 Ancient Egyptian multiplication relied on forming a table of multiples. Now, this does require us to multiply a number somehow, and so it's worth keeping in mind that the two easiest multiples of a number are twice the number, and 10 times the number. So let's multiply 6 times 57, and we'll construct a table of multiples of 57. Now, to avoid introducing modern concepts, we'll treat this like a grocery list. In this case, our grocery list says to pick up 6 of the 57s. So let's begin our tally. One of the 57s is 57. Well, that's not quite enough. We need more, and so now we have to do some multiplications. Remember, the easiest multiplications to do are to double or to multiply by 10. So if we double the 57, we get two of the 57s, and that's going to be 114. Well, we wanted 6, so if we double it again, that gets us 4 of the 57s, which will be 228. Since we want 6 of the 57s, we'll need the set of 4 and the set of 2. And in the Ancient Egyptian scribal notebooks, these are typically checked off. It's also worth pointing out that this one of the 57s is not actually needed at this point, and we could either cross it out or ignore it. And so that means if I want 6 of the 57s, it says 114 plus 228, that's 342. Well, let's take a look at this. We want to multiply 153 by 45, and it's important to note here that we can, and the Egyptians did take advantage of the fact that A times B is equal to B times A. And so the question you've got to ask yourself is, self, which should we do? 153 times 45 or 45 times 153. So to make our decision, it's worth noting that in 153 times 45, we're getting 153 of the 45s, while in 45 times 153, we're getting 45 of the 153s. Since the second requires getting fewer items, that's 45 instead of 153, we'll find 45 times 153. And so we want 45 of the 153s. So as before, we'll start with one of the 153s. Well, that's just 153. And we'll double 153 a few times. So two of the 153s is 306. Doubling again, we find that four of the 153s is 612. Now remember, we want 45 153s. And remember, the second easiest multiplication, or maybe the easiest multiplication, is to multiply by 10. So here, if I have four of the 153s, I know what 40 of the 153s is. It's going to be. Now to get 45, I need to pick up the set of 40. And five more. Well, that's a set of four and the set of one. So I'll put a check mark next to these to indicate that we do need these in our shopping cart. Meanwhile, we don't need this set of two. So we'll either cross it out or ignore it. So our shopping cart now has a 40, a four, and a one. And if we add the corresponding amounts together, we'll get a 45 153s. Now the usual question here is, how much should you write down? And in ancient Egypt, papyrus was very expensive. And so they didn't write down a lot of this. What they mostly wrote down was the multiplier, this column, and the product, this column. And they didn't bother writing down the words. So if paper is very expensive and it's very costly to write things down, you only need to write down the multipliers and the multiples. But if you want to understand the method, it's very helpful to write down the explanatory words as well. The important idea here is that we're recording how many of which objects. One other quick note. When we did this, we went through this analysis deciding which product we were going to find. What would happen if you didn't do this analysis? Would it be catastrophic? Well, let's find out. So what if we found 153 times 45 instead? We'd need 153 of the 45s. So let's start our table. One of the 45s is 45. Since we need 100, let's start out by finding 100 of the 45s. And that's like multiplying by 10 twice. So 100 of the 45s is going to be 4500. Since we need 50, we'll find 10 of the 45s first. That's an easy multiplication. 10 of the 45s is 450. And then we'll double our amount a couple of times. So 20 of the 45s is 900 and 40 is 1800. And we can stop doubling because we have our 50 from the set of 40 and the set of 10. And finally, we need 3. So let's take our 1 and double it a few times. And so let's identify the pieces we need. We need 100. We need 50. That's this 10 and this 40. And we need 3. That's 1 and 2. But we don't need the 4 or the 20. So we'll ignore them. And we'll add these together. And we see we get the same product regardless of which multiplication we actually performed.