Archive for the ‘Math Education’ Category.

My Students’ Jokes

The homework I give to my students (who are in 6th through 9th grades) often starts with a math joke related to the topic. Once, I decided to let them be the comedians. One of the homework questions was to invent a math joke. Here are some of their creations. Two of my students decided to restrict themselves to the topic we studied that week: sorting algorithms. The algorithm jokes are at the end.

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A binary integer asked if I could help to double its value for a special occasion. I thought it might want a lot of space, but it only needed a bit.

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Everyone envies the circle. It is well-rounded and highly educated: after all, it has 360 degrees.

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Why did Bob start dating a triangle? It was acute one.

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Why is Bob scared of the square root of 2? Because he has irrational fears.

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Are you my multiplicative inverse? Because together, we are one.

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How do you know the number line is the most popular?
It has everyone’s number.

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A study from MIT found that the top 100 richest people on Earth all own private jets and yachts. Therefore, if you want to be one of the richest people on Earth, you should first buy a private jet and yacht.

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Why did the geometry student not use a graphing calculator? Because the cos was too high.

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Which sorting algorithm rises above others when done underwater? Bubble sort!

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Which sorting algorithm is the most relaxing? The bubble bath sort.


Guess the Number in One Question

There are a lot of puzzles where you need to guess something asking only yes-or-no questions. In this puzzle, there are not two but three possible answers.

Puzzle. Mike thought of one of three numbers: 1, 2, or 3. He is allowed to answer “Yes”, “No”, or “I don’t know”. Can Pete guess the number in one question?

Yes, he can. This problem was in one of my homeworks, and my students had a lot of ideas. Here is the first list were ideas are similar to each other.

  • I am thinking of an odd number. Is my number divisible by your number?
  • If I were to choose 1 or 2, would your number be bigger than mine?
  • If I were to pick a number from the set {1,2,3} that is different from yours, would my number be greater than yours?
  • If I have a machine that takes numbers and does nothing to them except have a 50 percent chance of changing a two to a one. Would your number, after going through the machine, be one?
  • If I were to choose a number between 1.5 and 2.5, would my number be greater than yours?
  • If your number is x and I flip a fair coin x times, will there be at least two times when I flip the same thing?
  • I am thinking of a comparison operation that is either “greater” or “greater or equal”. Does your number compare in this way to two?

One student was straightforward.

  • Mike, please, do me a favor by responding ‘yes’ to this question if you are thinking about 1, ‘no’ if you are thinking about 2, and ‘I don’t know’ if you are thinking about 3?

One student used a famous unsolved problem: It is not known whether an odd perfect number exists.

  • Is every perfect number divisible by your number?

Then, I gave this to my grandchildren, and they decided to answer in a form of a puzzle. Payback time.

  • I’m thinking of a number too, and I don’t know whether it’s double yours. Is the sum of our numbers prime?


The Angry Wife

Here is the homework problem I gave to my PRIMES STEP students.

Puzzle. A man called his wife from the office to say that he would be home at around eight o’clock. He got in at two minutes past eight. His wife was extremely angry at this lateness. Why?

The expected answer is that she thought he would be home at 8 in the evening, while he arrived at 8 in the morning. However, my students had more ideas.

For example, one student extended the time frame.

  • The man was one year late.

Another student found the words “got in” ambiguous.

  • He didn’t get into his house two minutes past eight. He got into his car.

A student realized that the puzzle never directly stated why she got angry.

  • The wife already got angry when he said he would be home around eight, as she needed him home earlier.

The students found alternative meanings to “called his wife from the office” and “minutes.”

  • He had an office wife whom he called. But the wife at home was a different wife, and she was angry.
  • “Two minutes past eight” could be a latitude.

My Unique Christmas Card

One of the perks of being a teacher is receiving congratulations not only from family and friends, but also from students. By the way, I do not like physical gifts — I prefer just congratulations. Luckily, MIT has a policy that doesn’t allow accepting gifts of any monetary value from minors and their parents.

Thus, my students are limited to emails and greeting cards.

One of my former students, Evin Liang, got really creative. He programmed the Game of Life to generate a Christmas card for me. You can see it for yourself on YouTube at: Conway Game of Life by Evin Liang.

This is one of my favorite congratulations ever.


Four Sheep

I like including warm-up puzzles with every homework.

Puzzle. Farmer Giles has four sheep. One day, he notices that they are all standing the same distance away from each other. How can this be so?

The expected answer: The configuration is impossible in 2D. So, one of the sheep is on a hill or in a pit.

Some students thought big: The sheep could be placed at different locations around the Earth, forming a really big tetrahedron. In this case, we need to explain what it means for the farmer to “notice”, but this minor issue could be resolved in many ways.

Some of the students questioned the meaning of the word distance. They argued that if sheep are all touching each other, they are the same distance 0 from each other. One way this could happen is if the tails of the four sheep were entangled.



My PRIMES STEP program consists of two groups of ten students each: the senior group and the junior group. The senior group is usually stronger, and they were especially productive last academic year. We wrote four papers, which I described in the post EvenQuads at PRIMES STEP. The junior group wrote one paper related to the game SOS. The game was introduced in the following 1999 USAMO problem.

Problem. The game is played on a 1-by-2000 grid. Two players take turns writing an S or an O in an empty square. The first player who produces three consecutive squares that spell SOS wins. The game is a draw if all squares are filled without producing SOS. Prove that the second player has a winning strategy.

The solution is quite pretty, so I do not want to spoil it. If my readers want it, the solution for this grid, and, more generally, for any grid of size 1-by-n, is posted in many places.

My students studied generalizations of this game, and the results are posted at the arXiv: SOS. We tried different target strings and showed that:

  • The SOO game is always a draw.
  • The SSS game is always a draw.
  • The SOSO game is always a draw.

Then, we tried a version where the winner needed to spell one of two target strings. We showed that:

  • The SSSS-OOOO game is always a draw.

We tried several more elaborate variations, but I want to keep this post short.


How Many Cows Are Left?

Here is another STEP homework question, which is a famous riddle.

Puzzle. Peter had ten cows. All but nine died. How many cows are left?

The wording is confusing on purpose. So, the students who are in a hurry subtract nine from ten and answer that only one cow is left. This answer is wrong. All but nine means that one cow died. So, the correct answer is nine.

One of my students decided that nine is the name of one of the cows, though it should have been capitalized. This means that all the cows except for Nine died, and only one cow, Nine, is left.

This student managed to find a legitimate explanation for the standard wrong answer.


EvenQuads at PRIMES STEP

I love the game of SET, where you have a specialized deck of 81 cards. The image on each card has four features: color, shape, shading, and the number of objects. Each feature can have three possible values. Three cards form a set if, for every feature, the values on the cards are either all the same or all different. One of the best properties of the sets is that, given any two cards, we can calculate the third card that would complete a set.

If we look at the game mathematically, we can assign a number 0, 1, or 2 for every feature value. For example, green could be 0, purple could be 1, and red could be 2. Then, the condition that, given a feature, three cards are either all the same or all different is equivalent to saying that the sum of the values modulo 3 is zero.

A mathematician would wonder, can we make a new game where we take values modulo 4? Each feature value should correspond to 0, 1, 2, or 3. For example, we can add a yellow color corresponding to number 3. The new “set” should be four cards such that the sum of the values modulo 4 is 0. This condition guarantees that any three cards could be uniquely completed into a “set”. But such a game is inelegant. For example, one green, two purple, and one red card will form a set. But one green, one purple, and two red will not. The symmetry between colors is lost.

I decided that there is no good analog for the game of SET that is played modulo 4. I was wrong. Here is a new game called EvenQuads. I heard about it at the 2022 MOVES conference.

The deck consists of 64 cards with three features: color, shape, and the number of objects. There are four values for each feature. Four cards form a quad if, for every feature, the values are all the same, all different, or half-and-half.

Quad Example

For example, the figure above shows a quad where, for each feature, all values are different. To get familiar with quads, here is a puzzle for you. How many quads can you find in the picture below?

Find Quads

The game proceeds in a similar way to the game of SET. You can find out more about EvenQuads at the EvenQuads website.

I picked this game as a research project for my senior STEP group. As many of you know, I have a program where we conduct mathematical research with students in grades 7 through 9. The group started in the fall of 2022 and was extremely productive. We wrote FOUR papers in an academic year, which is obviously our record. The papers can be found at the arXiv.

  • In Card Games Unveiled: Exploring the Underlying Linear Algebra, we analyze four games related to linear algebra: SET, EvenQuads, Socks, and Spot It!. The games are so connected to each other that sometimes it is even possible to play one game using the cards from another game.
  • In Quad Squares, we study semimagic, magic, and strongly magic quad squares made out of EvenQuads cards. A semimagic quad square is a 4-by-4 square in which each row and column form a quad. You can guess what a magic quad square is. The idea was inspired by magic SET squares: 3-by-3 squares where each row, column, and diagonal form a set. A magic SET square has an additional property: there are always four more sets in such a square located on broken diagonals. In other words, consider three cards and their coordinates in a square. These cards form a set if and only if the values of each coordinate are either all the same or all different. This property is stronger than the definition of a magic SET square. In the case of the game of SET, these two definitions are the same. However, for EvenQuads, we get two different definitions. This is how we ended up defining strongly magic quad squares. You can find the details in the paper.
  • In EvenQuads Game and Error-Correcting Codes, we describe error-correcting codes based on a set of EvenQuads cards.
  • In Maximum Number of Quads, we calculate the maximum possible number of quads, given n cards from an EvenQuads deck. For example, the puzzle pictured above is actually an example of the maximum possible number of quads among 7 cards. We generalize this idea to decks of any size. Unfortunately, our formula is based on a conjecture. Though, we strongly believe that our formula is correct.

My students did a great job.


Childhood Memories

I was browsing my analog archives, my high-school notes, and stumbled upon a couple of problems from a math battle we had.

Puzzle. A square is divided into 100 pieces of the same area, in two ways. Prove that you can find 100 points such that each piece in the first division has a point inside, and each piece in the second division also has a point inside.

Puzzle. There is a finite number of points on a plane. All distances between any pairs of points are distinct. Each point is connected to its closest neighboring point. Prove that each point is connected to no more than 5 points.


The Linking Number

A link

A link is defined as two closed curves in three-dimensional space. The first picture shows an example of a link with one yellow curve and one blue. The linking number is a simple numerical invariant of a link. Intuitively, it represents the number of times that each curve winds around the other. For example, if it is possible to pull the two curves apart, the linking number is zero.

When I studied the linking number, I would look at a picture of a link trying to calculate this number. It was confusing. It only became easy after I started crocheting. For example, the second picture shows the same link as the first but slightly rearranged. I simply slid the yellow loop along the blue one until I could clearly see a piece of the blue loop as a straight segment and the yellow loop circling around it. Now, it is easy to see that the yellow loop winds around the blue one 3 times, making the linking number 3.

The only thing to remember is that while counting the number of windings, I need to consider the direction. It is possible for a loop to wind clockwise and then counterclockwise. In this case, the linking number is the difference between the two.

I crocheted a lot of links, and now my students and I have no problem calculating the linking numbers.

The linking number