Our next trick is that we can transform the known solution into solutions for new problems. And this will be illustrated by multiplicative magic squares. I will explain what it is. Just instead of sums we have products. So in magic square we have sums for rows, columns, and diagonals the same. And now we want products to be the same. And, of course, now we cannot hope to use all the sequential numbers, because can you see why it's impossible? It's easy. So for example, look at seven. Seven is somewhere in the square. I don't know where. And then it appears in some products. But in some other products, it doesn't appear. So we have no other numbers can give seven in combination. So these two products will be different and no hope. We forget about this requirement, we just require to have different integers. But they can be arbitrary. So, now the question, is there a multiplicative magic square? What do you think? This looks rather strange, it's just a formula: 2 in the power of x + y is 2 to the power of x times 2 to the power of y. And mathematicians will say that exponentiation transforms addition into multiplication. "It's a homomorphism of an additive group into a multiplicative group", but anyway we have this simple formula. Can you see what are the connections, how it can be used to transform the editive square into a multiplicative square? So this is the general slogan. And here is the trick. So instead of... each number is replaced by two power of this number. So if we multiply for example this three, then according to this rule, we need to just add the exponents. And the sum of the exponent is here and so all the products are here. The sum of 15 and so here the product is 2 to the 15. And we get a multiplicative magic square. Good. But the numbers are rather big, the maximal one is 512. And can you find smaller one? There is very simple observation. If you want numbers less than 300, what can you do? Actually, you see that the minimal one is 2. So we don't have zero here, so we have all this numbers start here from 2, 4, 8 and so on. And so we can just divide all the numbers by two and instead of 2 to the 9th, we have 2 to the 8th, which is only 256. So this is the trick. But it's not the best thing we can do, let me show you how we can do better. So, we can even get them less than 40 and this is again a use of exponentiation but in a more clever way. So let's start with a very strange magic square. It's magic in terms of the sum of all these three, but of course, the same numbers appear many times, so it's not magic. But still we can transform it into a multiplicative magic square but again, not magic but the same products. So why does it help? Let's do the same thing for a bit different square, we're just rotate it. And here we have the base 2 and here we have base 3. So we get 2 bad multiplicative magic squares, but the numbers are not different. So what do we do? We can multiply them. So actually, we just have 2 times 1 is 2, or 4 times 9 is 36. So in each cell we place the product of the numbers in the corresponding cells. So why it's good? Because if we now take the product for example here, actually, this is product of things here and here. And so here we have always 27, here we also have 8. So in every line we have 27 times 8. And so it's just a number, so we have the same number in all the lines, that's what we wanted. And additionally, you see that all the numbers are different. So it's multiplicative magic square and all its numbers are less than 40. That's what we promised and we achieved this in this funny way.