Palindromes

Look at 66, 848, 575, 797 and 1111. Each reads the same from left to right and from right to left. Such numbers are called palindromes.

The book asks for all the 3-digit palindromes using only the digits 1, 2 and 3 (a digit may be repeated). The first and last digits must match: 1, 2 or 3. The middle digit can be 1, 2 or 3. That gives 3 × 3 = 9 palindromes: 111, 121, 131, 212, 222, 232, 313, 323, 333.

A way to make palindromes: reverse-and-add. Add a 2-digit number to its reverse. Stop if you get a palindrome; if not, reverse and add again.

• 34 + 43 = 77: a palindrome at once.
• 29 + 92 = 121: also a palindrome.
• 48 + 84 = 132, not a palindrome, so again: 132 + 231 = 363. Done!

Most numbers get there quickly. But for 196, nobody has ever reached a palindrome, even with computers trying millions of steps. It is still a mystery.

The smallest and largest 5-digit palindromes? 10,000 is not a palindrome: backwards it reads 00001. The smallest is 10,001; the largest is 99,999. Their sum is 10,001 + 99,999 = 1,10,000.