How many numbers have a given number of digits, and why digit sums behave

0:00Start writing numbers. One, two, three, four, five, six, seven, eight, nine. And then you are stuck, because you have used up every symbol you have. There are only ten digits. Zero to nine. That is the entire alphabet of numbers. So to keep going you do something clever, and you did it so long ago that you have forgotten it was clever. You start using two positions. Ten. Eleven. Twelve. And off you go again.

0:31Your book puts this in a table. Five columns: how many numbers have one digit, two digits, three, four, five. It fills the first column in for you, and leaves the other four blank. Now, you could list them. But listing ninety thousand numbers is not a plan. We need to count without listing. First column. How many one-digit numbers are there? Almost everybody says ten, because there are ten digits.

1:03But your book's table starts at one, not zero. So the answer is nine. And that is not the book being fussy. It is the first appearance of the rule that runs through this whole section. A number does not start with zero. Write zero seven and you have not written a two-digit number. You have written seven, with something useless in front of it. So the digit you write first is special. It gets nine choices, not ten.

1:34Hold on to that, because it is about to do everything. Two-digit numbers. Instead of listing them, think about the two writing positions. Front position. What can go there? Anything from one to nine. Nine choices. Back position. What can go there? Anything from zero to nine, because now a zero is perfectly fine. Ten choices. And every front choice can pair with every back choice. Nine choices, then ten choices. Nine times ten. Ninety.

2:11Ninety two-digit numbers, and we never wrote a single one down. Check it if you like. Ten up to ninety-nine is ninety numbers. It agrees. Now the pattern just repeats, and it repeats beautifully. Three digits. Nine choices for the front, then ten, then ten. Nine hundred. Four digits. Nine, ten, ten, ten. Nine thousand. Five digits. Nine, ten, ten, ten, ten. Ninety thousand. So the table reads nine, ninety, nine hundred, nine thousand, ninety thousand.

2:50Every new position you allow multiplies the count by ten, because that new position has ten symbols available to it. And the nine at the front never changes, because the first digit never gets to be zero. One idea — fix a position, count what can stand there, multiply — filled in the entire table. Let's place those five-digit numbers, since we were on a number line last time. The smallest five-digit number is ten thousand. A one, followed by four zeros.

3:26The largest is ninety-nine thousand, nine hundred and ninety-nine. And then the very next number needs a sixth position, and the whole story starts over. Count them. From ten thousand up to ninety-nine thousand nine hundred and ninety-nine is ninety thousand numbers. The same answer we got by multiplying. Which is the point — two different routes, one number. Now the section turns, and a pupil called Komal notices something.

4:00She writes three numbers on the board. Sixty-eight. A hundred and seventy-six. Five hundred and forty-five. Three numbers with nothing obvious in common. Different sizes, different lengths. And she adds up the digits of each one. Six plus eight is fourteen. One plus seven plus six is fourteen. Five plus four plus five is fourteen. All three of them. Fourteen. And that total has a name. It is called the digit sum.

4:34So what is a digit sum actually doing? Here is the honest answer, and it explains everything that follows. When you add up the digits, you are deliberately throwing away where each digit stood. In a hundred and seventy-six, the one means a hundred. But in the digit sum it contributes one. Just one. Place value is the whole reason a one can mean a hundred, and the digit sum discards it on purpose.

5:04Which is why sixty-eight and five hundred and forty-five can land on the same total while being nowhere near each other in size. A digit sum is not a measure of how big a number is. It measures the symbols you wrote, ignoring where you wrote them. Your book asks for the smallest number whose digits add to fourteen. One digit cannot do it. The biggest single digit is nine, and nine is short of fourteen.

5:37So we need at least two digits. And to make the number small, you want the front digit as small as you can get away with. Call the front digit t and the back one u. They add to fourteen, and u cannot be more than nine. So t is at least five. Try five. Then u is nine. Fifty-nine. And fifty-nine works. Five plus nine is fourteen. Nothing below it can, because a smaller two-digit number has a front digit under five, and then the back digit would have to be more than nine. There is no such digit.

6:16Now the other end. The largest five-digit number whose digits add to fourteen. Being largest means starting as big as possible, so make the first digit nine. That uses nine of our fourteen, and leaves five to spread across four more positions. Same logic again. Make the next digit as big as it can be. Five. And that uses up everything we had. So the remaining three positions all take zero. Ninety-five thousand.

6:50Now the last part of the question, which is the interesting one. How big can a number with digit sum fourteen get? There is no biggest. Take ninety-five thousand and stick another zero on the end. Nine hundred and fifty thousand. Digit sum still fourteen, and it is ten times bigger. Zeros are free — they cost nothing in a digit sum, and you can always add one more. Next, write down the digit sums of every number from forty to seventy, and look at the shape they make.

7:27Forty is four. Forty-one is five. Forty-two is six. Up by one each time, all the way to forty-nine, which is thirteen. And then fifty. Which is five. It fell by eight. And it happens again from fifty-nine to sixty, and again from sixty-nine to seventy. Three drops, every one of them eight. The reason is exactly the place-value one. Going from forty-nine to fifty, the units digit falls from nine to zero, losing nine, while the tens digit rises by one, gaining one.

8:07Lose nine, gain one. A net drop of eight. Every single time. Question three. Three-digit numbers whose digits run consecutively upward. Your book's example is three four five. Its digit sum is twelve. Try another. One two three gives six. Two three four gives nine. Six, nine, twelve. Every one of them a multiple of three. Here is why, and it takes one line. The digits are some number, then the next one, then the one after that.

8:44Their total is three times the middle digit. Always. So it is always a multiple of three. And how many such numbers are there? Only seven. One two three, up to seven eight nine. Eight nine ten does not exist, because ten is not a digit. And zero one two is not a three-digit number, because it starts with a zero. Both ends are shut by rules we have already met.

9:14Last one, and there is a trap in the wording. A pupil called Dinesh has written out every number from one to a hundred, and wonders how many sevens that took. Notice what is being asked. How many sevens he wrote. Not how many numbers had a seven in them. Those are different questions, and here is the number that proves it. Seventy-seven. Seventy-seven is one number, but writing it takes two sevens.

9:44So between one and a hundred, nineteen numbers contain a seven — while twenty sevens actually get written. And you can count those twenty cleanly. A seven appears in the units position ten times: seven, seventeen, twenty-seven, and so on. And in the tens position ten times: seventy through to seventy-nine. Ten plus ten is twenty. No listing. From one to a thousand? Three hundred sevens get written, though only two hundred and seventy-one numbers contain one.

10:21Which brings the section full circle. Every count in it came from fixing a position and asking what can stand there. Next time, palindromes — numbers that read the same backwards, and a strange process that keeps landing on them.