# Let's Invent a Math Trick Together

## Метаданные

- **Канал:** polymathematic
- **YouTube:** https://www.youtube.com/watch?v=5u6sy7VCeMA
- **Дата:** 13.05.2026
- **Длительность:** 3:00
- **Просмотры:** 1,822
- **Источник:** https://ekstraktznaniy.ru/video/51304

## Описание

Let's build a math trick from scratch.

Check out the channel! @polymathematic 

Start with a column of products of numbers just above 200 and stare at them for a minute. The first thing worth noticing isn't the answer. It's how far each factor is from 200. For 203 × 207, the "extras" are 3 and 7. For 205 × 209, they're 5 and 9. For 202 × 211, they're 2 and 11. And so on.

Now glance at the actual products. 203 × 207 ends in 21. Hmm. 3 × 7 = 21. 205 × 209 ends in 45, and 5 × 9 = 45. 202 × 211 ends in 22; 2 × 11 = 22. 208 × 206 ends in 48; 8 × 6 = 48. The last two digits of these products are just the product of the two extras. We've got half a trick already.

Keep looking. The middle of each answer wobbles a little. 200 × 200 is 40,000, so all of these products live a little above 40,000. How much above? Take the extras again — 3 and 7 — add them to get 10, and then double it to get 20. Sure enough, 203 × 207 is 42,021. Try 205 × 209: extras 5 and 9 sum to 14, double to 28, and the pro

## Транскрипт

### Segment 1 (00:00 - 03:00) []

You know, I don't have such a trick. So, let's invent one. To invent a math trick, we want to begin by observing. We're going to invent a math trick for multiplying numbers just above 200. So, I've got several such products right here. 203 * 207 all the way down the list to 211 * 212. We want to pay special attention to how far away from 200 we are. So, in 203 * 207, we're 3 and 7 away. And do we notice anything over here related to 3 and 7? Yes. 3 * 7 is 21. And that happens to be the last two digits of our product. But maybe that's just a coincidence, right? We need to go down the list and see whether that happens again. Well, 205 * 209 makes something that ends in 45, which of course is 5 * 9. And so on and so on. 202 * 211 ends in 22. 208 and 206 ends in 48. 8 * 6 is 48. 211 * 212 ends in 32. Okay. Well, we'll come back to that. Now, it's all well and good to say we have the final two digits of our product, but there's a lot more product to go. So, where do we get some of these other digits? Take those same two numbers. We were three and seven bigger than 200. 3 + 7 is 10. And if you double 10, you get 20. Well, that's interesting. I wonder if that continues. 5 + 9 is 14. And if you double that, you get 28. 2 and 11 together make 13. If you double that, you get 26. And 8 and 6 together make 14. And yes, if you double that, you get 28. Now come back to this last one. 21 * 212. 11 * 12 isn't 32, but it is 132. And also 11 + 12 makes 23. 23 * 2 is 46. But our two digits here weren't 46, they were 47. The way in which this deviates from the trick actually reveals to us the deeper truth of what's going on here. If we use a tool to group our multiplication, here's what we can tell. Our answers were always a little bit bigger than 40,000 because again, we're multiplying two numbers a little bigger than 200. And 200 * itself is 40,000. 200 * 12 makes 2400. 200 * 11 makes 2200. And I can tell, oh, of course, I'm just adding up the 11 and 12. and then doubling because I'm literally doubling the 11 and 12 to get these values and then adding them together. But remember in this case we were off by one. Those two digits there weren't just double the sum of 11 and 12. They were one bigger than The reason for that is 11 * 12 is 132. And so we do have the 32 here in our final two digits, but we have to essentially carry over the extra 100 that we generated. And so now, not only do we understand this nifty trick to multiply numbers just above 200, we also understand why it's working and we can apply it in hopefully more scenarios.