Archive for the ‘Math’ Category.

Rulers to the Rescue

I recently posted a cute puzzle about poisoned wine. Today, I would like to discuss this puzzle’s variation with N total glasses, of which two are poisoned.

Puzzle. N glasses of wine are placed in a circle on a round table. S sages are invited to take the following challenge. In the presence of the first sage, N − 2 glasses are filled with good wine and the other two with poisoned wine, indistinguishable from the good wine. After drinking the poisoned wine, the person will die a terrible tormented death. Each sage has to drink one full glass. The first sage is not allowed to give any hints to the other sages, but they can see which glass he chooses before making their own selection. The sages can agree on their strategy beforehand. For which S can you find a strategy to keep them all alive?

What does this have to do with rulers, and what are those? I am grateful to Konstantin Knop for showing me a solution with rulers. But first, let’s define them.

A sparse ruler is a ruler in which some distance marks may be missing. For example, suppose we have a ruler of length 6, with only one mark at a distance 1 from the left. We can still measure distances 1, 5, and 6. Such a ruler is often described as {0,1,6} to emphasize its marks, endpoints, and size.

A complete sparse ruler is a sparse ruler that allows one to measure any integer distance up to its full length. For example, the ruler {0,1,6} is not complete. It can’t measure distances 2, 3, and 4. Thus, if we add the marks at 2, 3, and 4, such a ruler becomes complete.

A complete sparse ruler is called minimal if it uses as few marks as possible. In our previous example, the ruler {0,1,2,3,4,6} is not minimal. The distance between marks 1 and 4 is 3, so if we remove mark 3, we can still measure any distance. We can remove mark 2 too. The ruler {0,1,4,6} with marks 1 and 4 is minimal.

Oops. I forgot that we have a round table. This means we need to look at cyclic rulers: the idea is the same, but the numbers wrap around. For example, consider the cyclic ruler {0,1,4,6} of length 6, where 0 and 6 is the same point. This ruler has three marks at 0, 1, and 4.

Going back to the puzzle, suppose N = 6, aka there are six glasses around the table. The sages need to agree on a complete cyclic ruler, for example, the one described above. As this ruler contains any possible difference between the marks, the first sage can mentally place the ruler on the table so that the marks cover poisoned glasses. He signals the position of the ruler by drinking his glass. The sages can agree that the glass drunk by the first sage corresponds to position −1 on the ruler, and the other sages avoid the first, second, and fifth glass clockwise from the chosen glass.

In this case, three glasses are not covered by the ruler’s marks. This means three sages can be saved.

To summarize, the sages need a complete ruler, as such a ruler can always cover two glasses at any distance from each other with its marks. The number of sages that can be saved by such a ruler is N minus the number of marks. To save more sages, we want to find a minimal ruler.

There are actually more cool rulers. A ruler is called maximal if it is the longest complete ruler with a given number of marks. For example, the non-cyclic ruler {0,1,4,6} is maximal. A ruler is optimal if it is both maximal and minimal. Thus, the ruler {0,1,4,6} is also optimal.

There are other types of rulers called Golomb rulers. They require measured distances to be distinct rather than covering all possibilities. Formally, a Golomb ruler is a ruler with a set of marks at integer positions such that no two pairs of marks are the same distance apart. If, however, a Golomb ruler can measure all the distances, it is called a perfect Golomb ruler. As we can deduce, a perfect Golomb ruler is a complete sparse ruler. I leave it to the reader to show that a perfect Golomb ruler must be a minimal complete sparse ruler.

The rulers rule!

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Tricky Probabilistic Arguments

The Conway subprime Fibonacci sequence is defined as follows. Start with the Fibonacci sequence 0, 1, 1, 2, 3, 5, …, but before you write down a composite term, divide it by its least prime factor. Thus, the next term after 5 is not 8, but rather 8/2 = 4. After that, the sum gives us 5 + 4 = 9, which is also composite. Thus, you write 9/3 = 3, then 4 + 3 = 7, which is okay since it is prime, then 3 + 7 = 10, but you write 10/2 = 5, and so on. Here is the sequence: 0, 1, 1, 2, 3, 5, 4, 3, 7, 5, 6, 11, 17, 14, 31, and so on.

My coauthors, Richard Guy and Julian Salazar, and I studied this sequence in our paper Conway’s subprime Fibonacci sequences, in which we allowed any two integers to be the two starting terms. Our computation showed that for small starting terms, the sequence starts cycling. In our first draft, we had a probabilistic argument as to why the sequence should always cycle. Our argument was the following.

Flawed probabilistic argument. Consider the parity of the numbers in a particular subprime Fibonacci sequence. I will leave it to the reader to see that such a sequence can’t have all even numbers. As soon we get to an odd number, the even numbers in the sequence become isolated. Indeed, if two numbers have different parity, the next number must be odd. For subprime Fibonacci sequences, when we add two odd numbers, there is a 50 percent chance that after dividing by 2, the result will be odd. Assuming the result is odd, there a is 25 percent chance the next number will be odd as well. And this pattern continues. Thus, as soon as we get two odd numbers in a row, on average, we expect three odd numbers in a run of odd numbers. Hence, we would expect a typical subprime Fibonacci sequence to have three times as many odd as even numbers.
This means, on average, two consecutive terms sum to an even number half of the time. Therefore, while calculating the next term, we divide by 2 with probability 1/2 and by 3 with probability 1/6. Ignoring larger primes, on average, we expect to have divided by at least 1.698, while the usual Fibonacci growth is only by a factor φ, which is approximately 1.618. We see that the subprime Fibonacci sequence is divided faster than its expected growth rate. Thus, it has to cycle.

However, this argument doesn’t work. When we submitted the paper, an anonymous reviewer pointed out that the argument was flawed. Here I want to explain why. I will start with a counterexample suggested by my sons, Alexey and Sergei.

Counterexample. Start with two real numbers. Suppose the sequence is to add the two previous numbers and divide the sum by 1.8 (which is greater than the golden ratio φ). The resulting sequence grows as a geometric progression with ratio 1.07321.

Here is a variation of the above argument.

Counterexample Variation. Suppose we have a sequence where we add the two previous numbers and then divide the sum by 2. We are dividing by a noticeably larger number than the golden ratio. By our flawed argument earlier, the sequence should decrease. But if we start such a sequence with two identical numbers n and n, the sequence will be constant, disproving the argument.

Here is an additional observation. Obviously, whenever we add two numbers and divide the sum by a number bigger than 2, the sequence has to cycle. Indeed, the maximum of two consecutive terms decreases. Can we add probabilities to this argument? Below is the averaging version of this argument for the subprime Fibonacci numbers. Unfortunately, this argument is also flawed.

Another flawed probabilistic argument. We need to show that, on average, at each step, we divide our sum by a number bigger than 2. We can ignore divisions by 2 as they are fine. Let’s see what happens with sums that we do not divide by 2. With probability 1/3, we divide them by 3. If not, with probability 1/5, we divide them by 5. Otherwise, with probability 1/7, we divide them by 7. Hence, if we do not divide our sum by 2, then with probability 1/3, we divide it by 3, plus with probability 2/15, we divide it by 5, plus with probability 8/105, we divide it by 7. Thus, on average, if we do not divide the sum by 2, then we divide it by at least 31/352/1578/105 = 2.07.

Counterexample to the second argument. Why is this wrong? Let us, for example, consider a sequence with the following recursions: a2k+1 = a2k + a2k-1. And a2k = (a2k-1 + a2k-2)/x, where x is a very large number. Then on average, we divide the sum by a very large number. Would this sequence converge to zero? If we look at it more closely, the subsequence of odd-indexed numbers increases. So the answer is no.

We found yet another probabilistic argument that the sequences shouldn’t increase indefinitely. We are sure that one works. Our anonymous reviewer was happy with it too. You can find the argument in our paper: Conway’s subprime Fibonacci sequences.

But I wrote this essay to show how tricky probabilistic arguments can be.


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What are Your Math Research Interests?

For students applying to PRIMES, we have a question about their research interests. RSI asks a similar question from their applicants.

I am looking at all the submissions, and this essay will help our applicants to get projects that are well-suited for them.

We, at PRIMES and RSI Math, usually have research projects lined up in advance. That means, we are not creating projects to match applicants’ requests. We match existing projects to students’ backgrounds and interests.

If you are applying to one of these programs, here is my advice.

Don’t be too specific about what you want. Suppose you want to study the symmetries of an icosahedron. This request is too narrow: there is a high probability we do not have such a project. How will we match you to a project? Our hope, in this case, is to find clues in your essay. For example, we might discover that you heard a fascinating lecture on icosahedron’s symmetries, which is why you requested the topic. In this case, we assume that another fascinating lecture on a different topic might also excite you, and you will be matched with a random project. But if your description is broader, say, if you write that you like group theory or geometry, your match won’t be as random.

Specify things you do not want. Given our project distribution, you might not get a project in the area that is your first or even your second choice. On the other hand, if you write to us that you hate geometry, it is very easy to find a project without a geometric component. If there is something you definitely do not want, it is advantageous for you to mention it. Be precise about your advanced knowledge. For example, linear algebra is one of the most powerful tools in mathematics. Not surprisingly, we often have projects that require a serious background in linear algebra and specifically look for students who know it. Unfortunately, we often receive inadequate descriptions of students’ backgrounds. Even if you took a linear algebra course, it might be useful to mention which book you used and how many chapters you covered. This also applies to other advanced topics. An applicant might say they are proficient in cohomologies after half-listening to one half-hour lecture on the topic. This is not proficiency; it only indicates interest. If you claim advanced knowledge, specify the scope. The best way to start is by listing the books you have attempted to read. For each book, describe which chapters you only scanned, which chapters you read and understood, and for which chapters you solved all of the exercises.

Add more information if your first choice is number theory. Almost every year, we have several students requesting number theory. This might be explained by the successes of the Ross and PROMYS summer programs. The graduates from these programs love number theory and have a good number theory background. However, modern number theory is very advanced, and we seldom have these types of projects. So, if number theory is your top choice, there are two things you can do. First, mention your second choice. Second, specify what you like about number theory. For example, if you are into the more abstract parts of number theory, another abstract project might be a good fit.

Describe your priorities in broader terms. It is beneficial for every starting mathematician to figure out the area they like by asking themselves broader questions. If you know the answers to the questions below, it is helpful to write them on the application form.

  • Do you love or hate abstractions?
  • Do you prefer discreet or continuous problems?
  • Is the real-life impact or inner beauty of your project more important to you?
  • Do you enjoy having a visual component to your project?
  • Do you like when problems involve programs and calculations?

If you follow my advice, you might get a project that matches your tastes better. Not only that: figuring out the answers to these questions will help you build the life you love.


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Maximum Egalitarian Cost in the Stable Marriage Problem

Assume that we have n men and n women, all of whom want to get married. They rank each other without ties. After that, we can match them into n pairs for marriage. The matching is called stable if there are no rogue couples. A rogue couple is defined as a man and a woman who prefer each other over their assigned future spouses.

A famous theorem says that, whatever all the rankings are, a stable matching always exists. But how good could a stable matching be? There is a way to assign a quality score to a matching, called the egalitarian cost. The egalitarian cost of any matching is the sum of the rankings that each person gave their assigned partner. The best potential outcome is when all people are matched with their first choices. This corresponds to the minimum egalitarian cost of 2n. But what is the maximum egalitarian cost of a stable matching? I couldn’t find it in the literature, so I proved that it is n(n+1).

Proof. It is easy to see that the egalitarian cost of n(n+1) is achievable. For example, if all men gave an identical ranking to the women, and vice versa, the matching algorithm will end up with couples having mutual rankings (j,j) for different values of j. Another example is a Latin preference profile. Each woman ranks a man n+1−x whenever he ranks her x. In this case, every potential couple’s mutual rankings sum to n+1. Thus, any matching for such rankings ends up with the egalitarian cost of n(n+1).

The next step is to prove that the egalitarian cost can’t be greater. Suppose the cost of a stable matching is C and is greater than n(n+1). Then, for every person, we count the number of people who are better (ranked smaller) than their assigned partner and sum these numbers over all the people. The result must be C−2n, which is the number of pairs of people of opposite genders such that the first person prefers the second one over their assigned partner. Moreover, the result is greater than n(n−1).

The total number of possible couples is n2. Thus, the number of unrealized potential couples is n(n−1). We can conclude that we counted one of these unrealized couples twice. In such a couple, two people prefer each other over their assigned partners. Thus, they form a rogue couple, contradicting our assumption that the matching is stable.

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Whitehead Links for Ukraine

Whitehead Links for Ukraine

This is my second crocheting project to help finance our new program, Yulia’s Dream, in honor of Yulia Zdanovska, a young Ukrainian math talent killed in the war.

I made four Whitehead links in the colors of the Ukrainian flag. You can read about my other crocheting project in my previous post: Hyperbolic Surfaces for Ukraine.

Fun trivia about the Whitehead links.

  • Why are these links so famous? They are the simplest non-trivial links with the linking number zero.
  • What’s the linking number? The linking number is an invariant of a link. If two loops are not linked (they are called an unlink), their linking number is zero. If they are linked, then their linking number is usually not zero. Here the loops are linked, but the linking number is nevertheless zero. Thus, the linking number can’t differentiate this link from an unlink. To explain how to calculate the linking number, I need to explain another simpler invariant: the crossing number.
  • What’s the crossing number? The crossing number is the smallest number of crossings when projecting the link on a plane. The top two pictures have 6 crossings, and the bottom two pictures have 5. The top two pictures emphasize the symmetry of the link, and the bottom two pictures have the smallest possible number of crossings for the Whitehead link. So the crossing number of the Whitehead link is 5.
  • Can you now explain how to calculate the linking number? One way to calculate the linking number is to choose directions for the blue and yellow loops and select the crossings where the blue is on top. After that, following the chosen direction of the blue loop, at each crossing with the yellow loop, check the latter’s direction. If the direction is from right to left, count it with a plus and, otherwise, with a minus. The total is the linking number. In the case of the Whitehead link, it is zero.
  • Is there a more elegant explanation for why the Whitehead link has the linking number zero? Yes. The linking number only looks at the crossings of two different strands and ignores self-crossings. If you look at the two bottom pictures, there is one self-crossing of the blue loop. Now imagine you change the crossing by moving the top blue strand underneath. After such a transformation, the crossing number doesn’t change, but the loops become unlinked. Thus, the linking number of the Whitehead link must be zero.
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Hyperbolic Surfaces for Ukraine

Hyperbolic Surfaces for Ukraine

As you might know, my team started a project, Yulia’s Dream, in honor of Yulia Zdanovska, a young Ukrainian math talent killed in the war.

In this program, we will do what we are great at — help gifted youngsters pursue advanced math. To help the program, I started crocheting hyperbolic surfaces in the colors of the Ukrainian flag. These crochets are designed as gifts to encourage individual donors.

Fun trivia about these hyperbolic surfaces.

  • Why are these surfaces so famous? These surfaces prove that Euclid’s fifth axiom is independent of the other four axioms. The fifth axiom (also known as the parallel postulate) says that if there is a line L and a point P outside of L, then there is exactly one line through P parallel to L. On these hyperbolic surfaces, the first four of Euclid’s axioms hold, while the fifth one doesn’t: if there is a line L and a point P outside of L on such a surface, then there are infinitely many lines through P parallel to L.
  • But what is a line on a hyperbolic surface? A line segment connecting two points is defined as the shortest path between these points, known as a geodesic.
  • How can such a surface be crocheted? I crocheted a tiny circle and continued in a spiral, making 6 stitches in each new row for every 5 stitches in the previous row. This means that each small piece of the crocheted surface is the same throughout the thingy, making these thingies hyperbolic surfaces of constant curvature.
  • What is the constant curvature good for? Constant curvature makes it easy to find lines. You can just fold the thingy, and the resulting crease is a line.
  • Is this thingy a hyperbolic plane? No. A cool theorem states that a hyperbolic plane can’t fit into a 3D space, so whatever someone crochets has to be finite. On second thought, anything someone crochets has to be finite anyway. But I digress. This shape can be viewed as a disc with a hole.
  • The Ukrainian flag is half blue and half yellow, so why do the colors here seem so unevenly distributed? My goal was to use the same amount of blue and yellow yarn per thingy. I leave it as an exercise for the reader to calculate that regardless of how many rows of one color I crochet, to use the same amount of yarn in the second color, I need more than 3 and less than 4 rows of that color. Since I wanted the thingy to be symmetrical, sometimes I had 3, and other times, I had 4 rows of the second color. I also made 4 surfaces where I switched colors every row.

Overall, I crocheted 10 hyperbolic surfaces. If you are interested in donating to help Ukrainian students receive coaching from our program at MIT, the details will be announced on our website shortly.

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My Vision for Number Gossip

I run Number Gossip, where you can input a number and get some of its cute properties. For example, the number 63 is composite, deficient, evil, lucky, and odd. In addition, it has a unique property: 63 is the smallest number out of two (the other being 69), such that the common alphabetical value of its Roman representation is equal to itself. Indeed, the Roman representation of 63 is LXIII, where L is the 12th digit, X is the 24th, and I is the 9th. Summing them up, we get 12 + 24 + 9 + 9 + 9 = 63 — the number itself.

I have a list of about 50 properties of numbers that my program checks. Each number greater than one gets at least four properties. This is because I have four groups of properties that cover all the numbers. Every number is either odd or even. Every number is either deficient, perfect, or abundant. Every number greater than one is either prime or composite. Every number is either evil or odious.

In addition, I collect unique number properties. During the website’s conception, I decided not to list all possible unique properties that I could imagine but to limit the list to interesting and unusual properties. My least favorite properties of numbers are the ones that contain many parameters. For example, 138 is the smallest number whose base 4 representation (2022) contains 1 zero and 3 twos. If you are submitting a number property to me, keep this in mind.

Some parameters are more forgiving than others. For example, 361 is the smallest number which is not a multiple of 9, whose digital sum coincides with the digital sum of its largest proper divisor. In more detail, the digital sum of 361 is 3 + 6 + 1 = 10, while 19, the largest divisor of 361, has the same digital sum of 10. In this case, the parameter 9 is special: for a multiple of 9, it is too easy to find examples that work, such as 18, 27, and so on. Sequence A345309 lists numbers whose digital sum coincides with the digital sum of their largest proper divisor. The first 15 terms of the sequence are divisible by 9, and 361 is the smallest term that is not divisible by 9.

By the way, another number that is buried deep in a sequence is 945, which is the smallest odd abundant number. There are 231 abundant numbers smaller than 945; all of them are even.

A more recent addition to my collection is related to the sequence of distended numbers (A051772). Distended numbers are positive integers n for which each divisor of n is greater than the sum of all smaller divisors. It is easy to see that for distended numbers, all sums of subsets of divisors are distinct. The opposite is not true: 175 is the smallest number, where all sums of subsets of its divisors are distinct, but the number itself is not distended.

It is difficult to find special properties for larger numbers, so I am less picky with them. For example, 3841 is the number of intersections of diagonals inside a regular icosagon. The word icosagon hides a parameter, but I still like the property.

I invite people to submit number properties to me. And I received many interesting submissions. For example, the following oxymoronic property was submitted by Alexey Radul: 1 is the only square-free square.

The numbers below 200 that still lack unique properties are 116, 147, 155, 162, 166, 182, and 194. The earliest century that doesn’t have unique number properties ranges between 7000 and 7100. The next ones are 8000–8100 and 9100–9200. By the way, my site goes up to ten thousand.

I also have a lot of properties in my internal database that I haven’t checked yet. I am most interested in the proof of the following property: 26 is the only number to be sandwiched between any two non-trivial powers.

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Fun with Latin Squares

Last year, our junior PRIMES STEP group studied Latin squares. We invented a lot of different types of Latin squares and wrote a paper about it, Fun with Latin Squares. Recall that a Latin square is an n by n table containing numbers 1 through n in every cell, so that every number occurs once in each row and column. In this post, I want to talk about anti-chiece Latin squares.

First, what’s a chiece? A chiece is a portmanteau word made out of two words, chess and piece, and, not surprisingly, it means a chess piece. Given a chiece, an anti-chiece Latin square is a Latin square such that any two cells, where our chiece can move from one cell to the other, according to the rules of chess, can’t contain the same number. Let’s see what this means.

Let’s start with rooks, which move along rows and columns. An anti-rook Latin square can’t have the same numbers repeating in any one row or column. Ha, anti-rook Latin squares are just Latin squares. Anti-bishop and anti-queen Latin squares can’t have the same numbers repeating on any diagonal.

Now, here is a picture of an anti-knight Latin square in which no two identical numbers are a knight’s move apart. This particular Latin square also forms a mini-Sudoku: not only does each row and column, but also each 2 by 2 corner region, contains all distinct numbers.

Anti-knight Sudoku

Consider all instances of some number, say 1, in an anti-chiece Latin square. If the board is n by n, we get n instances of non-attacking chieces. A famous math puzzle asks to place eight non-attacking queens on a standard chessboard. So the instances of any one particular number in an anti-queen Latin square solves the problem of placing n non-attacking queens on an n by n chessboard. Thus, building an anti-queen Latin square is more complicated than solving the non-attacking queens puzzle. The former requires filling the chessboard with n non-overlapping sets of non-attacking queens. The picture below gives an example of an anti-queen 5 by 5 Latin square.

Anti-queen Sudoku

This square has some interesting properties. It can be formed by cycling the first row. It also happens to be one of the chiece Latin squares we study in our paper. A chiece Latin square is a Latin square such that for each number in a cell, there is another cell, a chiece’s move apart, containing the same number. You can check that our anti-queen Latin square is at the same time a knight Latin square.

I wonder, can anyone build an anti-queen Latin square on the standard 8 by 8 chessboard?


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Reversible Tetrahedra

Do you know what an isosceles tetrahedron is? I didn’t until recently. An isosceles tetrahedron has all of its faces congruent. Equivalently, all pairs of opposite edges are of equal length. Such tetrahedra are also called disphenoids. Here are some cute statements about them.

  • If the four faces of a tetrahedron all have the same perimeter, then the tetrahedron is a disphenoid.
  • If the four faces of a tetrahedron all have the same area, then the tetrahedron is a disphenoid.

Disphenoids have three nontrivial symmetries: 180-degree rotations around three lines connecting the midpoints of opposite edges.

Is it possible to have a tetrahedron with fewer symmetries than disphenoids? Yes, it is. Dan Klain just published a fascinating paper about such tetrahedra: Tetrahedra with Congruent Face Pairs. The results are so elegant and simple that I was surprised that they were new. I got curious and started to google aggressively. I found an official name for this kind of tetrahedron: phyllic disphenoid, but no theorems about them. Their name is quite unappealing: no wonder people didn’t want to study them much. Obviously, Dan wasn’t as aggressive at googling, so he didn’t find this official name and called them reversible tetrahedra. But my favorite name for them is the name Dan thought of but didn’t use in his paper: bi-isosceles tetrahedra.

Reversible Tetrahedron

Here is the picture of a bi-isosceles tetrahedron from Dan’s paper. It has two faces with sides a, b, and c, and two faces with sides a, b, and d. The edges of this tetrahedron are: a, a, b, b, c, and d. It has one nontrivial symmetry: a 180-degree rotation around the line connecting the midpoints of the unequal opposite edges c and d. The picture emphasizes this symmetry. The figure in the picture is a projection of a bi-isosceles tetrahedron onto a plane, such that the line of symmetry is projected onto the point of symmetry of the projection. The two cute statements above, about disphenoids, can be generalized to the case of bi-isosceles tetrahedra.

  • If the four faces of a tetrahedron can be split into two pairs with the same perimeter, then the tetrahedron is bi-isosceles.
  • If the four faces of a tetrahedron can be split into two pairs with the same area, then the tetrahedron is bi-isosceles.

The first property is trivial, while the second one is proven in Dan’s paper. Dan also shows how to calculate the volume of a bi-isosceles tetrahedron:

Reversible Tetrahedron Volume

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Zipper Math

Zipper Surfaces

Each time I teach my students about the Möbius strip, I bring paper, scissors, and tape to class. The students make Möbius strips, and then I ask them to cut the strips in half along the midline and predict the result.

Then we move to advanced strips. To glue the Möbius strip, you need one turn of the paper. An interesting experiment is to glue strips with two, three, or more turns. In this case, too, it is fun to cut them along the midline and predict the shape of the resulting thingy. My class ends with a big paper mess.

As you might know, I am passionate about recycling. So I have always wanted to buy Möbius strips that can be cut in half and then put back together. I didn’t find them, so I made them myself from zippers. Now I can unzip them along the midline and zip them back together. I hope to have less mess in my future classes. We’ll see.

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