Unit 2 · Topic 04 · Number Play
Anaya read 121 forwards. Then she read it backwards.
Same number. Kabir tried it with 123 — forwards and backwards gave two different numbers entirely.
One of those numbers had a name, and the other one didn't.
Ms. Rao wrote 121 on the board. "Read it left to right." Anaya read: one-two-one. "Now right to left." Anaya read again: one-two-one. "Same both ways."
"That's called a palindrome," said Ms. Rao. "A number — or a word — that reads identically forwards and backwards. 'MADAM' is a palindrome word. 121 is a palindromic number."
Kabir tried 123 the same way. Forwards: one-two-three. Backwards: three-two-one. "Different. Not a palindrome."
"Right. Now here's a genuinely useful trick," said Ms. Rao. "Take ANY number that isn't already a palindrome, reverse its digits, and ADD the reversed version to the original. Sometimes — not always — you land on a palindrome immediately."
She tried 47. Reversed: 74. Added: 47 + 74 = 121. "A palindrome, in one step."
Anaya tried 68. Reversed: 86. 68 + 86 = 154. "Not a palindrome yet." She repeated the process on 154: reversed is 451. 154 + 451 = 605. Still not. She tried again: reverse 605 to 506, add: 605 + 506 = 1111. "That's a palindrome! It just took three rounds instead of one."
"That process — reverse, add, repeat until you hit a palindrome — is called the reverse-and-add process," said Ms. Rao. "Most numbers reach a palindrome within a few rounds. A very few numbers, like 196, are suspected to never reach one at all, no matter how many rounds you try — nobody has ever proven it either way."
Kabir was struck by that. "So this is genuinely unsolved? Not just hard homework?"
"Genuinely unsolved," said Ms. Rao. "Mathematics still has real open questions, and this is one you can poke at yourself with nothing but addition."
A palindromic number reads identically forwards and backwards, like 121 or 1331.
The reverse-and-add process: reverse a number's digits, add the reversed version to the original, and check if the result is a palindrome. If not, repeat on the new result.
Most numbers become palindromes within a few rounds of reverse-and-add — some in a single step, others taking several.
A small number of starting values (196 is the most famous) have never been shown to reach a palindrome no matter how many rounds are tried — a genuinely open, unsolved question in mathematics.
Try the reverse-and-add process on 5 different two-digit numbers and count how many rounds each takes.
Find 3 palindromic numbers with 3 digits and 3 palindromic numbers with 4 digits.
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