Real Mathematics – Algorithm #3

Card Games

I remember vividly; at least once a week I was dragged to family meetings. All I wanted was to stay home and play Duke Nukem. But I had to go to those meetings and keep score of a card game which dads play. There were rumors about my high mathematics grades, and I really wanted to fail mathematics just because I might have escaped this responsibility. And game had absurd scores too: “300 to us, 4250 to them, did you write 20 points to us?”

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Everyone who had experienced campus life knows that living in a dorm meant both misery and lots of fun. Personally my favorites were the times when I sat down with three other guys and played card games for hours. After four years I became addicted to the very thing I despised as a child.

I love playing competitive games but if a game has algorithms on the background then I get addicted to that game. In some card games players develop algorithms in their heads. And in a group of close friends, everyone is aware of his/her opponents’ algorithms. This is why making simultaneous adjustments on your algorithm might win you the game.


Pico is a German card game that shows how mathematics can be used in card games perfectly. Even though Pico is a simple game it has astonishing mathematics that lies behind it.


  • In this multiplayer game there are eleven cards which have 2, 3, 4, 5, 6, 7, 8, 9, 10, 13 and 16 written on them.
  • Cards are shuffled and dealt. Each player gets five cards.
  • The only card left gets turned over so that both players know which cards his/her opponent has.
  • In every hand players select a card simultaneously.
  • The card that has the highest number on it wins the hand unless it is more than twice of the other card.
  • The winning card is placed next to the player who won the hand. The number on that card is the score he/she gets.
  • Losing card goes back to its owner.
  • Game continues until either one of the players holds exactly one card.

    In case there is a deck of cards in your home, you could assign K to 13 and J to 16 so that you can play Pico.


I thought about writing a new game which I called Seko:


  • Seko is a multiplayer game just like Pico. There are numbers from 2 to 50. Each player selects six numbers at once. Players roll a dice in order to determine who starts selecting first.
  • Twelve selected numbers are written on a paper.
  • Out of these twelve numbers, each player selects six numbers but this time one by one. Again they roll a dice in order to determine who starts selecting first.
  • In the first hand players select a number simultaneously.
  • If the difference of those numbers is odd, bigger number wins. If the difference is even and less than 20, smallest number wins.
  • The winning number is written in front of the winner. Loser gets his/her number back.
  • Game continues until a player has exactly one number left.

One wonders…

Instead of finding the difference in Seko, add the numbers. If the addition is odd, biggest number wins. If addition is even and less than 50, smallest number wins.

Can you find an algorithm to use in every game of Seko?

M. Serkan Kalaycıoğlu

Real Mathematics – Special Numbers #3

Lynx – Snowshoe Hare War

In North America there is a predator-prey relationship between lynx and snowshoe hare. Snowshoe hare is the primary food of the lynx which effects the populations of both species directly. For instance if lynx population is high in a forest, it is safe to assume that snowshoe hare population is (or at least once was) high in the same forest. And if lynx population decreases, one can conclude that one of its main reasons is the decrease in snowshoe hare population.

Ecologists have been conducting researches to understand the behaviors of this population change between lynx and snowshoe hare. Within time they found interesting results:

  • Whenever lynx population sees its highest point, snowshoe hare population hits its lowest point.
  • As its primary food supply almost runs out, population of lynx starts to decrease.
  • Decrease for lynx population directly effects the population of snowshoe hare as it increases rapidly.
  • Around the time when population of snowshoe hare hits its heights, lynx population bounces back from its lowest point.
  • More snowshoe hare meant more food for lynx and this leads lynx population to rise as snowshoe hare population to fall down once again.
Comparison of  the lynx (blue) population with the snowshoe hare (red) population between 1845-1935.

This is a cycle. Ecologists came up with a conclusion for the cycle between these two species. Their results showed them that one cycle lasts 8 to 11 years. It is still unknown what derives this time period. Ecology is filled with numerous secrets such as this one. Scientists showed that predator-prey relationship has a direct effect for the changes in the populations of lynx and snowshoe hare. Although this is not the only reason behind the cycle as climate, human factor and other predators also have an effect.

Periodic Living Beings

Population cycles are crucial for all the species living on Earth. Living beings had evolved so that they can maintain their lives and they managed it with evolving defense mechanisms. Some species don’t have enough strength so they had to come up with new defense mechanisms which are not physical. For instance some animals hide for long periods of time and only come out to breed. Hiding is their defense against predators. These animals could be called as “periodic living beings”.

Assume that the animal A comes out only once in every 10 years to breed for a couple of weeks.

Q: If A’s predator B is also a periodic living being, how long its period should be in order to catch A’s coming out?

If A comes out every 10 years, predator’s period should be a factor to A’s period so that B could catch A as soon as possible.

Factors of 10 are 1, 2, 5 and 10.

  • If predator’s period is 1 year, then (10/1=10) predator catches A in its 10th cycle.
  • If predator’s period is 2 years, then (10/2=5) predator catches A in its 5th cycle.
  • If predator’s period is 5 years, then (10/5=2) predator catches A in its 2nd cycle.
  • If predator’s period is 10 years, then (10/10=1) predator catches A in its 1st cycle.

There is still more bad news to come for A. These four numbers are not the only cycles a predator can have in order to catch A. For example if a predator’s cycle is 6 years, then it would catch A in their least common multiple. For 6 and 10, least common multiple is 30. Thus these two species meet once in every 30 years.

In the end, 10 years of cycle is not suitable for a living being.

Q: Is there a periodic living being on nature?


There are different types of cicada which only come out once in every 7, 13 and 17 years just for breeding. This is why cicadas are known as “periodic living beings”.

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Their cycles are 7, 13 and 17. These three numbers are not ordinary numbers: They are primes.

Prime numbers are special in their own way as they don’t have factors besides 1 and themselves.

It is still a mystery how these cicadas evolved into these specific primes. But we can speculate over this fact. A physicist named Mario Markus claimed that cicadas emerge in these prime numbers so that they will survive. This is a bold but interesting assumption.

Example: Let’s choose a kind of cicada that has 13 year-cycles. If its predator has a cycle of 5 years, there two creatures could only meet once in every (13*5=65) 65 years. Since 13 is a prime number, predator and prey can meet only at the multiples of 13.

M. Serkan Kalaycıoğlu


Real Mathematics – Numbers #2

Lottery Bust

No matter how hard you try to avoid them; one day you may need negative numbers. There are a respectable number of people from Britain who would agree with me on this matter. Back in 2007, a British lottery company released a new lottery game named Cool Cash. On every Cool Cash card, there were five boxes that you had to scratch. The box which was on the left bottom corner was called the “temperature of the day”.

Other four boxes were assigned to a prize. In order to win at Cool Cash, your box or boxes should have had lower temperature than the temperature of the day.


In this example our temperature of the day is given as -8 Celsius degrees and our four temperatures are -4, -6, -7 and -7 Celsius degrees. All four degrees are higher than the temperature of the day which meant that this Cool Cash card gives away no prize whatsoever.

Unfortunately this game caused huge problems for the lottery company. It turned out that British people had a poor understanding on negative number notion. Majority of the people called the company and claimed that they won even though they really did not. Ironically, some people thought they lost and threw away their cards which in fact had won a prize. After increasing number of complaints executives of the lottery company realized that they had two choices: Teach mathematics or stop producing Cool Cash cards. They had chosen the second option which was clearly the easiest of the options.

Smallest Number

Negative numbers had challenged mathematicians throughout the history. That is why it is not surprising to see kids having trouble when they try to make calculations with them. Although in most mathematics curricula negative numbers are ought to be thought for 11-12 years old students, some high school students have problems with negative number notion. Even in the 17th century Europe where modern science flourished, many scientists claimed that there can’t be a number that is less than zero.

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Chinese rod numerals.

When one searches for a counter example, it is possible to see the negative number concept in the early civilizations. For instance in ancient China there was a system for distinguishing positive and negative numbers. In ancient China, numbers were represented by rods in two different colors: Red rods were used for positive numbers as black rods for negative numbers. (Today we use red and black to show if an account is in plus or in minus.) Nevertheless Chinese mathematicians believed that a problem or an equation can never have a negative answer.

Adding Words

Ancient China and Egypt was using this technique quite often: Addition to the statement. People were adding extra words in their statements such that these words would change the whole set of the problem. In the end these extra words prevented people from the need of negative numbers. Let me explain this with an example:

Example 1-a: If one has 50 euro in his/her account and shops for 70 euro, that person should end up with -20 euro in total.

Example 1-b: If one has 50 euro in his/her account and shops for 70 euro, that person should end up with 20 euro debt in total.

Addition of the word “debt” changes a negative number into a positive one. This technique is still used today in our daily lives.


People also developed an opposition technique in their language to prevent the need of negative numbers: Choose the opposite of the term.

Example 2-a: A submarine is compared to sea level at -2400 meters height.

Example 2-b: A submarine is compared to sea level at 2400 meters depth.

Oppositions such as up-down, forward-backward, profit-loss, more-loss … are useful for turning a number from negative to positive.

Real Issue

The real issue for understanding negative number notion is that there are no negative numbered things in nature. This is really crucial as we don’t let students think over number concept. Actually whole number notion is not natural; we create numbers in our minds. You can’t go out to nature and run into something (living or not) that is in the shape of number 3. The number 3 makes sense to us when we see 3 cows in a cattle farm.

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Brahmagupta (598-668)

3 apples, 3 trees, 3 cats… Numbers make sense to us whenever you bring them with another concept. However, if someone asks you “what makes 2+1?” you would answer “3”. Here, “3” has no concept attached to itself, yet you used it perfectly.

Almost 1400 years ago, a brilliant Indian scientist named Brahmagupta used negative numbers as we use it today. He explained the negative number notion with another concept attached to it: “Negative numbers stay in the debt side of the number zero.” Brahmagupta also gave rules to negative numbers in calculations. He is the first known person who claimed that a negative multiplying another negative makes a positive.

One wonders…


This is the picture of +2 apples. How can you photograph -2 apples?

M. Serkan Kalaycıoğlu

Real Mathematics – Numbers #1

First thing first; mathematics and philosophy can’t be separated. I can back this idea with a historical fact: Compared to history of science, “scientist” is a relatively new title. For instance, Isaac Newton who is regarded as the father of modern physics was known as a “natural philosopher”. Use of the word scientist had made us forget the fact that a scientist is actually a person who does deep thought sessions in majority of his/her work.

Mathematics Education – “This is that, and hence the answer should be that.”

Unfortunately, in traditional mathematics education there is almost no time for one to think. I think I should explain what I mean with the term “traditional mathematics education”: Imagine that numbers is the subject of a specific mathematics class. Teacher follows the traditional way and gives definitions to number types.

Counting Numbers: These types are also known as whole numbers. They start from 1 and go to infinity one by one:

1, 2, 3, 4 …

Natural Numbers: They start from 0 and continue to the infinity one by one. Their only difference with counting numbers is the mighty 0 (zero).

0, 1, 2, 3, 4 …

Odd Natural Numbers: They are the numbers which has remainder 1 when divided to 2. First odd number is 1. Odd numbers increase by 2. So their sequence is:

1, 3, 5, 7, 9 …

Even Natural Numbers: They are the numbers which can be divided by 2 with remainder 0 (zero). First even number is 0. Their sequence is:

0, 2, 4, 6, 8 …

Integers: They are the sum of natural numbers and negative of counting numbers. They can be shown like an infinite line where left goes to negative infinity as right goes to positive infinity. Their sequence is:

… -3, -2, -1, 0, 1, 2, 3 …

Prime Numbers: They are greater than 1 and they can only be divided by themselves and 1 to give and integer answer. They are:

2, 3, 5, 7, 11, 13, 17 …

“What is 7 times 8? Bravo! You are so fast!”

We need to define these, so there is no problem up to this point. What is done wrong with traditional approach is that it rapidly focuses on problems. There is nothing wrong with asking questions to kids, although with the traditional approach important thing isn’t the question: It is the answer! After a while kids focus on the answer rather than the question. That is one of the reasons why when asked (I am not talking about mathematics per se) children try to answer even without thinking. Sometimes even families contribute that with awarding how fast the answer is given.

Even before they start high school most of the kids lose the ability to think in mathematics classes.

What needs to be done?

If you are good at observation, you probably realized that those definitions above include notions such as negative, zero and infinity. These three notions are relatively new to humans and they all possess deep meanings underneath them. For instance only 1350 years ago humans first explained the number zero mathematically. Actually first use of the number zero in Europe was around 12th century which was almost 500 years after its first discovery in India! (I will get back to this subject in another article.)

Nowadays we define negative numbers and we expect kids to do calculations with it immediately. Actually negative numbers were known to civilizations way before the number zero. Although in the 16th century important European mathematicians were referring to negative numbers as “wrong numbers”. It has been only 200 years or so since humans made peace with the negative number notion. Maybe it is clearer now to you how wrong it is to expect kids understand negative numbers instantly.

Infinity might seem the hardest of these three notions. There is a popular definition for it: Something that never ends, keeps going until the end of time, everlasting. However there are much more profound meanings for infinity in mathematics. For example, some infinities are countable and some are bigger than others.

Hilbert’s Infinity Hotel

This paradox was thrown out for consideration by David Hilbert, one of the leading mathematicians of 20th century. In Hilbert’s hotel there is infinite number of rooms. Imagine that you are the manager of this hotel and you work on commission. One day business was ticking and every room was filled with customers.

  1. Would you turn down if a new customer arrives? If you decide to give this customer a room, which room will it be? How can you decide that room number?
  2. A couple of hours later you see infinite number of customers in front of the hotel. How can you arrange rooms for this many people?
  3. Just when you thought you could relax for a bit, you hear honking coming outside of the hotel. You go outside and see infinite number of buses in the parking lot. Not only that, each bus has infinite number of tourists. WHAT WILL YOU DO?!


Answers to these three questions will help students comprehend different number notions. You should stay away from internet in case you have never heard of this paradox. Give yourself time, even if it is limited, at least 1-2 hours. Just think on these questions. Try to remember this: It is not about “when”, it is about “how”.

M. Serkan Kalaycıoğlu

Real Mathematics – Special Numbers #2

Sieve of Eratosthenes

Eratosthenes (whenever I write his name, I can’t stop thinking about a nice and crusty toast) was not only a one-job man: He did not only measure the circumference of Earth, he also contributed to various fields, mathematics in particular. Ancient Greek philosophers dealt with geometry a lot which led to a misunderstanding as if they didn’t work for other branches of mathematics. Reality shows us that lots of Greek philosophers worked for mathematics in a broader aspect. Eratosthenes was one of them and he discovered an ingenious method for finding prime numbers.

Today that method is known as the Sieve of Eratosthenes. This method is really easy to learn and very affective for finding prime numbers in a list. But method is rather slow and takes too much time in larger number lists.

Let me show you how Eratosthenes’ method works between numbers 2 and 100. (I am skipping 1. I’ll be talking about 1 and why it is not considered as prime in another article.)


Thus we start with the number 2, which is the smallest prime number. Here is how the method goes:

  • Move in the list until you find a prime number.
  • When you find your prime, stop there and square it.
  • Go to the square of the prime in the list, and eliminate all the numbers that are multiples of the prime.
  • When you finish eliminating, go to your prime in the list and move until you find your next prime number.
  • Repeat the same process until there are only primes left in the list.

Example: Let’s do it for 2-100.

2 is prime. Square it: 4. Start eliminating all numbers that are multiples of 2.

Go back to 2. Move to the next number, which is 3. It is prime. Square it: 9. Go to 9 and start eliminating all the numbers that are multiples of 3.

Go back to 3. Move to the next number; 4. It had been eliminated already, thus move to 5 which is a prime number. Square it: 25. Start from 25 and eliminate all the numbers that are multiples of 5.

Go back to 5. Move to the next available number which is 7 and it is a prime number. Square it: 49. Start from 49 and eliminate all the numbers that are multiples of 7.

Go back to 7 and move to the next available number which is 11. It is prime, thus square it: 121. 121 is out of our list. This means every available in our list is a prime number.


New In the Olympics: Walking the Square

Game 1:

  • Draw a circle and place eight dots on it.
  • You will walk around the circle from the dot that is on top. You’ll take 1 step each time you move, that means you will be moving to the neighbor dot.
  • You’d visit every dot with this method until you finish your walk.

Game 2:

  • Again start with the same circle and eight dots on it.
  • This time you will take 2 steps in each time. That means always move to the second dot.
  • When you finish your walk, you’ll see that you visited only half of the dots.

Game 3-4-5-6-7:

Try walking with 3, 4, 5, 6 and 7 steps.

One wonders…

In which steps you would visit all the dots? Do you see anything special about those numbers of steps?

Can you find a general conclusion from this game?

What would you see if you play the same game for 9, 10 and 11 dots?

M. Serkan Kalaycıoğlu

Real Mathematics – Special Numbers #1

Understanding Time

How did humans keep track of time before the invention of clock?

First thing comes to mind is that humans observed rise and fall of the Sun that is 1 day. People needed to know time in a broader aspect as civilizations emerged so that they could understand crucial time intervals such as rain season etcetera. For that, they used a cycle that was up there in the sky, changing its shape in a pattern: Phases of the moon. They observed and calculated that Moon phases are in cycles of 29 days.


A lunar month is roughly 29 days. There is about 1 day difference between lengths of a month and a lunar month.

29 is a whole integer.

If you try to divide it into two equal parts, it will give you 14,5 which is not a whole integer.

If you continue dividing 29 into equal whole integers, you would fail for every number between 2 and 28. Only there is one 29 and twenty nine 1s. Nothing else can divide 29 into equal whole integers.


In other words, there are only two whole numbers which gives 29 when they are multiplied: 1 and 29. These kinds of numbers are called “prime numbers”.

Today “one month” refers to 30 days. One of the main reasons behind turning 29 days into 30 is that 30 is not a prime number. It can be divided equally into whole numbers such as 2, 3, 5, 6 (2*3), 10 (2*5) and 15 (5*3).

Prime Numbers and Encrypting

Another beautiful property about 30 is that it can be shown as the multiplication of 2, 3 and 5. All these three numbers are prime. Therefore we can write any whole integer as the multiplication of prime numbers.

Let’s take this one step further: Every whole integer has one and only one representation what is a multiplication of prime numbers. Hence different integers have different prime number demonstrations.

This property has a key importance in our modern day. Assume that I have to create a password for my online banking account. Bank’s application requires me to create this password with using numbers only. Naturally I would like to create a password that would be very hard (preferably impossible) to crack. Also I would like to use an easy method when I create the password so that I can remember the method in case I forget the password itself.

Trapdoor Function: Roughly, it is a method that is easy to compute in one direction, yet difficult to compute in the opposite direction. You could imagine it as a machine that takes a number (input number) and easily processes it into another number (output number). Although it would be very hard or even impossible to use the output number and find the input number with it.

If I take one small and one huge (let’s say 6-digits) prime numbers, it would take only a few seconds to multiply them using a calculator. Resulting number, my new password, can only be found with the multiplication of those specific prime numbers. Then, if someone tries to log in to my bank account, he/she must know the primes I’ve chosen.

Creating 1115035 is very easy: Just multiply prime number 223007 with another prime 7.

Finding the prime multiplication of a random number is a really hard and long lasting thing to do.


Because, even today we don’t have a method that finds the prime multiplication of any given number. Thus, using prime numbers is an effective way to secure information.

Crack the Code

I’ve built an encryption system that uses prime numbers and alphabet. According to my system, every letter in the alphabet is assigned to a number:



  • I choose two prime numbers between 1 and 20.
  • I multiplied them.
  • Then I added the result to each letter in the alphabet.
  • For instance if I choose 3 and 5, letter A would go to (3*5=15, 15+1=16) the 16th letter that is the letter M. Hence letter M in my encrypt means that it is the letter A in reality.

Now, try to crack the encrypted sentence below:


M. Serkan Kalaycıoğlu

Real Mathematics: Numbers #3

Negative integers are a “game changer” when they sneak into our lives. Everything we learn about calculation gets a little bit more complex with the introduction of negative integers. Now we have rules that say crazy things such as “multiplication of two negative integers gives you a positive one”.  How come two negatives make a positive? Is there a sensible explanation?

Q: (-2).(-3) = ?

Algebraic Method: It is being taught that when two negative integers are being multiplied, absolute values 2 and 3 are being done and a plus sign comes to the top of the calculation. This gives +6 or 6 as a result. Almost all of the students learn how to do this calculation but most of them have no idea why it is being done like this.

There are numerous examples to explain the reason. I prefer giving a specific geometric explanation as an example.

Geometric Method: We teach that integers can be shown on an infinite straight line what we call “number line”.  Middle of this line is assigned to the number zero. Left of zero is for negative integers as right side is for positives.

While teaching multiplication with negative integers, you could imagine a line that is perpendicular to the original number line. Upwards would be assigned to the positive integers as negative integers go to downwards.



Assume that we have to multiply two numbers.

  1. Sign of the first number tells us which way we are facing: Upwards or downwards. Positive would mean “look up” as negative means “look down”.
  2. Sign of the second number tells us the way we are taking our steps: We could either go forward (which means the number is positive) or go backwards (meaning that the number is negative). You could imagine that backward steps are like how Michael Jackson was “moonwalking”.
  3. Value of the first number tells us how many steps are taken in each time.
  4. Value of the second number tells us how many times those steps are going to be taken.

Let’s solve our original problem with the geometric method: What is (-2).(-3) ?

  • First number is negative: We are facing downwards.
  • Second number is negative: We are taking our steps backwards.
  • Value of first number is 2: We are taking 2 steps in each time.
  • Value of the second number is 3: We are taking those steps 3 times.


In the end, we are facing down, moving backwards (like Michael Jackson’ moonwalk), 2 steps at a time and 3 times. Our arrival would be +6 as shown on the graphic.

One wonders…

  1. Try to prove that the algebraic method is true. (Hint: Start with assuming that the multiplying two negatives won’t make a positive.)
  2. Try to find another example for geometric method from real life.

M. Serkan Kalaycıoğlu

Real Mathematics: Pattern #4

Leonardo Pisano

Italian town Pisa was the home of an ingenious person named Leonardo Pisano, which means Leonardo from Pisa. He was not only essential to history of mathematics, but he was also influential for the birth of scientific revolution. It is not a surprised that Leonardo Pisano was from Italy as Italians were involved with Arabs through trading.

Arabs knew an amazing way of counting and calculating, which were done with a system called decimal system. I’ll talk about that story in another article.

Leonardo Pisano was the first known person who brought modern numbers Western Europe. Although this was an amazing accomplishment, his importance comes even more fascinating if you look at what he did for patterns.

The Rabbit Problem

If I wrote his name as Fibonacci, then majority of you would understand what problem I’ll mention in the following:

In a farm, there is one couple of baby rabbits. A rabbit couple can give birth to baby rabbits only after their 2nd month and they can continue giving birth each month after that. Leonardo Pisano tried to find out the number of rabbit couples after one year.


First month there is a baby couple. This couple will be adult in the second month and they will give birth to one couple baby rabbits in the third month.

In the fourth month first couple reproduces as the second couple becomes an adult.


In the fifth month first and second couples have new babies as third couple becomes an adult.


In the sixth month first, second and third couples have new babies as the fourth and fifth couples become an adult.


IMG_5690At this point we can point out a pattern in the number of rabbit couples. After second month, total of previous two months gives the number of rabbits in the next month. For example number of rabbit couples in the third month becomes the summation of first and second months, which is 1+1=2.

Fourth month = Second month + Third month = 1 + 2 = 3… and so on.


Then number of rabbit couples after one year (twelve months) is:


Beauty of Fibonacci

This number sequence is known as the Fibonacci sequence and it is visible to us in nature on so many occasions. I’ll be talking about the most popular examples of Fibonacci sequence in the following articles.

Real life examples of math subjects are crucial, especially the ones from nature itself. But most of the population lives in the cities and this force us math teachers to find out examples from modern life.

Stairs and Fibonacci

Imagine that you have to climb up from stairs inside your apartment.

  1. How many ways are there to climb 3 steps?
  2. How many ways are there for 5 steps, 6 steps, 8 steps and n steps?
  3. What is the relationship of this question and Fibonacci numbers?

M. Serkan Kalaycıoğlu

Real Mathematics: Numbers #8

Dotted Tic-Tac-Toe

Almost everyone knows how to play tic-tac-toe. Dotted tic-tac-toe is a multiplayer game to teach kids about number systems.


  • Players throw a dice in turns.
  • Each square in the board has the capacity of 9 dots.
  • Players should put dots into one of the boxes as many as their dice shows.
  • While putting down dots, players must be careful not to exceed the capacity of the box. In such situations they have to put dots to another box that has capacity.
  • Whoever gets right to left, upside-down or diagonal three boxes will be the winner.


Beauty of this game is although it is competitive, it requires help of your opponent.

An example:

In the first three rounds players get 5, 6 and 3 from the dice:

Fourth dice is 1 which gives second player a chance to complete a box.

In the next four rounds players get 5, 5, 6 and 4 which gives again second player to complete another box.

Game resolves in the next rounds as shown in the following:

Binary Tic-Tac-Toe

In this version of the game, use a coin instead of a dice.

Heads: 1

Tails: 0

  • Consider that a box can have either of the following: A head & tail (tail & head is the same) or a tail & tail.
  • A square can never had head & head, since 1+1=2 which is overload for the binary number system.
  • This means that every box is either 1+0=0+1=1 or 0+0=0.
  • So in this version of the game, x-o-x turns into 1-0-1. Right to left, upside-down or diagonal 1s or 0s wins the game.


Once again, although the game is competitive, you’ll need your opponent’s help.

An Example:

First two coin tosses are both tails.


Then four consecutive heads are tossed.

Game resolved after three consecutive tails.

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