Real Mathematics – Strange Worlds #18

Every year in December, each city changes drastically. Suddenly we find ourselves surrounded by decorations that remind us of the upcoming new year.

Steve the teacher starts to decorate his classrooms for the new year like he does every year. Though, Steve the teacher set his mind on using new year decorations for his mathematics lessons.

New Year Decorations Game (N.Y.D.G.)

Steve’s creation N.Y.D.G. is a multiplayer game. This is why the game is played in knockout stages/rounds. The winner of the game wins the new year decorations and gets to decorate the classroom as he/she wishes.

Content of N.Y.D.G.

  • In each knockout round, students are given 4 decorations as follows:
  • Players wind the decorations one another.
  • The winding procedure should be done secretly from the opponent.
  • Each player has at most four moves for winding.

Let’s use an example to explain what a “move” means during the winding procedure.

Assume that the first move is made with the red decoration as follows:

This counts as one move. The red one undergoes the blue and green decorations in this move. Let the next two moves are as follows:

In the second move, the yellow decoration undergoes the green and red ones, while the blue one passes over the green and yellow decorations. The illustration (up-right) shows us how the winding looks after these 3 moves.

In the end, winding gives us a braid.

The Goal of The Game

In any round, to knock your opponent out, you should solve the braid of your opponent faster than your opponent solves yours. (Solving a braid means, bringing the decorations to their first state. For instance, in the example given up, the first state is yellow-green-blue-red in order.)

Braids

Braids have a very important part in daily life. We encounter them not just in new year decorations, but also in a piece of cheese, a hairstyle, a basket or even in a bracelet:

In case you wish to understand what braids mean in mathematics; one can take a look at Austrian mathematician Emil Artin’s works from the 1920s.

Let’s call the following an identity braid from now on:

In Steve the teacher’s game, the ambition is to go back to the identity braid from a complex braid in the shortest amount of time. To do that, we can use Artin’s work on braids.

Example One: Solving two ropes.

Assume that we have two ropes tangles with each other as follows:

Red undergoes the green.

The inverse of this rope is:

Green undergoes the red.

If we combine these two ropes, when each rope to be stretched, the result will give us the identity braid:

Example Two: Solving three ropes.

Take three ropes and make a braid as follows:

There are three intersections in this braid:

1: Green over the blue.

2: Red over the green.

3: Blue over the red.

Now, you should repeat these steps, but from last to the first this time. Then, you should do these moves:

Move #1: Blue over the red.

Move #2: Red over the green.

Move #3: Green over the blue.

Finally, the combination will give you the identity braid. Try and see yourself.

Paper and Braids

Take an A4 paper and cut the paper using a knife like the following:

Then, hold the paper from its sides and rotate it 90 degrees to the left. You will end up with some kind of a braid:

One wonders…

  • How can you use Emil Artin’s work in the game of Steve the teacher?
  • In “example two”, rotate the ropes 90 degrees to the left. Start investigating the intersections from left to right. What do you notice?
  • Play Steve the teacher’s game with an A4 paper. (It is more than enough to use 3 or 4 cuts on the paper.)

M. Serkan Kalaycıoğlu

Real Mathematics – Graphs #7

Serkan’s System

Serkan the math teacher, hands out a specific number of problems to his students. Kids who can solve 1 or more of those problems would get a certain prize. At the beginning of each semester, Serkan and his students sit down and agree on what kind of prize is going to be distributed. For the current semester, oreo is chosen as the prize:

If Serkan the math teacher hands out 10 problems:

  • 10 Oreos for the kids who solved 10, 9 or 8 of those problems,
  • 5 Oreos for the kids who solved 7, 6 or 5 of those problems,
  • 2 Oreos for the kids who solved 4, 3 or 2 of those problems,
  • 1 oreo for the kids who solved 1 problem,
  • Absolutely nothing for the students who solved… well… none of those problems.

If you take a careful look at the numbers, you can see that Serkan the math teacher selected those numbers with a kind of logic: 10, 5, 2 and 1.

These are the natural numbers that can divide the number of the problems (that is 10) without any remainder.

Prize Distribution Machine (P.D.M.)

One month later…

Serkan the math teacher had faced some problems 4 weeks into the semester. He realized that it took hours to distribute the prizes since he has 10 classes in total.

Serkan the math teacher had to use almost all his free time in school to distribute the Oreos. This led him to think about a machine that would help him with the distribution:

  • P.D.M. will have 4 different compartments. (Because of 10, 5, 2 and 1.)
  • The volumes of those compartments will be measured with Oreos. They will be 10, 5, 2 and 1 Oreo-sized.
  • Oreos will enter the machine from the 10-Oreo-sized compartment. From there, Oreos will move to the other compartments using the connections that will be established.
  • Golden Rule: To establish a connection between any two compartments, the size of those compartments must be factors of one another.

Connections of the compartments for 10 problems:

  • For 10-Oreo-sized: 5, 2 and 1.
  • For 5-Oreo-sized: 10 and 1.
  • For 2-Oreo-sized: 10 and 1.
  • For 1-Oreo-sized: 10, 5, and 2.

Then, the sketch of the P.D.M. would look like the following:

Is this another graph?!

If you are familiar with graph theory (or if you read the graph section of the blog) you can recognize that the sketch of Serkan the math teacher’s machine is a planar graph:

You should connect the numbers (dots) using lines (connections) according to the golden rule.

One wonders…

What if Serkan the math teacher asks 12 problems?

For 12 problems, the numbers of prizes are going to be: 12, 6, 4, 3, 2 and 1.

In such a situation, can Serkan build his machine? In other words; is it possible to connect the dots for 12-sized P.D.M.?

Hint: First, you should consider where the lines should be. Also, you can arrange the dots in any order you’d like.

M. Serkan Kalaycıoğlu

Real Mathematics – Algorithm #7

How to fail your math test algorithm (H.T.F.Y.M.T.A.)

The late 90s…

At last, there is a computer at home. Now a new battle emerged between Steve and his brothers: “whose turn it is for the computer?”. Thanks to his high grades at school, Steve won this battle easily. After his victory, Steve started to crush zombies in Carmageddon, won the Champions League in Fifa 98, and did such things in Duke Nukem which I can only tell you face to face over a cup of latte.

Steve’s computer game madness went berserk after he met with a football simulation game called Championship Manager. On top of all these games, at least a few days a week, Steve continued to play football&basketball with his friends. A disaster was waiting for him at the end of this road. How didn’t he foresee this?! He was about to fail all his tests in school!

The first warning was the math test. There was less than a day left for the test. Steve should have studied, but he developed some habits since he had a computer. Now, instead of studying, he had a various number of chances to waste time:

1.Staying at home

When Steve decided to stay home, he would get stuck to his computer. Anyone can guess that he was not using his pc for his school. He was just playing one of the following games:
a. Fifa
b. Carmageddon
c. Championship Manager

2.Going out

Whenever Steve went out, he was not going to the library to study:

a. Chase any ball (football or basketball)
b. Behave like a bum with your friends (a.k.a. meet with your friends and do absolutely nothing productive)

Graph of H.T.F.Y.M.T.A.

In the previous posts, I mentioned what graphs are and how they can be useful in certain situations. In Steve’s situation, using graphs can be very helpful to understand what is going on. Since Steve chooses not to study for his math test, his decisions will lead him to fail the test:

What does the graph tell us?

In the graph above, lines represent Steve’s choices, and dots represent at what state he is in after his choices. The graph tells us two certain things: Steve decides not to study and eventually he fails the math test.

This is why the lines that show Steve’s choices have directions.

During making choices, some steps cannot be skipped. For example, in order to play Fifa, Steve first should sit beside his computer, and to do that, he first should decide to stay home.

Let’s assume that Steve’s choices are like the following:

Stay home -> Turn on the pc -> Play Fifa.

In such a situation, since Steve has limited time before the test, he cannot play Fifa and then return and study for the test. After his decisions, there is no other country than “fail-town”.

Another thing the graph tells us is that Steve cannot go back after making a decision. Mathematicians call these kinds of graphs as “acyclic/chain digraphs”.


Some Information

Acyclic Digraphs

An acyclic digraph does not have a cycle. In other words, once you start moving on an acyclic digraph, you can never go back to the point you previously were at.

An acyclic (finite) digraph has at least one “source” and at least one “sink”.

A point is called “source” if it has no lines leading into it from any other point(s). A point is called “sink” if there are no edges from that point to any other point(s). In the graph of Steve, “don’t study” is the source, and “Result: F for fail” is the sink.


One wonders…

We all use acyclic digraphs during our daily lives. To show an example, I will use Steve’s life once again.

Every school day, Steve takes a shower as soon as he wakes up and gets ready for school. His steps are more or less like the following:

Wake up

Get into the shower

Brush teeth after shower

Get dressed

(Steve’s school uniform consists of pants, shirt, tie, and a vest.)

Q: Draw the graph that shows the way Steve gets ready for school.

Ps. After the shower, Steve must complete his tasks in the right direction. For example, he cannot put on his boxer before pants, can he?!

M. Serkan Kalaycıoğlu

Real Mathematics – Algorithm #6

The year 2600…

Finally, we discovered a planet where we can live besides Earth. Scientists named this planet as T-489. The living conditions in T-489 seems very much like the Earth’s. Satellite views show that there is water on this planet. And scientists discovered that the atmosphere on this planet is almost the same as Earth’s atmosphere.

T-489 and its system.

Space agencies of the world’s biggest countries gathered a team to discover T-489. According to the plan, the first team that lands on T-489 will secure the landing point and build a headquarter. This point is set beforehand with the use of the satellite photos.

After the completion of the headquarters, three separate teams will be sent to T-489 to discover certain points on the planet that was set beforehand. These teams will be searching for the possible existence of life forms as well as testing the life conditions on each location.

The map: Headquarters is at the yellow point. Other points must be discovered by the teams. In an emergency, teams can use the roads and return to the headquarters.

The headquarter of T-489 should develop a map for those three teams which will be scouting the planet. This map is crucial since it will show how those teams should travel on the planet, and it will also explain how a team can return to the headquarters in case of an emergency.

In certain situations such as not knowing where a team is at a given time, headquarters must add an algorithm to the map. This way, in an emergency, any team can use this algorithm and return to the headquarters safely.

Road Coloring Problem

In 1970, Roy Adler suggested the road coloring problem. After almost 40 years, an Israeli mathematician called Trahtman solved this problem.

Yellow point the headquarters. Select a point and use the roads that in this order three times: blue-red-red. You will always end up at the yellow point.

Trahtman thought of the map as a graph shown above; each road had a direction and a distinct color. Trahtman created these directions and colors such that, following a certain algorithm, your travel would always end up on a specific point. In this example, the algorithm is following blue-red-red roads three times and always ending up at the yellow point. You can try at any point and see by yourself.

Chaos on T-489

You must develop a map for the teams that will be scouting the planet. Headquarters and the locations to visit are shown as follows:

M is the headquarters. A, B and C are the points to be discovered by the teams.

You should prepare for the worst possible scenario. If the connection breaks between the headquarters and the scouting teams, and if those teams can’t access to the map, your algorithm might save their lives.

The idea is: Place a sign on the entry of each point. These signs will have only two pieces of information: The color and the direction of the roads. (Why don’t you just hang the map on these signs? Because you are not sure if there is an alien life form on the planet. And in case they exist, such maps might put the headquarters and the whole mission at danger.)

Let’s say you created a map as follows:

Such a map would have an algorithm like this: Follow red-blue twice, and you will always return to the headquarters.

One wonders…

Create a map and an algorithm that will let you arrive at the point M in each time.

Let’s say that there is one more point added to the discovery. Create a map and an algorithm such that, whenever you use the algorithm you will end up at the headquarters.

(Try to use the least number of roads on your map.)

M. Serkan Kalaycıoğlu

Real Mathematics – What are the chances?! #8

Hooligans

This year’s cup final will be between two arc-rivals of the country; it is both a final and a derby game. Supporters of both sides are waiting anxiously for the game. Though, Steve has no interest in this game. His team was eliminated in the semi-finals.

Two of Steve’s close friends, Jack and Patrick, have been arguing over the final for the past two weeks. Jack thinks his team FC Ravens will be the winner as Patrick thinks his team AC Bolognese will be victorious. Meanwhile, Steve is thinking about what to eat at supper.

Jack

“We haven’t lost to them in years. The cup is 90% ours. But, this is football; anything is possible. Well, that makes 10% anyway. Yeah, I am pretty sure of my team. If you want, let’s bet on it!”

Patrick

“We are the best team of the season. Plus, our striker scored 69 goals in 27 games. The cup is 75% ours. But, we have been unlucky against them. Which is why I am giving them 25%. Steve, I can bet on it right this second!”

Smart Steve

Steve is aware of the fact that either of these teams will be victorious as this is a cup final and there is no chance for a tie.

He would like to take advantage of his friends’ blindness. Steve would like to set such conditions for the bets such that he would profit whatever the result is.

What would you do if you were in Steve’s situation?

M. Serkan Kalaycıoğlu

Real Mathematics – Strange Worlds #17

Topology On Your Head

Often you see me writing about changing our perspective. For example, when you encounter a baby first thing you do is to make baby sounds and try to make the baby laugh. Whereas if you’d looked carefully at baby’s hair, you could have seen a very valuable mathematical knowledge hidden on the baby’s head:

As shown above, there is a point on each baby’s head. You can see that the hair besides that point is growing in different directions. Can you tell me which direction the hair grows at that exact point?

Hairy ball theorem can give us the answer.

Hairy Ball Theorem

Hairy ball theorem asks you to comb a hairy ball towards a specific direction. The theorem states that there is always at least one point (or one hair) that doesn’t move into that direction.

You can try yourself and see it: Each time at least one hair stands high. This hair (or point) is a sort of singularity. That hair is too stubborn to bend.

Baby’s hair is some kind of a hairy ball example. (I use the expression “some kind of”because the hairy ball has hair all over its surface. Though the baby’s head is not covered with hair completely.) This is why the point on the baby’s head is a singularity. It is the hair that gives a cowlick no matter how hard you comb the baby’s hair.

Torus

Hairy ball theorem doesn’t work on a torus that is covered with hair. In other words, it is possible to comb the hair on a torus towards a single direction.

No Wind

Hairy ball theorem can be used in meteorology. The theorem states that there is a point on earth where there is no wind whatsoever.

To prove that, you can use a hairy ball. Let’s assume that there is wind all over the earth from east to west. If you comb the ball like that, you will realize that north and south poles will have no wind at all.

On Maps

The hairy ball theorem is a kind of a fixed point theorem. Actually, it is also proven by L. E. J. Brouwer in 1912.

One of the real-life examples of the fixed point theorem uses maps. For example, print the map of the country you live in, and place it on the ground:

You could use a smaller map though.

There is a point on the printed map that is exactly the same as the map’s geographical location.

“You are here” maps in malls or bus stops can be seen as an example of this fact.

One wonders…

Assume that all the objects below are covered with hair. Which one(s) can be combed towards the same direction at all its points? Why is that?

M. Serkan Kalaycıoğlu

Real Mathematics – Strange Worlds #16

The Walk

  • Select two points in the classroom.
  • Draw a line between them.
  • Send a student to one of those points.
  • Once the student starts his/her walk, he/she should arrive at the other point exactly 10 seconds later.
  • Everybody in the classroom would count to 10 to help the walker.
    Ask the student to do the same walk twice while recording the walk using a camera.

The goal of the experiment

After the experiment is done, the following question is asked to the classroom:
“Is there a moment during both walks when the student stands at the exact point?”
In other words, the student walks the same distance in the same amount of time at different speeds. The goal is to find if there is a moment in both walks when the student passes the exact point on the line.
First of all, we should give time to the students for them to think and brainstorm on the problem. Then, using the video shots, the answer is given.
The most important question comes at last: Why so?

Weeding out the stone

In my childhood, one of my duties involved weeding out the stones inside a pile of rice. To be honest, I loved weeding out. Because I was having fun with the rice as I was making different shapes with it.

Years later when I was an undergrad mathematics student I heard of a theorem that made me think of my weed out days. This theorem stated that after I finish the weed out, there should be at least one rice particle that sits in the exact point where it was before the weed out started. (Assuming that the rice particles are covering the surface completely.) In other words; no matter how hard to stir the rice particles, there should be at least one rice particle that has the exact spot where it was before stirring.

This astonishing situation was explained by a Dutch mathematician named L.E.J. Brouwer. Brouwer’s fixed point theorem is a topology subject and it is known as one of the most important theorems in mathematics.

The answer to the walking problem,

The walking problem is an example of Brouwer’s fixed point theorem. This is why the answer to the question is “yes”: There is a moment in both walks when the student stands at the exact point on the line.

I will be talking about Brouwer’s fixed point in the next article.

One wonders…

A man leaves his home at 08:00 and arrives at another city at 14:00. Next morning at 08:00 he leaves that city and arrives at his home at 14:00, using the exact roads.

Conditions

  • Starting and finishing points are the same, as well as the time intervals of both trips.
  • The first condition means that the man could travel in his choice of speed as long as he sticks to the first condition.

Is there a point on these trips where the man passes at the exact time during both trips?

Hint: You could assume that the distance is 600 km and the man must finish that in 6 hours. For instance, he could have been traveling 100 km/h the first day, and the next day 80 km/h in the first 2 hours; 100 km/h in the next 2 hours, and 120 km/h in the last 2 hours of the trip.

M. Serkan Kalaycıoğlu

Real MATHEMATICS – Geometry #20

Escape From Alcatraz

Imagine a classroom that has 5 meters between its walls in length. Tie a 6-meter long rope between these walls. Let the rope be 2 cm high off the ground. Since the rope strained to its limits, its 1-meter long part hangs from either side of the rope.

The ultimate goal is to escape from the classroom from under this rope, without touching the rope.

Rules

  • Escape should be from the middle point of the rope.
  • One should use the extra part of the rope to extend it.
  • One of the students will help you during the escape. He/she will strain the rope for you so that you can avoid touching the rope.
  • Each student has exactly one try for his/her escape.

Winning Condition: Using the least amount of rope for your escape.

Football Field

Legal-size for a football field is between 90 and 120 meters in length. Assume that we strain a rope on a football field that is 100 meters long. We fixed this rope right in the middle of both goals while the rope is touching the pitch.

The middle of the rope sits right on the starting point of the field. This is also known as the kick-off point.

Let us add 1 meter to the existing rope. Now, the rope sits flexed, not strained, on the field.

Question: If we try to pick the rope up at the kick-off point, how high will the rope go?

Solution

We can express the question also as follows:

“Two ropes which have length 100m and 101m are tied between two points sitting 100m apart from each other. One picks the 101m-long rope up from its middle point. How high the rope can go?”

If we examine the situation carefully, we can realize that there are two equal right-angled triangles in the drawing:

Using Pythagorean Theorem, we can find the length h:

(50,5)2 = 502 + h2

h ≈ 7,089 meters.

Conclusion

Adding only 1 meter to a 100-meter long rope helps the rope to go as high as 7 meters in its middle point. This means that a 1-meter addition could let an 18-wheeler truck pass under the rope with ease.

M. Serkan Kalaycıoğlu

Real MATHEMATICS – Strange Worlds #15

Human Knot Game

Inside a classroom divide students into groups such that each group has at least 5 students. Groups should stand up and form a circle before following the upcoming instructions:

      1. If there is an even number of student in a group:20190812_153400.jpg

  • Have each student extend his/her right hand and take the right hand of another student in the group who is not adjacent to him/her.
    20190812_153408.jpg
  • Repeat the same thing for the left hand.
    20190812_153415.jpg

      2. If there is an odd number of student in a group:

20190812_153325.jpg

  • Have each student (except one of them) extend his/her right hand and take the right hand of another student in the group who is not adjacent to him/her.
    20190812_153338.jpg
  • Then take the extra student and have him/her extend his/her right hand so that the extra student can hold the left hand of another student who is not adjacent to him/her.
    20190812_153345.jpg
  • Finally, repeat the process for the students whose left hands are free.
    20190812_153353.jpg

In the end, students will be knotted.

The-Human-Knot-Game-e1447920419118-663x375

Now, each group should find a way to untangle themselves without letting their hands go. To do that, one can use Reidemeister moves.

Reidemeister Moves

Back in 1926 Kurt Reidemeister discovered something very useful in knot theory. According to him, in the knot theory one can use three moves which we call after his name. Using these three moves we can show if two (or more) knots are the same or not.
For example, using Reidemeister moves, we can see if a knot is an unknot (in other words, if it can be untangled or not).

What are these moves?

Twist

First Reidemeister move is twist. We can twist (or untwist) a part of a knot within the knot theory rules.

Poke

Second one is poke. We can poke a part of a knot as long as we don’t break (or cut) the knot.

Slide

Final one is slide. We can slide a part of a knot according to Reidemeister.

One wonders…

After you participating in a human knot game, ask yourselves which Reidemeister moves did you use during the game?

M. Serkan Kalaycıoğlu