Find the Side

Another cute geometry puzzle was posted on Facebook.

Puzzle. An equilateral triangle in a plane has three vertices with known x-coordinates: a, b, and c. What is the side of the triangle?

I want to describe three different solutions that the readers of the Facebook channel posted. But before doing so, let’s look at the problem’s symmetries. We can immediately say that the answer should be a symmetric function of three variables: |a-b|, |b-c|, and |c-a|. It is possible to coordinate-bash the problem. However, I always prefer geometric solutions. Having said that, if one wants a calculation, using complex numbers might speed things up.

A solution using complex numbers. Suppose c is the origin, then the first vertex corresponds to a complex number a+xi. Then, the second vertex can be found after rotating the first vertex around the origin by 60 degrees. That means it is at (a+xi)exp(±2πi/6). Without loss of generality, we can assume that the second vertex corresponds to (a+xi)(1+i√3)/2. It follows that b = (a−x√3)/2. Thus, x = 2(a/2-b)/√3. And the side length is √(a2+x2) = √(4(a2-ab+b2)/3). Adjusting for the choice of the origin, we get that the length is √(2((a-b)2+(b-c)2+(c-a)2)/3).

A geometric solution. Draw a line through point A parallel to the x-axis. Denote the intersections of this line with lines x=b and x=c as P and Q, correspondingly. Let R be the midpoint of the side BC. Then, the triangle PQR is equilateral. To prove it, notice that angles ARC and AQC are right, which implies that points ARCB are on the same circle with diameter AC. It follows that the angles RCA and RQA are the same; thus, the angle RQA is 60 degrees. Given that the triangle PQR is isosceles as R has to be on the bisector of PQ, we conclude that the triangle PQR is equilateral. Now, we can calculate the height of PQR and, therefore, the height of ABC, from which the result follows.

Find the Side Solution

A physics solution. Without loss of generality, we can assume that a+b+c=0. Thus, the y-axis passes through the triangle’s centroid. The moment of inertia of the system consisting of the three triangle vertices with respect to the y-axis is a2 + b2 + c2. Now, we add the symmetry consideration: the inertia ellipse must be invariant under the 60-degree rotation, implying that the ellipse is actually a circle. This means that the inertia moment doesn’t change under any system rotation. Thus, we can assume that one of the vertices lies on the y-axis. In this case, the inertia moment equals L2/2, where L is the length of the triangle’s side. The answer follows.


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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.

* * *

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.

* * *

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

* * *

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

* * *

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

* * *

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

* * *

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

* * *

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.

* * *

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

* * *

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

* * *

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


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Fudge Likes Meatballs

Here is an interesting puzzle by Ivan Mitrofanov.

Puzzle. In front of my dog, Fudge, lies an infinite number of meatballs with a fly sitting on each of them. At each move, Fudge makes two consecutive operations described below.

  1. Eats a meatball and all the flies sitting on it at that time.
  2. Transfers one fly from one meatball to another (there can be as many flies as you want on a meatball).

Fudge wants to eat no more than a million flies. Assuming that flies sit still, prove that Fudge doesn’t have a strategy where each meatball is eaten at some point.


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My Family’s Jokes

I collect math jokes and, of course, show them to my family. From time to time, my family contributes. The first joke is by my son, Alexey.

* * *

When you board a train traveling East from Chicago to Boston at 60 miles an hour, you realize you are a part of the problem.

And this one is one of mine.

* * *

A dyslexic’s excuse: my god ate my homework.

My grandson, Alex, heard the following famous joke.

* * *

A logician rides an elevator. The door opens, and someone asks:
—”Are you going up or down?”
—”Yes.”

He created his own version.

* * *

A logician rides an elevator. The door opens, and someone asks:
—”Are you going up or down?”
“No,” replies the logician and walks out.

Alex, though he is 10, is very good with words. I liked the wordplay in one of his comments.

* * *

This puzzle is confusling.


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Two Lovely Puzzles

These two puzzles were given to me by Andrey Khesin.

Puzzle. My friend and I are going to play the following game at a casino. Each round, each of us (my friend, the dealer, and I) secretly chooses a black or white stone and drops it in the same bag. Then, the contents of the bag are revealed. If all three stones are the same color, my friend and I win the round. If not, we lose to the dealer. One extra caveat. I have a superpower: as soon as we sit down, I can read the dealer’s mind and learn the dealer’s choices for all future rounds. Unfortunately, at that time, it’s too late for me to give this information to my friend and win all the rounds. The only thing we can do is agree on a strategy before the game.

  • Design a strategy to win 6 out of 10 rounds.
  • Design a strategy to win 7 out of 11 rounds.
  • Is it possible to win 6 out of 9 rounds?

Puzzle. In a crowd of 70 people, one person is a murderer, and another person is a witness to said murder. A detective can invite a group into his office and ask if anyone knows anything. The detective knows that everyone except the witness would say nothing. The witness is a responsible person who is more afraid of the murderer than they desire to fulfill their civic duty. If the witness is in the same group as the murderer, the witness will be silent; otherwise, the witness will point to the murderer. The detective knows this will happen and wants to find the murderer in as few office gatherings as possible. What is the minimum number of times he needs to use his office, and how exactly should the detective proceed?


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Some Recent Puns Added to My Collection

* * *

—Why was the fraction worried about marrying the decimal?
—Because he would have to convert.

* * *

—How does a professional mathematician plow a field?
—With a protractor.

* * *

—How many bakers does it take to bake a pi?
—3.14.

* * *

—What did the witch doctor say after lifting the curse?
—Hexagon.

* * *

—What do you call a teapot of boiling water on top of a mountain?
—A high-pot-in-use.

* * *

—What do you call a K1 graph drawn at freezing temperature?
—An ice-olated vertex!


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Some Recent Jokes Added to My Collection

* * *

—What is the best way to pass a geometry test?
—Know all the angles.

* * *

—Did you hear about the over-educated circle?
—It has 360 degrees!

* * *

—What do parallel lines and vegetarians have in common?
—They never meat.

* * *

—Did you hear about the mathematician who’s afraid of negative numbers?
—He’ll stop at nothing to avoid them.

* * *

—What do you call a gentleman who spent all the summer at the beach?
—A tangent.

* * *

—What do mathematicians and the Air Force have in common?
—They both use pi-lots.

* * *

—Why can’t a nose be 12 inches long?
—Because then it would be a foot.

* * *

—Are monsters good at math?
—Not unless you Count Dracula.

* * *

—Why did the math professor divide sine by tan?
—Just cos.

* * *

Two is the oddest prime.

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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?

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Icosahedron’s Resistance

I rarely post physics puzzles, but this one is too good to pass on.

Puzzle. A wireframe icosahedron is assembled so that each of its edges has a resistance of 1. What is the total resistance between opposite vertices of the icosahedron?

While we are at it, another interesting question would be the following.

Puzzle. A wireframe cube is assembled so that each of its edges has a resistance of 1. What is the total resistance between opposite vertices of the cube?

And this reminds me of a question I heard when I was preparing for an IMO many years ago.

Puzzle. A wireframe infinite square grid is assembled so that each of its edges has a resistance of 1. What is the total resistance between two neighboring vertices?


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Ikigai

Have you ever heard of ikigai? The Japanese concept which gives four simple requirements for a happy career:

  • Do what you love.
  • Do what you are good at.
  • Do what impacts the world.
  • Do what you can get paid for.

I often think about it for myself and for my students. Is this good advice for finding a career path?

I like that ikigai separates the first two requirements: passion and gift. Many of my students do not see the difference, as passion and gift are highly correlated. When you love something, you practice it and become better at it. When you are good at something, it becomes easy and enjoyable.

Nonetheless, passion and gift are different. Unfortunately, I’ve seen students who are good at math only because their parents push them, but they do not love it. Some of them already found their passion but are afraid to tell their parents. Some haven’t yet found their passion, but it is perfectly clear that math is not it. So, a gift doesn’t imply passion.

What about the other way around? My programs are too selective, so I haven’t seen students who are not gifted in math. I will use myself as an example. I have always passionately loved dancing, but it is obvious that my dancing career would have been a disaster. I am very happy I closed that career path in fifth grade.

Anyway, the first two ikigai requirements are not the same, and both are necessary.

The third ikigai requirement is about doing what the world needs. Impacting the world is a great motivator and makes you feel good. And yet, I see happy and successful mathematicians who only care about the beauty of what they are doing and nothing else. This requirement is important but might not be a deal breaker for everyone.

The last ikigai requirement is crucial. If you are not being paid for your efforts, it is not a career; it is a hobby. I got attracted to it because it includes an important caveat: you need to find people who want to pay you for what you can offer. I recently wrote an essay Follow Your Heart? about many young aspiring opera singers who ignored this last requirement and ended up changing careers.

Nevertheless, the whole concept of ikigai bugs me. People who find their dream job might agree to work for much less pay than they are worth. It opens them up to potential exploitation by greedy employers.

Have I reached my ikigai? Judging by my low pay, I am close.


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