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

0:00Here are five numbers. Sixty-six. Eight hundred and forty-eight. Five hundred and seventy-five. Seven hundred and ninety-seven. And one thousand one hundred and eleven. They share a property, and your book wants you to spot it before it tells you. Notice first what it cannot be. They are different lengths. Two digits, three digits, four digits. So whatever the property is, it does not care how many digits a number has.

0:32And they are all over the place in size. Sixty-six and eleven hundred and eleven are not near each other. So it is not about size either. Have a proper look before I say it. Here it is. Read each one out backwards, digit by digit, and nothing changes. Sixty-six backwards is six six. The same number. Eight four eight backwards is eight four eight. The same. Five seven five. Seven nine seven. One one one one. All of them the same in reverse.

1:09A number like that is called a palindrome. And here is what makes this genuinely new, and worth stopping on. Every property you have met so far has been about the quantity. Even, odd, bigger, smaller, a multiple of three. This one is about the writing. It is a fact about the symbols on the page, not about the amount. That has not been true of anything else in this book.

1:38Your book gives you a small exercise. Using only the digits one, two and three, write down every three-digit palindrome you can. It shows you three of them to get you started. One two one. Three one three. Two two two. Now, you could hunt around for the rest. But we just spent a whole video learning not to hunt. Think about the positions instead. There are three of them. Front, middle, back.

2:07But being a palindrome ties the front and the back together. Whatever you put at the front, the back has to match it. So you really only get two free choices. What sits at the ends, and what sits in the middle. Front and back. You choose once, and you have three digits to choose from. Three choices. Middle. Three digits available, and completely independent of the ends. Three choices.

2:37Three times three. Nine. And here they are. One one one, one two one, one three one, two one two, two two two, two three two, three one three, three two three, three three three. Nine, exactly as predicted. Same move as last time. Fix a position, count what can stand there, multiply. The only new thing is that being a palindrome removed a position from the count, because the last digit was never free.

3:11Now the part of the section that is genuinely a machine. Take any number. Write it down. Write it again backwards. Add the two together. If what you get is a palindrome, stop. If it is not, do the same thing again to the answer. That is the whole procedure. It is called reverse-and-add. And the surprising thing, which your book wants you to discover for yourself, is how often it works, and how fast.

3:41So let's run it. Start with thirty-four. Backwards, that is forty-three. Thirty-four plus forty-three is seventy-seven. Seventy-seven is a palindrome. Done, in one round. Try twenty-nine. Backwards is ninety-two. Twenty-nine plus ninety-two is a hundred and twenty-one. One two one. Palindrome. Also one round. And that is not luck. Of the ninety two-digit numbers you could start with, nine are already palindromes, and forty-nine more finish in a single round.

4:21So more than half the time, one addition is all it takes. But not always. Try forty-eight. Forty-eight plus eighty-four is a hundred and thirty-two. One three two. Not a palindrome. So we go again. A hundred and thirty-two backwards is two three one. Add them, and you get three hundred and sixty-three. Three six three. Palindrome. Two rounds. Same story with seventy-six. Seventy-six plus sixty-seven is a hundred and forty-three. Not a palindrome.

4:59A hundred and forty-three plus three four one is four hundred and eighty-four. Palindrome. Two rounds. So why did those two need a second go when thirty-four and twenty-nine did not? Your book leaves that hanging. Let's answer it. Take any two-digit number. Call its tens digit a, and its units digit b. The number itself is ten a, plus b. Written backwards, it is ten b, plus a. Add those together and you get eleven a plus eleven b. Which is eleven times a plus b.

5:38And a plus b is just the digit sum. So a two-digit number plus its reverse is always eleven lots of its digit sum. Always. Check it. Thirty-four. Three plus four is seven, and eleven sevens is seventy-seven. Correct. Twenty-nine. Two plus nine is eleven, and eleven elevens is a hundred and twenty-one. Correct. Forty-eight. Four plus eight is twelve, and eleven twelves is a hundred and thirty-two. Correct. One line of algebra, and every printed example is accounted for.

6:18Now we can say exactly when it finishes in one round. Eleven times something is symmetric by construction. Eleven sevens is seventy-seven — a seven in the tens column and a seven in the units column. A palindrome for free. But that only holds while the digit sum stays under ten. Because eleven twelves would want a twelve in the tens column and a twelve in the units column. And a column cannot hold twelve. It overflows into the next one along, and the symmetry breaks.

6:56So: digit sum under ten, and you finish in one round. Digit sum of ten or more, and you will usually need another. Here is what eleven times ten and upwards actually gives. A hundred and ten. A hundred and twenty-one. A hundred and thirty-two, and so on, up to a hundred and ninety-eight. Look at that list. Exactly one of them is a palindrome — a hundred and twenty-one. And that is the whole explanation.

7:29So how long can this run? Your book warns you that some starts take a while. Among two-digit starts, most are done in one round or two. A handful take three, four, or six. And then there are two that take twenty-four. Eighty-nine. And ninety-eight, which is just eighty-nine backwards, so of course it behaves the same way. Twenty-four rounds. The numbers get enormous on the way, and then it lands.

7:59If you have a spare page, that one is genuinely worth doing by hand. Now the part that makes this section special, and it is hidden in a footnote at the bottom of the page. The book asks whether a two-digit start always ends up at a palindrome. For two-digit starts, yes. All ninety of them get there. And then it says something remarkable. For three-digit starts, nobody knows. That is not a textbook being cautious. That is genuinely unsolved mathematics, printed in a Class Six book.

8:37And it names the suspect. One hundred and ninety-six. People have run one ninety-six for hundreds of millions of rounds. It has never produced a palindrome. And nobody has proved that it never will. The question is still open. So you have just met the edge of what is known, using nothing but adding up. Last, a puzzle, and it is a good one, because every clue takes something away.

9:07Five boxes, labelled from the left: ten thousands, thousands, hundreds, tens, units. Clue one: the answer is a five-digit palindrome. So the ten-thousands digit equals the units digit, and the thousands digit equals the tens digit. Clue two: it is odd. So the last digit is odd. Clue three: the tens digit is twice the units digit. Clue four: the hundreds digit is twice the tens digit. Now chain them together. The units digit is odd, so it is one, three, five, seven or nine.

9:47The tens digit is twice that, and it has to fit in one box. Twice five is ten, which is too big. So the units digit is one or three. The hundreds digit is twice the tens, which is four times the units. Four threes is twelve, too big again. So the units digit is one. And now everything falls out. Units, one. Tens, two. Hundreds, four. And by the palindrome rule, thousands is two and ten-thousands is one.

10:20Twelve thousand, four hundred and twenty-one. And it is the only number that fits, because every other route ran into a digit bigger than nine. Next time, a process that does not just usually land somewhere. It always lands on exactly the same number. Six one seven four.