Question 2

We will play a number game. We reverse a number and add it to itself. We keep doing this. We want to see if we always get a special number called a palindrome.

Step 1 — What is a Palindrome?

A palindrome is a number. It reads the same forwards and backwards. For example, 121 is a palindrome. 363 is also a palindrome. 45 is not a palindrome.

Step 2 — How to Play the Game

Let us learn the rules of our number game. First, we pick a 2-digit number. Next, we flip its digits around. This gives us a new number. Then, we add the original number and the flipped number. We look at the sum. If the sum is a palindrome, we stop playing. If the sum is not a palindrome, we play again. We use the sum as our new number. We flip its digits and add them. We keep doing this until we find a palindrome.

Step 3 — Playing with 56

Let us start with the number 56. We reverse its digits. The reverse of 56 is 65. Now we add these two numbers.

$56 + 65$

$= 121$

The number 121 is a palindrome. It reads the same both ways. So, we found a palindrome in 1 step.

$\boxed{\text{Palindrome found: 121}}$

Step 4 — Playing with 29

Let us try another number. We pick 29. We reverse its digits. The reverse of 29 is 92. Now we add these two numbers.

$29 + 92$

$= 121$

The number 121 is a palindrome. It reads the same both ways. So, we found a palindrome in 1 step again.

$\boxed{\text{Palindrome found: 121}}$

Step 5 — Playing with 84

Let us try the number 84. We reverse its digits. The reverse of 84 is 48. Now we add these two numbers.

$84 + 48$

$= 132$

The number 132 is not a palindrome. It reads 132 forwards. It reads 231 backwards. So, we play again with 132. We reverse the digits of 132. The reverse of 132 is 231. Now we add these two numbers.

$132 + 231$

$= 363$

The number 363 is a palindrome. It reads the same both ways. So, we found a palindrome in 2 steps this time.

$\boxed{\text{Palindrome found: 363}}$

Step 6 — Our Findings

We played this game with different 2-digit numbers. We saw that we found a palindrome each time. This game usually ends with a palindrome. Most 2-digit numbers will lead to a palindrome. It might take a few steps.

However, there are some special numbers. For these numbers, it is very hard to find a palindrome. Or maybe it never ends. One famous example is the number 196. People have tried many, many times. They have not found a palindrome for 196 yet. But for most numbers, this game works.

Answer

(i) Yes, reversing and adding numbers repeatedly, starting with a 2-digit number, will eventually give a palindrome in most cases. (ii) The process involves taking a number, reversing its digits, and adding it to the original number. If the sum is not a palindrome, we repeat the process with the sum. (iii) For example, starting with 56, we get 56 + 65 = 121, which is a palindrome. Starting with 84, we get 84 + 48 = 132. Then 132 + 231 = 363, which is a palindrome. (iv) A known exception is the number 196, for which a palindrome has not been found through this process.