Construction of a Mathematical Clock – Using Three Ones

It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.

Pierre-Simon Laplace

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We all know the structure of a clock (here, we are considering the normal clock). We can see the numbers 1, 2, 3, . . . , 12 in the standard clock.

Now, our task is to create one mathematical clock, where in place of these numbers, we will put three 1’s and various mathematical symbols. In other words, we need to produce the numbers 1, 2, 3, . . . , 12 using only three 1’s and various mathematical symbols like +, −, ×, ÷, ! (factorial), . (decimal) etc.

Using simple addition and subtraction, we can write 1 = 1 – 1 + 1.

We know that 2 = 1 + 1. In view of 1 = 1 × 1, we can write 2 = 1 + 1 × 1.

If we add three 1’s, we obtain 3 = 1 + 1 + 1.

To represent the other numbers, we require few mathematical operations.

In view of this, we find that 1/x = 9.

Since 4 = 1 + 3, we can write

Now we will use the factorial operation. We define

Using the above definition, we get

6 = 3 × 2 × 1 = 3! = (1 + 1 + 1)!

The idea of generating 6 will be helpful to generate 7 as below:

Now, our task is to generate 8. We can write

We can easily write that 10 = 11 – 1.

Since 1 is the multiplicative identity, we write 11 = 11 × 1.

Finally, we have 12 = 11 + 1.

Putting the numbers in the respective places, we finally obtain one mathematical clock as below.

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