Numbers #12

Subitizing: The ability to recognize(or guess) the number of a small group of objects without counting.

The name subitizing comes from the Latin word “subitus” which means “sudden”.

Subitizing can be seen in many every-day activities. One of them is a six-pack soda. No matter how they are lined up, we know that the number of soda bottles is 6. We inherit this knowledge without counting the bottles. And if we decide to drink one of them, we automatically know(without having to count them) that the number of soda bottles left is 5.

You don’t have to count the dots on the surfaces. You just know that it is 5.

Another example of subitizing can be given from the game backgammon. Assume that two dices are rolled and you identify them as 2 and 5. The process of identifying the dices can be measured in milliseconds. This can be even shortened as you spend more time playing the game. In short; subitizing is a skill that can be developed if one spends time and work on it.

Research studies showed that 6-month olds can differentiate, visually (a top bounces 3 times) and from sounds (clapping hands 3 times), between 1, 2, and even 3. In other words; humans start developing the number concept when they are just infants.

Kebab Truck & Subitizing
Subitizing is hidden behind the number of customer groups in the game of Kebab Truck. As the game is played, scores become higher and higher. The reason behind this is that players’ subitizing skills are improving.

Let’s check this scene from Kebab Truck:

In the beginning, you will be making certain moves during the game. Nevertheless, in time, your moves will differ substantially. The biggest reason behind this is that your subitizing skills were improved while you were playing the game.

Kebab Truck also helps the players to develop their basic arithmetic skills. These improvements are not limited to adding and subtracting the number of customers. Once you understand how the scoring system formulated, you will realize that (to maximize your scoring) multiplication is an important part of this game as well.

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 – 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

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.

Berkut’s home.

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 – 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 – Geometry #19

How Much Chocolate?

It is midnight and my stomach is talking to me. I hope to find something to eat in the kitchen and I see a chocolate bar:

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Immediately made myself a cup of coffee and broke a piece of the chocolate bar:

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After I “killed” the broken piece I started having second thoughts about my decision: Oh God; Did I eat too much chocolate?

I placed the leftover on a grid. This way I found where both whole and the broken piece lies on the grid:

çiko1

The broken piece is shaped like a simple polygon. My goal is to calculate the area of that piece. There are several ways I could calculate the area. Although, the first thing comes to my mind is a theorem called “Gauss’ shoelace theorem”.

Gauss’ Shoelace Theorem

The shoelace theorem can only be applied to simple polygons. In order to use the theorem, I have to find where the edges of the simple polygon lie on the grid:

çiko2

Theorem uses these points just like shoelaces. But first we have to define the edges and make a list of them:

çiko3

lak1

 

 

Do not forget to add the first edge to the bottom of the list.

Now you can multiply the numbers diagonally; from right to left and left to right. Then add left to right and subtract it from right to left ones:

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{(0*0) + (5*1) + (4*3) + (5*3) + (0*0)} – {(0*5) + (0*4) + (1*5) + (3*0) + (3*0)}

{32} – {5}

27.

The area of that simple polygon can be found by dividing the result above:

27/2

13,5

M. Serkan Kalaycıoğlu

Real MATHEMATICS – Game #12

Triangle Invasion

Invasion is a multiplayer game which needs only a pencil and a pen. To start the game one should draw a big triangle on a paper whilst assigning the corners with 1-2-3:

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Then the big triangle should be triangulated (without any rules):

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Once the construction is over, we can start labeling the triangle corners with the direction of the following rules:

  1. One can label 1-2 side with either of 1 or 2.
  2. One can label 1-3 side with either of 1 or 3.
  3. One can label 2-3 side with either of 2 or 3.
  4. One can label the inside of the original triangle with any of 1, 2 or 3.

Progress of the Game

  • Assign the corners according to the rules.
  • In order to invade a triangle, player must assign the final unattached corner of that specific triangle.
  • Goal is to avoid invading a triangle that has corners 1-2-3. Winner is the player who invaded least number of such triangles.

Sperner’s Triangle

Idea of the invasion game comes from the Sperner’s triangle which is discovered by Emanuel Sperner in the 20th century. Triangulated shape in the invasion game is an example of Sperner’s triangle.

After labeling the corners a Sperner triangle will always give a small triangle that has corners 1-2-3:

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Actually, Sperner’s triangle always has odd-numbered 1-2-3 triangles.

This is why invasion game never ends with draw.

Dead End

Sperner’s triangle can be used to construct many games. For example, let’s say in a Sperner’s triangle all the sides of the little triangles that are connected between 1 and 2 are doors. And all the other sides are walls:

If one tries to walk through these doors, that person will end up in two situations:

  1. The person will end up inside a 1-2-3 triangle:
  2. The person will find him/herself outside of the triangle:

    M. Serkan Kalaycıoğlu

Real MATHEMATICS – Game #11

Sprouts

Sprouts is a multiplayer game which was created by M. S. Paterson and brilliant J. H. Conway back in 1967. All you need for playing sprouts are just a piece of paper and a pen/pencil.

  • Game starts with 3 dots on a paper:
    20190429_140440.jpg
  • Players take turns and draw lines from one dot to another (a line can be drawn to the same dot as well). Lines don’t have to be straight and a new dot must be placed on each line:
  • Lines can’t cross one another:
    20190429_140912.jpg
  • A dot is called “dead” if it has 3 lines coming out of it. In other words any dot can be connected to at most 3 lines:

    On right: A, B and E have 3 dots. This means A, B and E are all “dead”.

  • Player who draws the last possible line is the winner.

Brussels’ Sprouts

Brussels’ sprouts is a different kind of sprouts game. It is a multiplayer game just like regular sprouts and all it needs are a paper and a pen/pencil as well. But this time game starts with dots that have thorns. Assume that we will start a Brussels’ with two dots with 3 and 4 thorns on them:

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Players take turn to draw lines between thorns. When a player draws a line, he/she should mark a new dot that has two thorns on it:

Just like regular sprouts, lines can’t cross in Brussels’ sprouts. And the player who draws the last line wins the game:

Euler and Sprouts

You might wonder how on Earth I get to mention Euler in a game that was created about 200 years after he passed away. I recommend you to check Euler characteristics article.

Euler says:

Let’s imagine that V dots and E lines (which don’t cross one another) are sitting on a plane. If the number of faces on this shape is F (don’t ever forget to count the whole plane as one face), then the equation

V – E + F = 2

will always be satisfied.

Take a finished Brussels’ sprouts game on hand:

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Find the numbers of the dots, lines and faces:

Apply Euler’s formula:

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Euler will always be right!

Four Colors

Now take any Brussels’ sprouts sheet and color the faces on it. (Neighboring faces have different colors.)

You will see that four different colors will be enough to color any Brussels’ sheet:

One wonders…

  1. At most how many turns can there be in a regular sprouts game that starts with 3 dots?
  2. Is there a winning strategy for sprouts?
  3. Start a Brussels’ sprouts with 3 dots. If each of them has 3 thorns, at most how many turns can there be?

M. Serkan Kalaycıoğlu