beauty of mathematics

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:

The inverse of this rope is:

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 – Game #13

Crossing the bridge

Years and years ago, there was a village known as Togan. It was located in Mesopotamia’s fertile lands, next to rivers and beautiful waterfalls. There were only a few hundred people who lived in Togan. All of the Togan were farmers, except one: Berkut.

Berkut was the oldest man in the village. He had a long white beard. His story had become a kind of a legend within time. According to the people of Togan, whoever entered his property would never be seen again. This is why Berkut’s house was the only house that stayed on the other side of the river of Togan.

For kids, Berkut was a mystery. Whenever Berkut was out on his garden, kids would gather and watch him from the other side of the river.

People of Togan were hardworking farmers. They would be working on their farms starting from their childhood. Like the rest of the Togan, Ali starting helping his family at an early age. Ali would work from sunrise to the sunset.

For Ali and his friends, Berkut’s situation was one of the hot topics. One day, these four friends decided that they would skip working and go across the river to investigate Berkut’s house. This legend had to be questioned!

The next day, while he was out on his garden, Berkut realized that four kids were about to cross the river, using the old bridge. He watched them crossing the bridge in pairs, as the old bridge was not strong enough to carry more than 2 of them at the same time.

He, then, went inside his house as Ali and his friends were approaching the house. When kids were inside the garden, they saw that a jar of cookies and a steamy teapot was waiting for them. While they were checking the table, Berkut went outside and greeted them. Kids, dumbfounded, started screaming as they all dispersed out of the garden in different ways.

They found each other only after it was dark. Now, kids, who had only one lamp with themselves, had to cross the bridge as fast as they can. But they couldn’t risk crossing the old bridge at once. Each time, only two of them could cross the bridge.

Crossing Times:
Jane: 1 minute
Ali: 2 minutes
Tom: 6 minutes
Jenny: 10 minutes

Since they had to cross the bridge in pairs while sharing a lamp, their speed would be at the rate of the slowest of the pair.

Now, you must solve this problem:

What is the fastest route to the other side of the river?

M. Serkan Kalaycıoğlu

Real Mathematics – Numbers #11

A very long time ago in Mesopotamia, a few hundreds of people lived together in the village of Badaks. Badaks were very hard-working people, and they were among the first farmer communities. Their lives depended on two things more than anything: Their farms and sheep.

There was a lot of sheep in the village of Badaks. Thanks to them, people of Badaks were able to protect themselves from cold weather. Their milk and meat were also important to Badaks as food sources. Because of their importance, the person in charge of the sheep had to be wise and trustworthy.

Zaylin, head of Badaks, was in charge of this crucial duty.

Every day, with the first sunlight, Zaylin took the sheep out of their pens for them to explore the hills and graze the green grass of the village of Badaks. Before the sun is gone, Zaylin had to gather the sheep and be sure that every sheep returned to the pens.

Even though Badaks were one of the most progressive communities of their time, they didn’t know the use of numbers like the rest of humanity.

At this point, Zaylin had a bit of a problem: How did he know that he returned with the same number of sheep as left in the morning? Don’t get me wrong; Zaylin was an intelligent person for his time. But like everybody else, he didn’t know how to count.

One wonders…

Put yourself in Zaylin’s shoes: Is it possible to detect if you lost any sheep or not when you finish a day without counting or any use of numbers?

M. Serkan Kalaycıoğlu

Cookies

You can reach the 2-player game cookies through the link below:

https://www.geogebra.org/m/sxmvcj7n

Have fun!

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:

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 – What are the chances?! #7

How Close?

Game: In a group, everyone is asked to pick a number between 0 and 100. Even though it is possible for more than one person to pick the same number, it is forbidden for participants to communicate with each other.

Winner: Winner is the person who is closest to the two-thirds of the average of the picked numbers.

Question: Is there any way for you to optimize your chance to win the game?

At first glance, one might think that it is not important which number you pick between 0 and 100. Because the winning number depends on the choices of others’. Although, if there is a player who has probability knowledge, he/she could maximize his/her chance for winning the game.

Step #1
Assume that we have a group of 12 people, and every individual selects 100. Then the average becomes:

12*100/12 = 100.

The winning number is the one that is closest to the two-thirds of the average. That means 100*2/3 = 66,666…

66,666… is the highest winning number for this game. If you are aware of this fact; then you would select a number that is between 0 and 66.

0-66

Obviously, there is a chance for you to win the game even though you select a number higher than 66. Then again; why would you select such a number if you know that the winning number is between 0 and 66?!

What if everyone realizes…?!

Let’s assume that you are aware of this fact. Then while others will pick a number between 0 and 100, you will be picking a number between 0 and 66. This is a huge advantage. But, suddenly you realized something else: What if everyone came to the same conclusion?

Step #2

If everyone knows that the winning number can’t exceed 66,666…, then no one will choose a number higher than 66. Hence, everyone will choose a number between 0 and 66.

In this situation, the highest average can be 66:

66*12/12 = 66 average.

66*2/3 = 44 is the highest winning number.

0-44

This means that if everyone selects between 0 and 66; the winning number can’t exceed 44. Then, why would you choose a number that is higher than 44?!

Step #3

If everyone comes to the same conclusion, then no one within the group will select a number that is higher than 44. This causes a new calculation. Since everyone knows that the winner will be between 0 and 44, the winning number can at most be:

44*12/12 = 44 (average)

44*2/3 = 29,333…

This means that the winning number is at most 29. Then no one will choose a number that is higher than 29.

0-29

If one follows the same logic, at the end of the 11th step he/she will find 0 (zero) as a result. This is why picking zero for everyone is the most logical move for the whole group. Using your probability knowledge, one will eventually conclude that zero is the most reasonable choice for each individual.

Conclusion

Mathematics can help a group find a solution that benefits everyone within the group, even though there is no communication inside that group of people.

One wonders…

You know a person inside the group who isn’t good at mathematics. In this situation, would you change your logic? Give your answers using probabilistic calculations.

Ps. You can click here and create yourself an example set.

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:

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.
Repeat the same thing for the left hand.

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

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.
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.
Finally, repeat the process for the students whose left hands are free.

In the end, students will be knotted.

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