How to order 43 Chicken McNuggets - Numberphile

BRADY HARAN: 43 is a very special number with chicken nuggets. It's the highest number that you can't make by using the standard packs that you get from McDonald's. The packs are 9, 6 and 20. There's fewer of those numbers that you can't make. You can't make one, because they don't sell them in ones. You can't have 2 or 3. But 43 is the biggest. Yeah. So 43 isn't the only number that you can't make using these Chicken McNugget numbers. There's actually 22 different numbers you can't make. So the first one is 1. You can't make 1. They don't sell them in packs of 1. 2, we can't make. We can't make 3 or 4 or 5. But we can make 6. They do sell packs of 6. So we can make 6. But then we can't make 7 and 8. But we can make 9. So here's the pack of 9. And the others are 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 28, 31, 34, 37. And the biggest number that you can't make is 43. And all other numbers, you can make. So if you wanted to make the next number, 44, we would have 20-- a box of 20, two boxes of 9, and a box of 6. There we go. That makes 44. So obviously, McDonald's were very concerned about this problem. This happens a lot, people wandering about, ordering 43 nuggets. So, they've changed the rules. And this is a new development. They now sell the Happy Meal nugget box, which has 4 nuggets in it. They now sell that separately. So you can now buy boxes of 4, 6, 9 and 20, which changes everything. So this was our list of things we couldn't buy. But now that we can buy 4, we can do that no problem. And we can get 8, as well. No problem. Or could have 4 and a 6, so 10 disappears. In fact, most of these disappear now. And we can now order all these McNugget numbers. So the highest number now that you can't order is 11. Maybe that's something other people can try as well, OK. So if you don't want to eat 43, like we're going to have to do now, try it out with 11. See what we do. So the girls at McDonald's were quite concerned about this. We asked for 43, and they said, well, we can't sell packs of 3. We can't do 20's and 3's. And they said, well, we can give you 44, maybe two 20's and a 4. And I was very tempted to lean over and help them out. Because there is a way now that we can do 43. Three packs of 9. There's 9, 9, and we need another 9. Three packs of 9. Two packs of 6. 5, 6. And another 1, 2, 3, 4, 5, 6. And oh, a pack of 4, there. And that's 43. We've got one extra. Can I have this now? It's good. I haven't eaten yet. These McNugget numbers are great fun. But in the world of math, they're called Frobenius numbers. And this is the largest number that you can't make using-- well, whatever numbers that you decide. So for the McDonald's, it was 6, 9, and 20. Another name for the same problem is the coin problem. So if you have coins of different values, what can you make? What can you buy? What can't you buy? So we've ended up with 44 of these nuggets. But I want to know how many sachets of barbecue sauce they think someone with 44 nuggets needs. No. 1, 2 3, 4, 5, 6, 7, 8-- it's not enough. It's not enough. Come on. Disappointing. MALE SPEAKER: 8 seems like excessive to me.