## Real Mathematics – Social Situations #3

Battle of the Couples

Bruce: I have waited for this game since the start of the season. We will be hosting our arch rivals in the final game of the season. And like that is not enough excitement for me, if we win we will crown as champions: It is either us or them.

Every year when I decide to buy a season ticket, I dream of going to this game. I haven’t missed a game since 2006 and this season’s game will be cherry of top in case we win. We have to win and I must witness it.

Jane: My favorite band is retiring and they are on the road for the last time. Luckily for me they will be visiting my town this time: I will have the opportunity to watch them alive in my city for the first and the last time.

When I was 11, I bought a random album. I went home, pressed start on my music box and fell in love with them instantly. It was their best-selling album from 1992. I have been laughing with their songs, I have been crying with their songs. I had spent my youth with them. And I will say farewell to them in person.

Bruce and Jane have been together for some time, but they are in a pickle now. Derby game is postponed and both events will be started at the same day, same hour.

Ideal Couple

In this scenario we assume that Jane and Bruce are equally (or at least equivalently) in love with each other.

Since they are ideal couple, spending time together is the most important thing for them. That is why we’ll give them 1 point for being together if they choose to go to the same event.

We’ll give them 1 extra point for their favorite event: For Jane concert is 1 extra point and for Bruce game is 1 extra point.

If they are not together, they get -1 point as they are unhappy for being apart. This is why in case they go to separate events they will get 0 point each.

Then we can construct the game matrix as follows:

In this game if players choose to be selfish and only consider their happiness, Jane would choose to go to the concert as Bruce would choose to go to the game. We already know that the outcome of concert-game is (0,0). In order to test if this is the Nash equilibrium or not, we must test players one by one. First let Bruce chooses first. He will decide to go to the game.

If Jane knows about it, she will have to choose going to the game also since outcome of the game (1) is larger than the outcome of the concert (0) for her. In this case the equilibrium is game-game.

Now let Jane do the first selection. She will choose to go to the concert:

In this case Bruce has to choose to go to the concert as its outcome (1) is larger than game’s outcome (0). Here, equilibrium is concert-concert.

We just found out that there are two Nash equilibriums in this game: Concert-concert or game-game. Both cases could happen if only Jane and Bruce are willing to cooperate. Otherwise if they act selfish, they will get no happiness whatsoever.

Personal Thought: Even with an ideal couple Bruce should compromise since “she” will always win.

Jane Doesn’t Love Bruce

Jane thinks she could do better. And she has a point: Bruce is a 43-year old unsuccessful computer engineer who is slowly going bald. On the other hand Bruce is very much so in love with Jane and deep inside he knows that she is a catch.

For Jane, being in the concert means more.

For Bruce, being with Jane and going to the game are equally important.

Let’s build our game with this information.

If they both decide to go to the concert: Jane will receive 2 points (0 from being with Bruce, 2 from being at the concert). Bruce will get 1 point and that comes from being with Jane.

If they both decide to go to the game: Jane will receive -1 point (0 from being with Bruce, and -1 from not being in the concert). Bruce will get 2 points: 1 from being with Jane, the other from being at the game.

Jane to the concert, Bruce to the game: Jane will get 2 points in total, all coming from being at the concert. Bruce will get 1 point in total and that comes from being at the game.

Jane to the game, Bruce to the concert: Jane will be furious and get -1 points as Bruce will get 0 point.

Now we can construct the matrix of the game as follows:

If both players are selfish, Jane would choose to go to the concert as Bruce would choose to go to the game. In this scenario concert-game becomes the result of the game. Its outcome is (2,1).

We should check if concert-game is the Nash equilibrium for this game.

If Jane knows that Bruce is going to the game, she would have two options: 2 and -1. Obviously she would choose 2; that is going to the concert. Then in this game result becomes concert-game.

If Bruce knows that Jane is going to the concert, his choices would get him 1 point in either case. This is why Bruce would have two identical choices. Concert-concert and concert-game will have the same probabilities.

In conclusion, there are two Nash equilibriums for this game: Concert-concert and concert-game. Both cases have the same outcome that is (2,1).

One wonders…

Find the matrix of the game and its Nash equilibrium when Bruce doesn’t love Jane while Jane does love him.

## Real Mathematics – Life vs. Maths #4

Where am I?

Mathematicians love to make generalizations. Personally I don’t enjoy that either, but generalizations are very useful in case you’d like to make some mathematical magic. Is there anything cooler than magic?!

Let’s assume (beware of the generalization that is coming towards you) that we had taken N random steps in one dimension. Even though it is very small there still is a possibility that those steps could well be taken into the right hand side (or left). That would have meant that after N random steps, we are standing at +N (or –N). In this case we understand that we are N steps away from the starting point.

If we take half of the N random steps to the right, and other half to the left hand side we would be standing right on the starting point. In that case we would be 0 steps away from the starting point.

These two scenarios are the furthest (N steps) and closest (0 steps) destinations to the starting point after N random steps are taken. Thus, we are finally aware of the fact that after N random steps in one dimension, we have to stop at 0 to N steps away from the starting point.

Q: Is there an algorithm to find out how far we would be to our starting point even before we take a certain number of random steps in one dimension?

For N random steps the answer is the square root of N. For instance if we take 100 random steps in one dimension, we would be √100 = +/- 10 steps away from the starting point.

Now you are wondering: Where and how can we use this information in life?

There are 16 teams participating in the Euroleague, which is the most prestigious tournament of Europe. In the regular season of the Euroleague teams get to play with one another twice. In the end of the regular season top 8 teams advance into the playoffs where champion of the season is decided.

Let’s assume that you are supporting a team that is average which means your team would like to fight for the top 8 positions. Just before the season starts you look at the calendar and try to guess how many games your team could win in order to stay in this fight: “If we beat Barcelona at home, and Darüşşafaka on both games…”

You really don’t have to do that. Obviously I will show you how you can use mathematics in order to guess how many wins your team should get.

There are two possible outcomes for a basketball game: Win or lose. It doesn’t matter how strong your opponent is, a game will have two outcomes whatsoever.

Similarities with Random Walks

In one dimension we know that there are two outcomes for a random walk: Right or left. And this is why basketball games can be treated as a random one dimensional walk.

In the regular season each team will play 15×2=30 games. This is same as taking 30 random steps in one dimension.

Then the difference between win and loss column after 30 games can be calculated with taking square root of 30.

√30 = 5,47…

We will call it 6 games. Outcome of these 6 games depend on luck. Your team can win or lose each and every one of them. It proves that after 30 games you either won 6 games more than you lost, or you lost 6 games more than you won:

Conclusion

This information we found using one dimensional random walks tells us that if a team wins between 18 to 12 games in the regular season, that team will be fighting for the playoff positions.

These pictures show how two previous regular seasons ended in the Euroleague. As you can see teams that won between 18 to 12 games fought for a playoff spot.

One Wonders…

Try to apply what you learned for one dimensional random walks to a football team that is participating among 18 teams. If it is an average team, what kinds of predictions can you make for the team?

Don’t forget to include the third possibility: Win, lose or draw.

M. Serkan Kalaycıoğlu

## Real Mathematics – Geometry #9

Finding Pi

Humans discovered a connection between a circle’s circumference and its diameter around 4000 years ago. This connection, what we call the number pi now, affected everyone who was involved in mathematics and/or engineering in the early times. Countless people, including historical figures, worked in order to find what this number is, but all of them failed to find such answer.

Some of them tried to find an approximation and got really close. When it was their time to shine for geometry, one ancient Greek whom we are familiar with had taken approximation methods to a different level. This famous Greek dude was Archimedes.

Archimedes is believed to live between 287 BCE and 211 BCE. If you take a look at the history of mathematics it was common to see approximations for the number pi, especially in Babylon, Sumer, ancient Egypt and China civilizations.

Although, Archimedes was different in an important way: Before him, method to find the number pi was not that complicated. A certain circle would be inspected and compared to a similar regular polygon. Their areas or perimeters were thought to be same which would give an approximation for pi.

Let’s try to find an approximation for the number pi using this method.

Draw a circle and divide it into 12 equal parts, like a pizza.

Cut the 12th slice into two equal parts and lay down all the slices side to side.

Slices will seem like a rectangle which is a regular polygon. At this point circle and rectangle are believed to have same shapes.

Circumference of circle is found with pi*R, where R is the diameter of the circle. It is obvious to the naked eye that the long sides of the rectangle would give the circumference of the circle. Hence, we can find an approximation for the number pi.

Try this for another circle that has different diameter length.

And take the arithmetic mean of both results. There is the number pi… Well, it is just an approximation.

Archimedes’ Method

Even though there were so many engineering marvels as well as important geometry knowledge, Archimedes was not happy with the methods used to find the number pi. He developed his own method that required inscribing and circumscribing the unit circle with regular polygons. Archimedes did something different than the methods which were used before him: He continued using polygons with more sides and compared those results until he found the best polygon.

His method today is known as “the method of exhaustion”. It is the earliest known version of calculus.

You could draw a unit circle yourself. Try to inscribe and circumscribe a square. You’d get pi between 2,8 and 4, which is not bad for the first try. If you continue with polygons that have more sides, you’d get even better intervals.

The thing is, Archimedes didn’t try to find an exact value. Instead he found an interval which was 99,9% accurate!

Archimedes found this interval with starting a hexagon, and he finished his method at a 96-sided polygon. You could also see that 22/7, which is given to students for pi in school, is the upper limit that Archimedes found nearly 2200 years ago!

M. Serkan Kalaycıoğlu

## Real Mathematics – Game #4

The Last Biscuit

I think I was making use of my hunter-gatherer genes when I was a child.

I admit it; I’ve always loved junk food and it was a big problem in my childhood since we had a big family. And on top of that I was among the youngest children in that crowd which meant I had a physical disadvantage against other kids about matters such as getting to eat Pringles first. Also, almost everyone around me was competitive which made it harder for me to get junk food.

In the end my hunter-gatherer genes helped me. Although I wish I was craftier so that I could have created games which only I’d win.

Tea Party

Tea is almost there, biscuits are ready and willing. I am warming up my wrist so that when I dunk my biscuits into my tea I will have enough agility to save the biscuit from getting crumbled into my tea cup. Oh boy! Someone at the door… Now I have a guest!

I have two kinds of biscuits: Cacao and regular. I love the ones with cacao more than the regular ones. Well, who doesn’t?! The problem is that I only have four cacao biscuits along with six regular ones. Neither my friend nor I can decide who will get the cacao biscuits. We could share, but there is no fun in it! That is why we let a game decide our faiths.

Rules:

• Each player will take turn.
• In each turn a player could either take any number of biscuits from just one stack or take the same amount from both stacks.
• The player who takes the last biscuit(s) is the winner.

Example: Host vs. Guest

There are six regular and four cacao biscuits in stacks.

Guest starts first and takes one from cacao.

Host takes one from regular. In second turn guest takes one from each stacks.

Host takes three from regular biscuits.

Guest takes one from cacao.

Now there is one from each biscuits and host takes them both to claim his/her victory.

One wonders…

1. Does it matter who goes first in the host vs. guest example?
2. When did you understand that the guest lost? Would it change anything if guest played his/her move differently in the final turn?
3. Can you find a method that makes you the winner if you play this game with different numbers of biscuits?

M. Serkan Kalaycıoğlu

## Real Mathematics – Puzzle #3

Story of Magic Baklava

Some of you may have heard the story of Excalibur which was about the magical sword of King Arthur of Britain. It was a special sword indeed because the person who was in possession of Excalibur would gain superhuman powers. This is why Excalibur could be counted as a mythological story.

In this article I will be talking about another mythological story. A story which I found in a very old book… A story that might sound unfamiliar to you: Serkan’s magic baklava.

This is a true story that was lived 2700 years ago in the soils of modern Turkey.

Serkan’s father was born in Mardin, an ancient city that is located in the heart of Mesopotamia where first known civilizations flourished. He was the most popular baklava chef of his time until a traumatic event occurred. When Serkan was born, his father made free baklava to celebrate the birth of his first child. Unfortunately eight people were poisoned and hospitalized because of bad baklava…

Serkan’s father could not overcome this horrific event: Eventually he swore that he will never touch another baklava.

Birth of Magic Baklava

Years passed. Serkan was working with a shoe master to help his family financially as he was also an average senior in high school. University entrance exam was just one week away and Serkan’s guidance teacher was handing out candy to everyone in his class. (In modern Turkey it is a tradition to have candy with you in exams as elders believe that it helps you to get a better grade.)

There were two kinds of candies: Orange and mint. Serkan wanted orange but was left with mint candies. When his father saw Serkan with a sad face, he thought that it was time for him to help his beloved son. Next morning he woke his son:

“Son! I broke my vow once and for all and made a magic baklava for you. After eating this magic baklava, you will have superhuman powers and every question shall bow before your pencil! Harvard or Oxford, all the universities in the world will be yours thanks to this magic baklava.

Although, you have to prove that you are worthy of this baklava. This is why I prepared a riddle for you.

• Baklava is inside one of the four boxes.
• Until the exam morning, you can select one box a day.
• Every day I’ll be changing the box of the baklava.
• I’ll be putting the baklava only to its neighbor box. Eg. if baklava is inside box number 1, I should move it to its only neighbor box number 2. If it is inside box number 2, then I should move it to either one of its neighbors box number 1 or 3.
• Find the baklava before your exam.”

This story is known as the most realistic story of mythology. Even after 2700 years, it is still being told.

Questions

1. How can Serkan find his magic baklava?
2. Is there a sure algorithm for that?
3. What if there were 5 boxes? Or n boxes?

M. Serkan Kalaycıoğlu

## Real Mathematics: Numbers #7

Self-Aware Numbers

There are some numbers that are special. Although Sometimes a bored mathematician could specify a few rules and give away definitions. And this is how world would have new special numbers.

Self-aware numbers are like that. It doesn’t matter if they have any use or not, there are self-aware numbers.

How to find them?

When you have a number, take a closer look at each digit. The digit that is at the left end gives us the number of zeros that exist in the number. Next digit gives us the number of ones, and the one on its right gives us the number of twos etc.

If such a number exists, we call them “self-aware numbers”.

Example 1:

1210 is a self-aware number. Let me break it down into its digits and we get:

1 = # of zeros,

2 = # of ones,

1 = # of twos,

0 = # of threes.

Since they are all correct, 1210 is a self-aware number. Good for you 1210!

Example 2:

10 is not a self-aware number.

1 = # of zeros, which is correct.

0 = # of ones, which should have been one!

10 is a bad number… Shame on you 10.

Example 3:

How about 141110; is it self-aware?

1 = # of zeros. Correct.

4 = # of ones. Correct as well.

1 = # of twos. There is no two in the number. Which makes 141110 not a self-aware number. You are bad 141110.

Select and Eliminate

Let’s assume that we have a square full of numbers like the following:

1. Select a number. After circling it, eliminate all the other numbers which stay in the same column and row with the number you selected.

2. Select another number from the remainders. Circle it and repeat the same process.

3. Now select a third number from the survivors. Circle and repeat the process.

4. You’ll see one number remained. Circle it too.
5. Sum of the numbers you circled gives you exactly 10.

Better check it

• Are there any self-aware 10-digit numbers?
• Analyze the square from Select and Eliminate. Is there a specific algorithm for the numbers inside the square?

M. Serkan Kalaycıoğlu

## Real Mathematics: Game #3

Chocolate Box

A dear friend of mine started a new business recently. I bought a box of chocolate with the intention of visiting his store. Eventually I did visit his store. Although I could not stop myself opening the box before my visit. In the end I murdered almost half of the box. Sorry matey!

Today I checked what is left in the box and I realized something: Why aren’t same types of chocolates in the adjacent compartments?

Being Neighbors

In order to avoid neighborhood between same types of chocolates, we should be careful not to place them as shown in the photo.

Let’s call this rule “sufficient chocolate density”, or “S.C.D.”.

Question 1: Assume that you have four different types of chocolates and seven chocolates from each type. Could you have S.C.D. in such a box like the following photo?

Question 2: What is the least number of different chocolate types you can place inside this box?

Question 3: In case you have a box like shown below, at least how many different types of chocolate do you need to maintain S.C.D.?

Question 4: If you had four different chocolate types and nine from each type, could you be able to construct a box that has S.C.D.? If yes, what would it look like?

M. Serkan Kalaycıoğlu

## Real Mathematics: Game #2

In school, mathematics is being used to help kids gain problem-solving skills. Even though I love arithmetic aspect of it, problem-solving is usually focused on arithmetic more than developing strategies for the problem itself. This causes kids to focus on the answer and act without thinking. I can’t emphasis this more: Thinking is integral if you’d like to learn mathematics.

When a kid solves all 100 problems from his/her math textbook, it really doesn’t tell much about his/her problem-solving skills. It only shows that kid knows how to do arithmetic. Unfortunately for that kid, arithmetic is not enough when he/she will face an original problem in future.

Space Racing

Creating new strategies has a positive impact on problem-solving. In order to achieve that one should stop worrying about arithmetic so much and focus on thinking about the problem itself.

It is very important to show kids problems that don’t include arithmetic within themselves. Space Racing is a kind of game that looks like it has nothing to do with mathematics. But in truth, it is a real mathematics problem. You should always remember this: a math answer can be just a paragraph.

Space Racing is a multiplayer game which requires only a paper and a pen. Players put X on empty boxes in order. Player A wins if the last two empty boxes are adjacent. Player B wins if the last two empty boxes are apart from one another.

• Does it matter if player A starts first or not?
• Does it matter how many empty boxes there are in the drawing?
• Would anything change if players put two Xs in each turn?
• Is there a strategy for player A to maximize his/her chances to win the game?
• Is there a strategy for player B to maximize his/her chances to win the game?
• Would it be possible to guess the outcome of the game after certain number of turns?

Example

M. Serkan Kalaycıoğlu

## Real Mathematics: Game #1

Circle of Numbers

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Most of the countries in the world use these symbols for the remarkable decimal system. The system was designed so perfect, even though symbols might change in some regions, logic of the system is conserved all around the world. Decimal system is one of those things that are “universal”.

If you are willing to participate in today’s modern world, you’d better know how to use numbers. You could go on and try to live a day without using any numbers: You’d see that it is impossible to finish even one single day. Actually, using numbers is not enough: you should be able to understand what a given number represents.

Understanding Numbers

What do I mean by “understanding numbers”?

If I show you 2125555555, can you associate it with anything?

How about 212 555 55 55?

Now most of you realize that 212 555 55 55 is a telephone number. Using three blank spaces inside the number 2125555555 changed the way you look at it.

Numbers take too much space in our daily lives. I think this is a great reason to work and develop a good understanding of them. Circle of Numbers is a game that helps children cultivate the ability of using numbers.

In order to play Circle of Numbers, all you need is a pencil and a pen. This game can have various numbers of versions.

Circle of Numbers 1.0.0

Draw a circle.

Place four boxes on the circle.

You have to place the numbers 0, 1, 2 and 3 inside the boxes such that difference of two adjacent boxes will be an odd number.

Example

Circle of Numbers 1.0.1

Again draw a circle with four boxes on it.

This time each of the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 should be used.

In order to do that, you will need ten boxes which means you will have to add six more boxes on the circle.

Difference of two adjacent boxes should be odd.

Example

Let’s assume 4 and 7 are placed as following.

Player knows no matter which number is chosen, there will be number pairs that will not have odd difference. For instance if 5 is chosen, even though 5-4=1 is odd, 7-5=2 will be even. Player should add a new box to avoid this problem. Following will show you a working strategy step-by-step.

Circle of Numbers 2.0.0

Assume that there are five boxes on the circle and you are allowed to use the numbers 1, 3, 5, 8, 9. Can you construct a valid circle?

Check it out

I just gave three different versions for the game Circle of Numbers with three different names: Circle of Numbers 1.0.0, Circle of Numbers 1.0.1 and Circle of Numbers 2.0.0. What do these numbers remind you? What would Circle of Numbers 2.0.1 look like?

M. Serkan Kalaycıoğlu