Cyclic Numbers


Mathematics has some strange nooks and crannies that are fun to explore. One such idea is that of cyclic numbers.

A cyclic number is like a snake chasing its tail. When you multiply a cyclic number with another number, the product has the same digits as the cyclic number. And these digits are in the same order as the original number.
Take for example, 142857.

Write it like a snake chasing its tail:

Now let us multiply it by 4: 142857 × 4 = 571428

Now in this number 571428, the head is 5 and the tail is now 8:

In the same way, if you multiply 142857 with any number between 2 and 6, the body of the snake stays the same. But the head and tail changes:

142857 × 1 = 142857
142857 × 2 = 285714
142857 × 3 = 428571
142857 × 4 = 571428
142857 × 5 = 714285
142857 × 6 = 857142

The other interesting idea in this is that you don’t actually have to do the arithmetic to get the answer. There is a very simple trick to figure out where the head of the number is and where the tail.

Say, you want to find the answer to the multiple of 6:
Step 1: Take the digit at the unit’s position ? 7.
Step 2: Multiply 6 with 7 ? 42.
Step 3: Look at the units place of the product from step #2? 2.

Step 3 tells us where the tail of the snake is. The snake ends at 2. So the head of the snake must be the number after 2 (because the snake is eating its tail, remember?) ? 8.

And the answer….magically…. is 857142.
Remember this works only when multiplying the cyclic number with multiples between 2 and 6.
That was easy. Now let’s look at a really large cyclic number. This one has 96 digits:
Now I know that the first digit is 0 and some of you may argue that it’s technically a 95 digit number. But stay with me and watch the magic.
As with 142857, the trick works only with certain multiples. This time the range is larger and the trick works for all multiples between 2 and 96.
The steps to find the tail of the snake work the same way as above::
Step 1: As earlier, take the digit at the unit’s position ? 7.
Step 2: Multiply 7 with your chosen multiple.
Step 3: Look at the units place of the product from step #2 and this is the tail of the snake.
So if the multiple is 23, the tail of the snake is at 1 [ 3 X 7 = 21 ]. But 1 figures 10 times in our long snake. So which one is it?
The answer is in Step 4: The head of the snake is the same as the tens digit of the multiple. So the head is at 2 (from the multiple 23). Can you spot the breakpoint in the old snake? Remember the tail comes before the head in the old snake. So look for ‘12’.

The answer of Big Snake X 23 =
Want to check this? Try
What happens when we multiply with a number less than 10? Then the head of the snake is at 0.
There is an additional rule that you must remember: When the multiple has either 8 or 9 in the units place, change Step 2 to this: Multiply 7 with your chosen multiple and add one to it.
Why this Kolaveri Di?

Say the multiple is 68.
Step 1: Digit at unit’s place ? 7.
Step 2: Multiply 7 with your chosen multiple ? 7 X 8 = 56.
Step 3: Tail: Look at the units place of the product from step #2? 6.
Step 4: Head: The head of the snake is the same as the tens digit of the multiple: 6
So the split point is 66. But the big snake has no 66 in it.
So add Step 4a: Add one ? 6 + 1 = 7.
So the split point is 67.
Old snake:
Old snake X 68:

Was that fun?
Try this yourself: What is the number if you multiply the 96 digit snake with 51?

B D Bhargava was born in Ajmer, Rajasthan in 1932. He graduated from Ruia College Bombay (now Mumbai) and did M Sc ( Pure Maths) from the University of Bombay. He retired as Secretary Marketing from L.I.C. Central Office in November 1990. Now 84 years old he lives in Agra. He loves recreational and innovational work involving words and numbers. He has developed the world’s largest dictionary of Bacronyms – where a word is described in a sentence comprising words that start with the letters of the original word, in sequence.
His work was displayed at the Taj Literature Festival in Agra in February 2013, and has been featured in numerous publications throughout the world. He has also developed many extensions of magic squares and cubes.