INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More

Do you know that ** CLOCKS **add numbers in a different way than we do? Do you know that

Consider the clock on earth.

So, there are 12 numbers {1,2, ..., 12 } are written on the clock. But let's see how clocks add them.

What is 3+ 10 ?

Well, to the clock it is nothing else than 1. Why?

Say, it is 3 am and the clock shows 3 on the clock. Now you add 10 hours to 3 am. You get a 13th hour of the day. But to the clock, it is 1 pm.

So, 3 + 10 = 1.

If you take any other addition, say 9 + 21 = 6 to the clock ( 9 am + 21 hours = 6 pm ).

Now, you can write any other **Clocky **addition. But you will essentially see that the main idea is :

The clock counts 12 = 0.

Isn't it easy? 0 comes as an integer just before 1, but on the clock, it is 12 written. So 12 must be equal to 0. Yes, it is that easy.

This is a handsome and sober way to write the arithmetic of a set. It is useful if the set is finite like the numbers of the **CLOCK **Arithmetic.

Let me show you by an example.

Consider the planet **Cheenta**. A day on Cheenta consists of 6 earth hours.

So, how will the clock on Cheenta look like?

Let's us construct the Cayley Table for **Cheenta's Clocky Arithmetic**. Check it really works as you wish. Here for Cheenta Clock, 3 = 0.

: Draw the Cayley Table for the Earth (24 hours a day) and Jupiter (10 hours a day).Exercise

Nice, let's move on to the Rotato part. I mean the arithmetic of Rotation part.

Let's go through the following image.

Well, let's measure the symmetry of the figure. But how?

Well, which is more symmetric : The **Triskelion **or the **Square **(Imagine).

Well, Square seems more right? But what is the thing that is catching our eyes?

It is the set of all the symmetric positions, that capture the overall symmetry of a figure.

For the Triskelion, observe that there are three symmetric operations that are possible but that doesn't alter the picture:

- Rotation by 120 degrees. \(r_1\)
- Rotation by 240 degrees. \(r_2\)
- Rotation by 360 degrees. \(r_3\)

For the Square, the symmetries are:

- Rotation by 90 degrees.
- Rotation by 180 degrees.
- Rotation by 270 degrees.
- Rotation by 360 degrees.
- Four Reflections along the Four axes

For, a square there are symmetries, hence the eyes feel that too.

So, what about the arithmetic of these? Let's consider the Triskelion.

Just like 1 interact (+) 3 to give 4.

We say \(r_1\) interacts with \(r_2\) if \(r_1\) acts on the figure after \(r_2\) i.e ( 240 + 120 = 360 degrees rotation = \(r_3\) ).

Hence, this is the arithmetic of the rotations. To give a sober look to this arithmetic, we draw a Cayley Table for this arithmetic.

Well, check it out.

Exercise: Can you see any similarity of this table with that of anything before?

Challenge Problem: Can you draw the Cayley Table for the Square?

You may explore this link:- https://www.cheenta.com/tag/level-2/

And this video:- https://www.youtube.com/watch?v=UaGsKzR_KVw

Don't stop investigating.

All the best.

Hope, you enjoyed. ðŸ™‚

Passion for Mathematics.

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

JOIN TRIAL
This is absolutely amazing. Come on guys, let all of us fall in love with mathematics.

I am amazed by the beauty of maths...it really is a beauty and a language which connects us with the whole world. Hope the others have liked it.Thx.

Keep doing great mathematics ðŸ™‚

That is absolutely amazing...what is the content in the brain of a mathematian.Amazing!