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.

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.

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.

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.

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.

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

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

Poke

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

Slide

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

One wonders…

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

In my youth, I would never step outside without my cassette player. Though, I had two knotting problems with my cassette player: the First one was about the cassette itself. Rewinding cassettes was a big issue as sometimes the tape knotted itself. Whenever I was lucky, sticking a pencil would solve the whole problem.

The second knotting problem was about the headphone. Its cable would get knotted so bad, it would take me 10 minutes to untangle it. Most of the time I would bump into a friend of mine and the whole plan about listening to music would go down the drain.

The funny thing is I would get frustrated and chuck the headphone into my bag which would guarantee another frustration for the following day.

A similar tangling thing happens in our body, inside our cells, almost all the time.

DNA: A self-replicating material that is present in nearly all living organisms as the main constituent of chromosomes. It is the carrier of genetic information.

DNA has a spiral curve shape called “helix”. Inside the cell, the DNA spiral sits at almost 2-meters. Let me give another sight so that you can easily picture this length in your mind: If a cell’s nucleus was at the size of a basketball, the length of the DNA spiral would be up to 200 km!

You know what happens when you chuck one-meter long headphones into your bags. Trying to squeeze a 200 km long spiral into a basketball?! Lord; knots everywhere!

Knot Theory

This chaos itself was the reason why mathematicians got involved with knots. Although, knot-mathematics relationships existed way before DNA researches started. In the 19th century, a Scottish scientist named William Thomson (a.k.a. Lord Kelvin) suggested that all atoms are shaped like knots. Though soon enough Lord Kelvin’s idea was faulted and knot theory was put aside for nearly 100 years. (At the beginning of the 20th century, Kurt Reidemeister’s work was important in knot theory. I will get back to Reidemeister in the next post.)

Is there a difference between knots and mathematical knots?

For instance the knot we do with our shoelaces is not mathematical. Because both ends of the shoelace are open. Nevertheless a knot is mathematical if only ends are connected.

The left one is a knot, but not mathematically. The one on the right is mathematical though.

Unknot and Trefoil

In the knot theory we call different knots different names with using the number of crosses on the knot. A knot that has zero crossing is called “unknot”, and it looks like a circle:

A rubber band is an example of an unknot.

Check out the knots below:

They look different from each other, don’t they? The one on the left has 1 crossing as the one on the right has 2:

But, we can use manipulations on the knot without cutting (with turning aside and such) and turn one of the knots look exactly like the other one!

This means that these two knots are equivalent to each other. If you take a good look you will see that they are both unknots. For example, if you push the left side of the left knot, you will get an unknot:

Is there a knot that has 1 crossing but can’t be turned into an unknot?

The answer is: No! In fact, there are no such knots with 2 crossiongs either.

How about 3 crossings?

We call knots that have 3 crossings and that can’t be turned into unknots a “trefoil”.

Trefoil knot.

Even though in the first glance you would think that a trefoil could be turned into an unknot, it is in fact impossible to do so. Trefoil is a special knot because (if you don’t count unknot) it has the least number of crossings (3). This is why trefoil is the basis knot for the knot theory.

One of the most important things about trefoils is that their mirror images are different knots. In the picture, knot a and knot b are not equivalent to each other: In other words, they can’t be turned into one another.

Möbius Strip and Trefoil

I talked about Möbius strips and its properties in an old post. Just to summarize what it is; take a strip of paper and tape their ends together. You will get a circle. But if you do it with twisting one of the end 180 degrees, you will get a Möbius band.

Let’s twist one end 3 times:

Then cut from the middle of the strip parallel to its length:

We will get a shape like the following:

After fixing the strip, you can see that it is a trefoil knot:

To be continued…

One wonders…

1. In order to make a trefoil knot out of a paper strip, we twist one end 3 times. Is there a difference between twisting inwards and outwards? Try and observe what you end up with.
2. If you twist the end 5 times instead of 3, what would you get? (Answer is in the next post.)

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:

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

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:

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:

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

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:

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:

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

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

One can label 1-2 side with either of 1 or 2.

One can label 1-3 side with either of 1 or 3.

One can label 2-3 side with either of 2 or 3.

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 20^{th} 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:

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:

The person will end up inside a 1-2-3 triangle:

The person will find him/herself outside of the triangle:

Three friends will be sharing three slices of pizza.

Pizza Slices:

Pizza 1: Four cheese.

Pizza 2: Chicken.

Pizza 3: Pineapple.

Three friends could share the slices in six possible ways:

P1:Pizza 1, P2:Pizza 2, P3:Pizza 3.

Eventually answer is one of those possibilities.

How much do you want it?

Choosing a slice of pizza seems like a very basic procedure. Though, there are many factors behind this simple decision. First of all finance is involved. And obviously “preferences” is an important factor: Not everybody has the same taste for food.

Let’s add these factors to our pizza slice situation. I will assume that 6 dollars is paid in total for the pizza slices, and three friends Ali, Steve and Jane have preferences as follows:

Ali’s first choice is four cheese. If he can’t get that, his second and third choices are chicken and pineapple in order.

Steve’s first choice is chicken. His second and third choices are pineapple and four cheese in order.

Jane’s first choice is also chicken. Her second and third choices are four cheese and pineapple respectively.

In such situation, how can these friends decide fairly who gets which slice for how much?

Rental Harmony Problem

Two (or more) friends decide to rent an apartment. Rooms of the apartment are different in sizes and in some other factors (getting sunlight, having its own bathroom and such). For years mathematicians showed great interest towards the problematic of this situation: How can the rent and rooms be divided?

In 2004, three Turkish scientists Atila Abdulkadiroglu, Tayfun Sönmez and Utku Ünver published a paper that had a solution for the rental harmony problem. This paper shows an ingenious auction algorithm for the solution. (In fact, a famous website that is created to solve rental harmony problems named Spliddit had used this algorithm for almost a decade.)

The Algorithm

Roommates write their offers for every room in a closed envelope. For each roommate, total of the offers must be equal to the total rent of the apartment. (For instance, if the rent is 3000 dollars and total of the offers for each person must be equal to 3000 dollars.)

Envelopes are opened in front of everyone. A room goes to its highest bidder.

In the end, all the winner bids are added together. If it is equal to the rent, then winning bids are the amounts that are going to be paid. If it is exceeds or falls under the total rent, each offer gets to be corrected proportionally.

Answer to the Pizza Slices

We can use the auction algorithm in order to find a solution to the pizza slices problem. Assume that we have offers from Ali, Steve and Jane as follows:

Highest bids:

For the pizza 1 is Ali with 4 dollars.

For the pizza 2 is Steve with 4 dollars.

For the pizza 3 is Jane with 2 dollars.

Total of the winning bids is

4 + 4 + 2 = 10

which is higher than 6 dollars. Hence we have to correct the amounts that need to be paid for each of the friends:

In conclusion Steve and Ali would pay 2,4 dollars which is less than their offers. Jane would pay 1,2 dollars and it is also less than her offer. Auction algorithm helped these friends to select their pizza slices. The algorithm also helped them to pay less than what they were willing to. Hence, everyone is happy and envy-free with the conclusion.

Is there a bullet-proof method for cutting a cake fairly?

This became a legitimate problem in mathematics back in 1944. A mathematician named Hugo Steinhaus published a paper for fair cake-cutting. According to him solution is trivial for two people. Method for two people is today commonly known as “one cuts, other picks”:

First person cuts the cake in half (on his/her point of view) and other person gets to pick any piece he/she wants. In this method both of them are happy and envy-free as first person believes the pieces are equal and second person picks the biggest piece on his/her view. Steinhaus’ answer is the first envy-free answer for fair cake division.

What if there are three people? Is there an envy-free method for cutting a cake for three people?

Incredible but answer wasn’t that obvious. Only 18 years after Steinhaus’ work (1962) J. H. Conway and J. Selfridge (independently) found an answer.

Ali, Steve and Jane

General method:

Ali cuts the cake into three equal (that he considers equal) pieces.

Steve evaluates the pieces. If he decides to do nothing Jane takes the turn.

Jane selects any piece she wants. Steve picks as second, Ali gets the last piece.

Obviously things are never this easy. Let’s examine the method in details and show how ingenious it is:

Step #1: What should Ali do?

First step is easy: Ali cuts the cake into three equal pieces. Since he cuts them, he will be pleased to receive any of the pieces.

Step #2: What should Steve do?

Steve checks the pieces. Here, there is more than one possibility depending on how Steve sees Ali’s cuts:

If Steve decides that at least two of the pieces are the best ones (best as in biggest) he does nothing and process continues with Jane selecting her piece.
a. In case Steve thinks that all three pieces are equal, whichever Jane selects, he will get the best piece according to him:

Steve will select his piece as second and he will be pleased to get whichever piece as he thinks they are all equal in sizes.

b. In case Steve thinks that two of the pieces are equal and best, even if Jane selects one of them, he will get to select the other best piece. He will be happy in either case.

If Steve thinks one piece is bigger than the other two, he needs to do something. Otherwise Jane would get the biggest piece:
If Steve does nothing here, Jane will get the middle piece for sure.

In such situation Steve trimmers the biggest piece which will result in a trimmed piece and a leftover:

This way trimmed piece will be equal to at least one of the two remaining pieces which means at least two (trimmed and one of the other two) of the cake pieces are the best ones. Now Steve is ready to let Jane select her piece.

Step #3: What should Jane do?

Here, Jane is free to choose whichever piece she’d like to:

If Steve did nothing, Jane will select any of the pieces. Steve goes after her and Ali will get whatever is left. Everyone will be content with his/her choice.

(Steve trimmed a piece.) In such situation Jane is still free to choose. But now her choice determines the rest of the game. There are two possibilities at this state:
a. Jane selects the trimmed piece. Then Steve and Ali will get their pieces in that order. In this situation Steve gets to cut the leftover into 3 equal pieces. Then Jane, Ali and Steve each select a piece of the leftover in that order:

b. Jane selects any of the untrimmed piece. In that case Steve must select the trimmed piece and Jane gets to cut the leftover into 3 equal pieces. Then Steve, Ali and Jane each select a piece of the leftover in that order.

One wonders…

Is this really envy-free?

(Give yourself some time to think before you read the answer.)

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*

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Answers

For Ali: Yes, because Ali cuts the cake in three equal pieces on his view. He will be content with any of the pieces. Also he will not mind getting a part of the leftover.

For Steve: Yes, because Steve decides that there are at least two best pieces and he will be content with any of them. Also he will get to cut the leftover into 3 equal pieces which means he knows that he can get any of the leftover pieces.

For Jane: Yes, because she gets to select first for the original and leftover cake.

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:

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:

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:

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:

Find the numbers of the dots, lines and faces:

Apply Euler’s formula:

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…

At most how many turns can there be in a regular sprouts game that starts with 3 dots?

Is there a winning strategy for sprouts?

Start a Brussels’ sprouts with 3 dots. If each of them has 3 thorns, at most how many turns can there be?