Real MATHEMATICS – Puzzle #4

Naughty Students

Among all friendships, being classmates has a special part. Inside every classroom, each student has a friend who would cause trouble if they sit adjacent (side to side, front-back and diagonal) to each other. This is why teachers change the sitting-order to find the optimal situation for each classroom.

Steve the teacher and his problem

Teacher Steve realizes in one of his classes that in total 8 students cause trouble during lessons whenever they sit adjacent (from now on I will refer to being adjacent as “being neighbor”).

The Situation

  • Neighbor students are the students who sit either side to side, front and back or diagonal to each other.
  • If two students cause trouble whenever they are neighbors, there is a <–> sign between their names.
  • Deniz <–> Ali <–> Kirk <–> Jane <–> Poseidon <–> Rebecca <–> Lucreita <–> Bran
  • Sitting plan for these 8 students is shown in the following:

sekiz

Steve the teacher doesn’t want to change other students’ sitting plan. Hence his problem becomes as follows:

“How can I find an order for these 8 students so that there won’t be neighbor students who will become naughty?”

Hint: Assign numbers to the students.

I will explain the answer in the next post.

 M. Serkan Kalaycıoğlu

Real Mathematics – Puzzle #1

Crossing the River

A farmer is walking back home with a wolf, a goat and some lettuce. As he was on his way a river appears in front of him. Luckily for him, he sees an abandoned boat. Though it turned out the boat was not big enough for all of them. Farmer had to choose either one of the wolf, goat or lettuce in each time he crosses the river.

But there was a catch here: If they are left alone wolf would eat the goat and goat would eat the lettuce. What is farmer’s strategy in order to cross this river? (P.s: It is not possible to swim across the river.)

farmer-wolf-goat-cabbage

The Answer

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Between these three only wolf and lettuce can be left alone. Then farmer should start the process with carrying the goat to the right side of the river.

lahank

Then he should return the left side and picks up either of the wolf or the lettuce. (Let’s assume that he picks the lettuce.) Farmer reaches the right side of the river with the lettuce. Now before he leaves he realizes that lettuce and goat can’t be left alone. So he takes the goat with him to the left side.

kurtk

Now, lettuce is alone on the right side of the river as farmer, goat and the wolf are all on the left side. At this point farmer should carry the wolf to the right side and leaves it there with the lettuce as lettuce and wolf can be left alone without a problem.

hepsik

In the end farmer can go back to the left side of the river and takes the goat to the right side. Hence the problem gets solved.

There wouldn’t be a solution for this puzzle if wolf was eating the lettuce. But, in order to understand that we had to check every possible variation. This would have been a waste of time and energy. Thus we haven’t found an actual method for the solution of this puzzle.

River Algorithm

Graph Theory is a relatively new subject of mathematics. In Graph Theory what matters is the placement and the connection; not the shape. In this part of mathematics objects are described with dots and connections with lines. Then wolf, goat and the lettuce can be represented as dots. Only connection between these three is whether they eat one another or not. In the puzzle goat eats lettuce and wolf eats goat. Hence connections (or lines) should be between them.

If dot disappears, then the connections within disappears as well. In the puzzle we can take only one dot at a time. In the first move we choose the goat. This can be shown in graph as follows:

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Deleting the goat dot means to delete the lines connected to it. Graph has no line after the first move which is what we want to accomplish and we managed to do it with deleting only 1 dot. This means another thing: In order to solve the puzzle there must be as many spaces as the deleted dots in the boat.

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Let me explain this with an example: Assume that we have to take at least n dots away from the graph in order to get rid of all the lines. This means if boat has n empty spaces the puzzle can be solved.

Let’s add a rabbit to this puzzle.

Second River Puzzle

Now farmer has a goat, a wolf, some lettuce and a rabbit. On top of the previous puzzle rabbit eats the lettuce and wolf eats the rabbit.

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In the first puzzle we took goat’s dot and saw that there was no line left in the graph. But in the second puzzle none of the four dots is able to extinguish all the lines of the graph.

This is why in order to extinguish all the lines; we must try to remove at least two dots. For instance:

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As seen above when we take goat and the rabbit, all the lines inside the graph disappears. Using graphs we found an algorithm and with that algorithm we know that the boat must have at least 2 spaces in order to solve the second puzzle.

One Wonders…

  1. Solve the second puzzle.
  2. Let’s add carrot, cat and mice to the second puzzle. Goat, rabbit and mice eats the carrot and cat eats the mice. Try to find how big the boat should be and solve the puzzle for that boat.

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.

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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.

Unlucky Dad

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…

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A baklava master in the modern day.

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.)

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Oh orange candy…

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.

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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.
    madobak
  • 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.
    maddo
  • 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: Puzzle #2

There are certain movies I remember from my childhood which are among my favorites of all times. It would be though to make a top-10 list but if I were to make such a list, Die Hard 3 would make the list with ease. Crazy German terrorist Simon Gruber against our heroes John McClane and Zeus Carver whom were performed by Bruce Willis and Samuel Jackson respectively… Now, that is action!

die-hard-vengeance-laptop

There is a scene from this movie which will be the main topic of this article. In this particular scene, Simon hands out another life-or-death assignment for our heroes. Click here for the scene.

Simon says…

“There is a timed bomb in this briefcase. You have a 3-liter and a 5-liter jug that you can fill from the fountain. In order to stop the timer you must use those jugs and fill exactly 4 liters. It has to be precisely 4 liters because once you fill the jug; you should put it on the briefcase which has a scale.”

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Billiard Table

We could use try-and-see method to solve this riddle, but we might run out of time which would mean a sudden death for us. Obviously we don’t want that. To come up with a systematic method we have to set a few ground rules.

Rules

  • Let’s assume that vertices represents water amount in liters and lines represents how quantity can change.
  • Vertices will have number pairs. First number is for the 5 liter jug, second for the 3 liter jug.
  • Both jugs start from the vertex (0,0) as they are empty.

Example: Vertex (1,2) means that there are 1 liter in the 5-litered jug, 2 liters in the 3-litered jug.

According to our rules, we would have the following graph.

dayhard1

Solution with the method

We used equilateral triangles to build the billiard table for a reason. Using a billiard ball in such shaped table, ball would travel according to our rules. And that is making a regular reflection. In other words, ball moves on the table like a light ray reflecting from a mirror.

Ball’s paths are clear. When it starts travelling from the starting point which is (0,0) it could only follow either (0,3) or (5,0) path. Let’s assume we hit the ball into the direction of (0,3). Ball would pass from the vertices (0,1) and (0,2) and reach (0,3) until it makes a regular reflection which is in the direction of the vertices (3,3) or (3,0).

From now on I will refer to 5-liter jug as “jug A”, and 3-liter jug as “jug B”. To start the solution, I choose the direction of the vertex (5,0).  Then let’s follow the direction of the vertex (2,3).

dayhard2

From the vertex (2,3) let’s use the path to the vertex (2,0).

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I will hit the ball towards vertex (0,2) and (5,2) respectively.

Final step is to make the reflection into the path of the vertex (4,3). This vertex means jug A has 4 liters of water which was the amount Simon wanted us to achieve.

Solution with words

  • Fill jug A.
  • Pour it into jug B.
  • Empty jug B.
  • Pour the 2 liters (which was inside jug A) into jug B.
  • Fill jug A completely and pour it on top of what jug B has (at the point jug B has only 1 liter of capacity).
  • Jug A has exactly 4 liters of water.

If only our heroes knew the power of vertices and lines, they would have solved this problem in a much shorter time. Although I would have forgotten my name if I was in front of a bomb. So, I congratulate you boys!

Serkan says…

  1. Is there a second way to find the 4 liters with our method?
  2. Find both ways to get 1 liter of water.
  3. If jugs were 6 and 15 liters in capacity, would it be possible to have 5 liters of water? Give your answer with a proof.

M. Serkan Kalaycıoğlu