Palindromes, and why reverse-and-add usually lands on one

These teaching notes are for members

What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

Create a free account Sign in

What to assume they know

How many numbers have a given number of digits, and why digit sums behave: digit sums, and reading a number as a string of digits rather than as a size
Column addition of two- and three-digit numbers, including carrying
Reading a number aloud in words up to five digits
Odd and even numbers

What they should be able to do

Decide whether a given number is a palindrome
Write down all the palindromes of a given length using a restricted set of digits
Count those palindromes by choices per position rather than by listing
Carry out one round of reverse-and-add and say whether to stop
Repeat the procedure until a palindrome appears, keeping a record of the rounds
Show that adding a two-digit number to its reverse always gives eleven lots of its digit sum
Use that to explain why some starts finish in one round and others do not
State what the book's footnote claims for two-digit and for three-digit starts, and mark which of the two is settled
Solve a digit-clue puzzle whose answer is a five-digit palindrome, and justify that the answer is the only one

Where it usually goes wrong

"A palindrome has to have an odd number of digits so it has a middle." 66 and 1111 are in the book's own opening list and neither has a middle digit.
"Palindromes are rare." Between 1 and 1000 there are quite a few, and the restricted-digit exercise produces nine from three digits alone. Rare is the wrong intuition; structured is the right one.
"Reverse-and-add is a trick with no reason behind it." For two-digit starts the reason is one line of arithmetic. Show it, or the section is just button-pushing.
"If it did not work in one round it will never work." 48 and 76 both fail the first round and succeed on the second. The book puts them beside the one-round cases precisely to head this off.
"Two-digit starts finish quickly." Most do. Two of them do not, and the book warns about this in the sentence before the Explore box — see Notes for how far the worst case actually runs. An explanation that shows only 34 and 29 misrepresents the section's own hedge.
"The footnote says reverse-and-add always works." It says so for two-digit starts only. For three digits it says the opposite: nobody knows. Conflating those two claims is the single easiest error to make on this page.
"196 has been proved never to work." It has not. The footnote's word is suspected, and it is the right word.

Questions to check understanding

"Is this number a palindrome?" across several lengths
Write all the palindromes of a given length from a restricted digit set
Perform reverse-and-add for a given start and state how many rounds it took
Given a start, predict in advance whether one round will be enough
Explain, for a two-digit start, why the sum is always eleven digit-sums
Digit-clue puzzles whose answer is a palindrome
Items asking for the largest and the smallest palindrome of a stated length — the chapter-end block asks exactly that for five digits (p.65)
Items on palindromic clock times, which reuse this idea in Clock times and calendar dates: patterns the format lets you have

Examples worth working on the board

The five opening examples (§3.5, p.61). 66, 848, 575, 797 and 1111. Note that they are of three different lengths — the property does not care.
The restricted-digit exercise (§3.5, p.61). Using only 1, 2 and 3, write every 3-digit palindrome. The book prints 121, 313 and 222 as samples. The full set is nine numbers, because the outer digit may be any of three and the middle digit may be any of three, independently.
The four printed reverse-and-add chains (§3.5, p.61). These are drawn as handwritten sums beside the text and do not extract.
- 34 + 43 = 77 — one round
- 29 + 92 = 121 — one round
- 48 + 84 = 132, then 132 + 231 = 363 — two rounds
- 76 + 67 = 143, then 143 + 341 = 484 — two rounds
The eleven-times fact. A two-digit number and its reverse add to eleven lots of the two digits' sum. Check it on the printed cases: 3 + 4 = 7 and 11 × 7 = 77; 2 + 9 = 11 and 11 × 11 = 121; 4 + 8 = 12 and 11 × 12 = 132; 7 + 6 = 13 and 11 × 13 = 143. This is not printed in the book; it is the explanation the section invites and never gives.
Which digit sums finish first. Eleven lots of a digit sum under ten is a two-digit palindrome every time. From ten upward the results are 110, 121, 132, 143, 154, 165, 176, 187, 198 — of which only 121 is already a palindrome. That single list is the whole account of why some starts need a second round.
The Explore prompt and its footnote (§3.5, p.62). The prompt asks whether a two-digit start always produces a palindrome. The footnote at the bottom of p.62 answers: yes for two-digit starts; for three-digit starts it says the answer is not known, and that 196 is suspected never to reach one.
Puzzle time (§3.5, p.62). Five empty boxes labelled tth, th, h, t and u, with a box below for writing the answer in words, and four clues: it is a 5-digit palindrome; it is odd; its t digit is twice its u digit; its h digit is twice its t digit.

Figures to have open

The four printed reverse-and-add sums (§3.5, p.61). They are handwritten-style artwork and carry all the arithmetic; redraw them as clean column additions but keep all four, since the pairing of one-round with two-round cases is the argument.
The Puzzle time strip of five boxes with its printed place labels, and the words box beneath it (§3.5, p.62). The labels are what the clues refer to, so they must be legible.
A column-addition schematic in which the units column overflows, for section 9. Standard schematic, not in the textbook.
No table, dataset or photograph is required.

Where this sits in the book

NCERT Class 6 Mathematics (Ganita Prakash), Chapter 3 "Number Play": §3.5 Pretty Palindromic Patterns, pp.61–62
The definition and the five opening examples, p.61
The bold sub-headings All palindromes using 1, 2, 3 and Reverse-and-add palindromes, both p.61, printed without numbers
The four printed reverse-and-add chains, p.61
The Explore block, the Puzzle time block, and the footnote about 196, p.62
Forward link: §3.6, pp.62–63 (Kaprekar's 6174: a process that always ends in the same place), which begins on the same page with a second digit-rearranging machine
Forward link: the chapter-end Figure it Out question that asks for the largest and the smallest palindrome of five digits, p.65