Real Mathematics – Geometry #12

Cogito Ergo Sum*

*I think, therefore I am. You probably heard these words before. They belong to Rene Descartes. Descartes is believed to be the person who invented modern philosophy. Actually he was a groundbreaker for mathematics.

One of his best discoveries for mathematics has a mesmerizing story. Even though it is unknown whether this story is true or not, I find it amusing.

It is a known fact that Descartes had a rough childhood. He was constantly battling with various illnesses which were major setbacks for his early school days. Every day he was able to attend to his school around noon. This has become a habit for Descartes. For the rest of his life he spent his mornings in bed. (Except final days of his life in Sweden)

Story of Descartes’ discovery is based on this fact. Allegedly one morning as he was lying down, he saw a fly on his ceiling. He started thinking about fly’s position. There was one question in his mind: “How can I describe this fly’s position to someone who hasn’t been with me in this room?”

Descartes began his answer with assumption. If we assign a corner of the ceiling as the starting point, it is possible to reach the fly with only two directions for movement from that point: To the width and to the length.

This is the story of how Cartesian coordinate system was first discovered. Analytic geometry was born with Descartes’ publication on the issue.

Analytic Geometry = Cartesian Geometry = Coordinate Geometry

Cartesian Coordinate System

Let’s say that left bottom corner of Descartes’ ceiling is the starting point. There are two available paths for us: Up or right.

According to Descartes one can reach any point of the ceiling with two specific movements from the starting point: X amount to the right, Y amount to the up.

In that case the point (or the object) would have a position which could be expressed with two different numbers. If X represents amount of movement to the right, and Y to the left, then position can be shown as (X, Y).

Example: Assume that unit of movement is cm, and fly sits on the ceiling as shown:

If we sit at the starting point and make 4 cm to the right & 3 cm to the left, then it is safe to call the position of the fly (4,3).

Catching the Thief

Materials:

• Cartesian coordinate system that has size 6×6.
• Dice.
• Pencil/pen and paper.

Catching the thief is a multiplayer game. One player gets to be the detective as the other one gets to be the thief. Detective’s mission is the catch the thief as soon as possible.

Thief rolls the dice twice and determines its position: (First dice, Second dice).

Detective makes its first guess. If he/she is right, then the game is done.

If detective is wrong in his/her first guess, he/she receives a text message from the thief. Message reads a number. This number represents the total of the differences between these positions.

Let’s assume that the thief is at the position (2,2) and detective makes his/her guess as (4,1). Differences of the positions are: 4-2=2 and 2-1=1. Add them together and we’ll get 2+1=3. Hence thief sends the number 3 in his/her message.

Detective makes his second guess according to this information and this process continues until thief is caught.

Example

Thief rolls the dice. First try reads 3, second reads 4. Hence thief’s position is determined: (3,4).

Detective makes his/her first guess with (2,2) and misses.

Thief calculates the differences of the positions: 3-2=1 and 4-2=2. Thief adds them together: 1+2=3.

Detective receives the message: “3”. Now detective is in a slightly better situation.

Possibility 1: Detective adds 3 to the x. Guess: (5,2).

Possibility 2: Detective adds 2 to the x, and 1 to the y. Guess: (4,3).

Possibility 3: Detective adds 1 to the x, and 2 to the y. Guess: (3,4).

Possibility 4: Detective adds 3 to the y. Guess: (2,5).

Possibility 5: Detective subtracts 2 from the x, and 1 from the y. Guess: (0,1).

Possibility 6: Detective subtracts 1 from the x, and 2 from the y. Guess: (1,0).

Possibility 7: Detective subtracts 2 from the x, and adds 1 to the y. Guess: (0,3).

Possibility 8: Detective subtracts 1 from the x, and adds 2 to the y. Guess: (1,4).

Possibility 9: Detective adds 2 to the x, and subtracts 1 from the y. Guess: (4,1).

Possibility 10: Detective adds 1 to the x, and subtracts 2 from the y. Guess: (3,0).

After the message from the thief, detective knows for sure that the thief is at one of these 10 positions. Detective continues to make his guesses and uses the same strategy until he/she catches the thief.

One wonders…

1. Determine how many possibilities can the detective have in case he/she guesses (3,0)?
2. I am thief and you are the detective. Your first guess is (3,3) and I send you the text “2”. Where am I? Leave your answer to the comments.

M. Serkan Kalaycıoğlu