Real Mathematics – Game #6

Oreo Placement Game – OPG

Every week Steve and Tanya meet for coffee and play a game in the coffee shop with the Oreos they brought. They take turns placing those Oreos on a napkin that lies on the table before them. While they do that Oreos must not overlap. Also Oreos must stay within the napkin’s boundary: Overflow is forbidden. The player who places the last Oreo wins the game.

Rules:

  • Two players take turns and place one Oreo each time on a napkin.
  • Napkin has a circle shape.
  • Oreos can’t overlap.
  • Oreos can’t overflow the boundary of the napkin.
  • Winner is the player who puts down the final Oreo.

Is there a winning algorithm/strategy for either of the players?

Yes, there is an algorithm Steve and Tanya can use in order to win at OPG. But this algorithm works only for the player who puts down the first Oreo.

Winning Algorithm:

  • Be the player who starts the game.
  • Place your Oreo at the center of the napkin in the first move.
  • For the rest of the game wherever your opponent places his/her Oreo, play your next move with taking its symmetry by the center Oreo.
  • Eventually second player will run out of space on the napkin.

Place it to the center.

Wherever your opponent places his/her Oreo, rotate that location 180 degrees and place your Oreo there.

Continue the same strategy. Eventually your opponent will run out of space and you will win.

Other Shapes

In OPG, you can use the same algorithm with a napkin that has a regular polygonal shape. But first player might need adjustments in certain situations. Assume that you play OPG in a triangular-shaped napkin.

If you can fit even number of Oreos on the napkin, placing the first Oreo on the center of the triangle will be a losing strategy:

When the first Oreo placed on the center and exactly four Oreos fit inside the triangle, you will not win no matter what.

In order to avoid that you must place the first oreo slightly above the center:

Placing the first Oreo slightly above the center will divide the rest of the triangle into two isolated spaces where you can fit one Oreo each. First player will win no matter what happens with this strategy.

For pentagonal napkins you will have to use the same strategy which you used for triangular napkins:

It is possible to fit 6 Oreos on this pentagon. If you start first and place your Oreo on center, you will lose.

However, when you place the first Oreo slightly closer to one of the corners, algorithm will guarantee a win in every single time.

Q: Is there a winning algorithm in OPG when you don’t get to start the game?

Yes, there is. But in order to accomplish that second player must select a shape that is not convex. Let’s say the napkin is shaped as follows:

Here, first player tries to place his/her Oreo such that the triangle will be divided into two isolated places. No matter what the second player does, first player will win.

Second player must decide the shape of the napkin like following:

nankon4

In these situations winner should leave the napkin in two isolated regions such that these regions must have space for only one Oreo. Thus, second player must play his/her second to last move in such way:

No matter where the first player places his/her Oreo, second player will win if he/she leaves two isolated areas on the napkin.

One Wonders…

Would the same algorithm work if Oreos were in squared shapes? Test this with napkins that are in triangular and pentagonal shapes.

M. Serkan Kalaycıoğlu

Real Mathematics – Game #5

Me versus You

I am going to play a game against you. Yes, I am talking to you, my beloved reader. I promise you that I will personally hand it to you whatever you win at the end of the game.

Rules:

  • You will start the game from top-left box. You will be making mini-runs. In other words you will take as many steps as I tell you to.
  • Each time you can take a step towards one of these four directions: Up, down, left and right.

    hepsis
    In case you take a step from 1000-Euro box, you can either go right to the 1000-dollar box or down to the Iphone-box. You can’t go from 1000-Euro box to the Starbucks-coffee box with one step as going diagonal is not allowed.
  • As long as you follow the second rule you can move freely. It is allowed to visit the same box more than once. Also you can go backwards too.
  • You should start your next step wherever you stopped in the previous run.
  • After every run I will select a box or two and eliminate them from the game. In case I eliminate a box where you are currently sitting, you will be declared as winners.
  • I repeat: Your steps should be either of these directions: up-down-right-left.

We’ll start whenever you are ready. Good luck.

Game

1.You will be moving inside these 9 boxes starting from the top-left which is the 1000-Euro box. Take 5 steps.

hepsi

2. I know that you didn’t stop at the 1000-Euro box. So I am eliminating it. Now, wherever you stopped in the previous run, start taking 7 steps. Did you do it? Okay, you can go to the next.

hepsi2

3. After those steps I am 100% certain that you didn’t stop at the 1000-dollar box. Thus I am eliminating it. Now continue the game with taking 11 more steps.

hepsi3

4. This time I will make a bold decision and eliminate two boxes: Left-bottom corner (a bag of money box) and car key box. You are sitting either one of the following five boxes. Now take 5 more steps.

hepsi4

5. I know that you are not standing on the Iphone box. I also know that you are not on the 100-dollar box. After eliminating them, take 1 last step from wherever you are standing.

hepsi5

Result: Now I can eliminate Starbucks coffee and 500-dollar boxes as you are standing on the zero box. I am afraid this is what you won: A big, fat zero.

hepsi6

One wonders…

You can play this game from top over and over again. In the end, you will get the same result. How is that possible? Why am I this confident?

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.

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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.

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Guest takes one from cacao.

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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: 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?

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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.

çoko

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.?

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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.

Boşluk yarışı

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.

There are so much to think about this problem:

  • 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.

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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?

IMG_5176

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