College Mathematics I.S.I. and C.M.I. Entrance Math Olympiad USA Math Olympiad

Clocky Rotato Arithmetic

Do you know that CLOCKS add numbers in a different way than we do? Do you know that ROTATIONS can also behave as numbers and they have their own arithmetic? Well, this post is about how clock adds numbers and rotations behave like numbers. Let’s learn about clock rotation today

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.

Cayley’s Table

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?

Cheenta Planet clock

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.

Clock rotation explanation

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

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

Let’s go through the following image.

clock rotation

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

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

clock rotation

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:-

And this video:-

Don’t stop investigating.

All the best.

Hope, you enjoyed. 🙂

Passion for Mathematics.