Number Play | IT

Question 13

Try the same procedure for some other numbers, and perform the same steps. Stop if you get a palindrome. There are numbers for which you have to repeat this a large number of times. Are there numbers for which you do not reach a palindrome at all?

Solution

We take a number, reverse its digits, and add it to the original number. We repeat this until we get a number that reads the same forwards and backwards.

Step 1 — Try with 59

Let us start with the number 59. We reverse its digits.

$59 \text{ reversed is } 95$

We add the original number and the reversed number.

$59 + 95$

$= 154$

154 is not a palindrome. It does not read the same forwards and backwards.

Diagram 1

Step 2 — Continue with 154

We take the new number 154. We reverse its digits.

$154 \text{ reversed is } 451$

We add the number and its reversed number.

$154 + 451$

$= 605$

605 is not a palindrome. It does not read the same forwards and backwards.

Diagram 2

Step 3 — Continue with 605

We take the new number 605. We reverse its digits.

$605 \text{ reversed is } 506$

We add the number and its reversed number.

$605 + 506$

$= 1111$

1111 is a palindrome. It reads the same forwards and backwards. So, we stop here.

Diagram 3

Step 4 — Numbers needing many steps

Let us try another number, 79. We start with 79.

$79 + 97 = 176$

176 is not a palindrome. We continue.

$176 + 671 = 847$

847 is not a palindrome. We continue.

$847 + 748 = 1595$

1595 is not a palindrome. We continue.

$1595 + 5951 = 7546$

7546 is not a palindrome. We continue.

$7546 + 6457 = 14003$

14003 is not a palindrome. We continue.

$14003 + 30041 = 44044$

44044 is a palindrome. We stop here. This took 6 steps. This is a large number of steps.

Step 5 — Numbers that might never reach a palindrome

Some numbers might never reach a palindrome. Mathematicians have tried this for many numbers. For example, for the number 196, no one has found a palindrome yet. They have done the steps millions of times. The numbers become very, very big. But still, no palindrome has been found for 196. So, it seems some numbers might never reach a palindrome.

Answer

(i) Yes, there are numbers for which you have to repeat this a large number of times. For example, starting with 79, it takes 6 steps to reach the palindrome 44044. (ii) Yes, there are numbers for which we do not reach a palindrome at all. For example, for the number 196, mathematicians have not found a palindrome even after millions of steps.

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