The Land of Tens: each place is ten of the one before

0:00Imagine a calculator with one button. It adds. That is all it does, and it always adds the same amount. Here is the first one. Its button adds one thousand. Press it three times and the screen says three thousand. Press it a hundred times and you have one lakh. Here is the second. Its button adds ten. And the third adds one hundred. Three machines, one button each, and a question that is going to turn out to be about something much bigger than calculators.

0:36Let's get a feel for them. On the thousands machine, how many presses to reach one lakh? A hundred. A hundred thousands make a lakh. On the hundreds machine, the same target takes a thousand presses. And on the tens machine, ten thousand presses. Same destination, three very different amounts of work. It also runs the other way. If somebody presses the thousands button a hundred and fifty three times, the screen reads one lakh fifty three thousand.

1:12Four hundred and thirty five presses of the tens button gives four thousand three hundred and fifty. You can read the number off the count, and the count off the number. Now a better question. Not how long, but whether at all. Can the thousands machine show three thousand seven hundred? It cannot. It goes three thousand, four thousand. Three thousand seven hundred is not on the way. Can the tens machine show it? Yes. Three hundred and seventy presses.

1:47So the machines are not just slower and faster than each other. They reach different sets of numbers. And you can settle that without pressing anything. A machine that adds a hundred can only ever land on multiples of a hundred. That is a fact about division, and it decides the question before you start. Here is a claim worth testing. The hundreds machine says: there are numbers I can show that the other two cannot.

2:18Take a moment with that before I answer it. Against the thousands machine, it is right. Three thousand seven hundred is thirty seven presses for it, and impossible for the other. But against the tens machine, it is wrong. Completely wrong. Every multiple of a hundred is also a multiple of ten. So anything the hundreds machine can display, the tens machine can display too. It just takes ten times as long.

2:50The claim sounded reasonable and it does not survive one line of reasoning. That is worth more than being told the answer. Now a machine with seven buttons. One, ten, a hundred, a thousand, and so on upwards. Let's make three hundred and twenty one. Thirty two presses of ten, and one press of one. That is three hundred and twenty one, in thirty three presses. Or: two presses of a hundred, twelve of ten, and one of one. Also three hundred and twenty one, in fifteen presses.

3:29Both are correct. Neither is wrong. So how many different ways are there altogether? For three hundred and twenty one, there are exactly seventy two. That is a lot. But notice — it is a number. Not endless. Seventy two, and then you have found them all. So there is no one right recipe. But there might be a cheapest one. Before we hunt for it, let's write recipes down properly.

3:58Take five thousand and seventy two. One recipe: fifty presses of a hundred, seven of ten, two of one. Written out, that is fifty times a hundred, plus seven times ten, plus two times one. Fifty, plus seven, plus two. Fifty nine presses. Here is another: three presses of a thousand, twenty of a hundred, seventy two of one. Same number on the screen. Three, plus twenty, plus seventy two. Ninety five presses.

4:31Two recipes for one number, and one of them costs thirty six presses more than the other. So let's try to be efficient on purpose. Five thousand and seventy two again. Five presses of a thousand gets us to five thousand. No hundreds needed. Six presses of ten, and twelve of one. Five, plus nothing, plus six, plus twelve. Twenty three presses. That is much better than fifty nine. But is it the best?

5:04Try again. Five thousands. Seven tens. Two ones. Five, plus seven, plus two. Fourteen presses. Fourteen. And nothing beats fourteen. Hold on to that number, because something is about to happen. Why is fourteen the floor? Here is the whole mechanism, and it is one idea. Take any recipe and change it. Remove one press of a hundred. The screen is now a hundred short, so put it back with tens. That takes ten presses.

5:39You removed one press and added ten. Your total went up by nine. That is always true, at every level. Trade one thousand for ten hundreds, and you pay nine more presses. Trade down, pay nine. Every single time. Which means the cheapest recipe is the one that never trades down. Use the biggest button that fits, as often as it fits. And that is not a clever trick. It is forced.

6:11So what is the cheapest recipe for five thousand and seventy two? Five thousands. No hundreds. Seven tens. Two ones. Now look at the number itself. Five, zero, seven, two. They are the same thing. The cheapest recipe is not similar to the written number. It is the written number. That is what our notation has been all along. Not a way of writing numbers down, but the shortest instructions for building one.

6:44Which gives you a shortcut you can use immediately. The fewest presses is just the digits, added up. Sixty six thousand six hundred and sixty six. Five sixes. Thirty presses, and you did not have to try anything. Now let me break it, because a rule you have only seen work is a rule you do not understand yet. Here is a machine with just two buttons. Ten thousand, and one hundred. Nothing in between and nothing above.

7:17Twenty thousand eight hundred is fine. Two presses of ten thousand, eight of a hundred. Ten presses, exactly as the digits promise. Now ask it for sixty five lakh thirty thousand. Its biggest button is ten thousand. There is nothing bigger to carry into. So it needs six hundred and fifty three presses. For a number whose digits add to fourteen. The rule did not fail because arithmetic changed. It failed because the machine has no button for the big places.

7:53The digits only tell you the cost when there is a button for every place. That was the hidden condition all along. One last thing, and it runs the idea backwards. Suppose I want a three digit number that takes exactly thirty presses. Not fewer, not more. Every number has a cheapest count, which is its digits added up. And every trade down adds nine. So the counts a number can have are its digit sum, then nine more, then nine more.

8:29Nine hundred and ninety three costs twenty one at its cheapest. Twenty one plus nine is thirty. That works. And it is the largest three digit number that does. The smallest is one hundred and two. Three at its cheapest, then nine, then nine, then nine. But nine hundred and ninety nine can never be done in thirty presses. Its digits add to twenty seven, and from there you can only reach thirty six, or forty five.

9:01Thirty is simply not on its list. Here is my question for you. Take the year you were born, and find its cheapest recipe. Then tell me the next three press counts it can possibly have. You already know how to work them out.