 So let's take a look at problem solving in ancient Egypt. One of the oldest problems in mathematics from the Rhine of the Pyrus is the following. A number and its seventh make nineteen find the number. To solve this problem, I'll close for down three multiplication tables. We'll put those down here. And again, note that the material that we placed in the brackets is not part of the actual text, but it's material that makes the text a little bit more comprehensible. And so the important question is how should we interpret this? Well, let's take a look at this first multiplication table. So we know the problem is that a number and its seventh make nineteen. So notice here that Achmos has taken a number, found its seventh, and presumably added them together. And so we might read this first table as we guessed seven, we took its seventh, and together we found eight. In other words, we guessed a solution. And if we're fantastically lucky, it would actually be the solution, but it's not. We might look at the result as follows. We got a total of eight, but we wanted to get nineteen. So how much do we need to scale our guess by? Well, it's whatever we need to multiply eight by to get nineteen. And that's where the second multiplication table comes in. This is what do we need to multiply eight by to get nineteen? And so we find that we need to multiply our guess by to a quarter and an eighth. And so that means we need to find seven times to a quarter and an eighth. And so that's our last multiplication table, to a quarter and an eighth, multiplied by seven. Which gives us our answer, sixteen a half and an eighth. Now Achmos does many examples of this type of problem, and in the text this approach is usually accompanied with the word aha, which means, no, it's not the name of a Norwegian band. The word in ancient Egyptian actually refers to heap. And in modern terms what Achmos is doing is using a form of proportional reasoning called single false position. And that works in modern terms as follows. We guess a solution, see if it works, and if it doesn't, we'll scale it by an appropriate factor. So for example, a quantity and its half makes sixteen, let's find the quantity. So our method of single false position starts off by guessing a solution. And so we guess a solution, and the only restriction on this guess is that it should be something we can take half of. So let's guess, oh I don't know, how about eight? So we're supposed to take the quantity and its half, so let's take the quantity, that's eight, half of the quantity, that's four, and so together the quantity and its half make twelve. Now if we were fantastically lucky, or an extremely good guesser, we could have guessed the actual solution, but we didn't. Our guess of eight only gave us twelve, but we wanted sixteen. So we need to scale our guess. And the question you've got to ask yourself is, what times twelve makes sixteen? So we'll form that multiplication table for twelve. So one gets us twelve. We want to make sixteen, so we'll take the aliquote part one-third. Now we have the pieces we need, and so that tells us one and a third times twelve makes sixteen. And this gives us our scaling factor, one and a third, and so now we multiply our guess by this scaling factor. Now we could either take eight, one and a thirds, or one and a third eight, but it'll generally be easier to multiply our fractional amounts. So we'll start with one, one and a third, then double it a few times, and since we actually wanted eight, one and a thirds, our product will just be this last row. And that gives us our solution, ten, and two-thirds. Or how about a number, its half, and its third make thirteen find the number. So we guess, whatever we guess, it's got to be something we're willing to take half and a third of. So let's guess six. So we have our number, we find half and a third, and so the number is half and its third make thirteen. Oh, eleven, not thirteen. So again, if we're fantastically lucky, we might have actually guessed the solution, but we probably won't. And that's okay because we can scale our guess. It's helpful to think about this in the following way. Our guess got us eleven, but we wanted to make thirteen. And so the next question we have to ask ourselves is what times eleven makes thirteen? In other words, what do we have to multiply our result by to get what we want? So let's answer that question. We'll form a table of multiples of eleven. The only aliquot part is the eleventh. We can double the eleventh by looking up our two-over-end table. That turns out to be one-sixth, one-sixty-sixth. And to make thirteen, we'll need these two pieces. And so the way to read this is if we multiply our result eleven by one-a-sixth and a sixty-sixth, we'll get thirteen, which is what we want to make. And so that means we want to multiply our guess by this amount. So we'll form a table of multiples of one-one-sixth, one-sixty-sixth. We'll double. We'll double again. And we used our table again to find the double of one-thirty-third. Since we want six times, we'll take the two and the four, and that gives us our answer. And so our solution is seven, a twenty-second, a thirty-third, and a sixty-sixth.