meter long. The ants immediately start running along the stick, choosing a random direction, either left or right, with the same constant speed equal to 1 meter per second. When two ants bump into each other they immediately reverse direction, and continue running with the same speed. When an ant reaches the end of the stick it falls off. What is the expected time for the stick to become completely free of ants?


PRIMES STEP (Solve–Theorize–Explore–Prove) is your step into the world of mathematical discovery. It is a program for curious middle-schoolers who like solving challenging problems and are ready to explore advanced math topics.

The goal of our program is to use mathematics to expand the students' minds and encourage their creative thinking. We will not repeat the middle-school math curriculum. We will solve a lot of problems that are tricky, logical, fun, and might potentially be asked at a Wall Street interview. See some examples in the left column.

In addition to solving problems like these, we will cover some ideas and methods for solving olympiad-style problems. We will also try to simulate the research process by inventing new problems and being the first to solve them. We will even learn an occasional magic trick. The main goal is to have fun!

The instructor is Dr. Tanya Khovanova, an active researcher in the field of recreational mathematics. There are two groups of students, which meet on Mondays or Thursdays, 6-8 pm, on MIT campus.

Tanya Khovanova is Lecturer in the Mathematics Department at MIT. She received her Ph.D. in mathematics from Moscow State University in 1988, under Israel Gelfand. Her research was in representation theory, integrable systems, quantum groups, and superstring theory. Currently she is interested in combinatorics and recreational mathematics. Dr. Khovanova is Head Mentor of math research at MIT PRIMES and MIT's Research Science Institute (RSI). She has created the popular Number Gossip site.


PRIMES-STEP is open to students entering 7th-9th grade in September 2019. Home schooled students are also eligible.

Applications for the 2019/20 academic year are now closed. Admissions decisions will be made by June 10.

Applications for the 2020/21 academic year will open in mid-April 2020. The deadline will be in late May 2020. One letter of recommendation will be required. Applicants on the short list will be invited to MIT for an entrance quiz.

The tuition will be $2,000 per year, including $100 registration fee (due in late June) and two installments of $950 (due in September and in January). Need-based scholarships will be available for families receiving public assistance.

For preparation, we recommend studying books in the Art of Problem Solving series, as well as the 2015 STEP entrance test and its solutions, and the 2016 and 2017 entrance tests (later tests will not be posted).

If you have any questions about your application, please contact PRIMES Admissions Assistant Jonathan Harmon at

With general questions about the program, please contact PRIMES Program Director Dr. Slava Gerovitch at