In this part we have several different unrelated puzzles, each of them rather simple but they show different sides of this search process. Our first puzzle is to find a six digit number which starts by 100 and is divisible by 9,127. So this is our goal. We should prove that there exists such a number. And actually, it's not so many candidates. So there are three, here we should have something starting from 000 up to 999. So if you are lazy, you can just write a program, and then the program will print you the list of all candidates. So it's programmer's solution. But if you are a mathematician or if you are lazy in a different way, you do not want to program, then you can think of it. Okay, let's try to find a pencil and paper solution. Let's divide just the minimal possible candidate by this 9,000 something. And then we get something, which is close, smaller than 11, but close to 11. And so the most natural thing to try is to try 11. And then indeed, you get a number, which is of the required for. And by definition it's divisible by this number, and the quotient is 11. And you can see that the next one is too big. So if you have something divisible by 9,127 the next one be 9,127 apart. So it will be much bigger, this digit will be a 9. So actually this problem has only one solution. If you tried this programming, if you run this code, you will see the solution, hopefully. So this is easy. Next problem is to find a three digit number that produces a remainder 1 when divided by 2, 3, 4, 5, 6, 7. In all cases, that should produce remainder 1. And why we ask for a three digit number? Of course, if we're just, otherwise, we can just use 1. It has only one digit but otherwise it's okay, because if 1 is divided by anything starting from 2, we get remainder 1. So it doesn't work. But how can we find this N? Again, we can try a computer, but let's think of it more. So N gives a remainder 1 when divided by something. So what about N-1? What will have remainder? Of course, it will be 0, because if we delete this outstanding 1, then the rest should be divisible by 2, 3, 4, 5 and 6. And what is divisible by 2, 3, 4, 5, and 6? It's common multiple, it's called. And you see that this number works, 2, 3, 4., 5, and 6. So you can find here all the numbers, and actually, this is the minimal possible. So in number theory you will see that this is the minimum possible number. And all others are multiple of this. And this number is 400. So now recall that it's N-1 and so for N we have 421. And you can check. Then if you divided by all this, 2, 3, 4, 5, 6, 7, you get remainder 1. What about others, other three digit numbers. Yeah, so we have already said that you can take any multiple of this number. So the next one will be 840. So if N- 1 is 840, then you get the next one here in this computation. And the next one is too big. So there are actually two candidates for this solution. But we need only one, if we want to prove that there exists something, it's one example is enough. Good, now next, our next puzzle. We want to find a perfect square, which means that the square of a natural number that starts with this strange digit. It's actually first digits of pi, but doesn't matter. We can take any sequence here. So we want to find an integer, which such that n squared starts with a given sequence. We're going to prove that such a number exists. How do we prove it? And then is a very important observation, it's enough to find a decimal fraction. So if we have some decimal fraction, and there is a decimal point in something here. And then if you multiply this fraction by 10 this means that we move the decimal point one position to the right. And in the n square, if we multiply x by 10, then the square is multiplied by 100. So in the square we have moved two positions to the right. But, anyway, it's just the position changed and the digits are the same. So if we move the decimal point we do not change anything in the sequence of digits. Now, everything is very easy, we can just calculate the square root using a pocket calculator. It's useful, but still if you don't have it you can do it by hand. So you get this. And then you can try to move this point somewhere and see what happens. So let's see. Just we take all these things and we see that we get almost what we wanted. But we have something very close to this but just a bit smaller. So we need to take something bigger. Let's try something, probably we don't need so many digits. So here is what we can do. So 177.243 square is it starts in the right way. But this is not an integer but we know what to do. We just move this thing three positions and this thing moves six positions and we are done. And here is the answer. So we have an integer, n and n squared starts with what we wanted. Good, here is just the next one. It's also a key. But there is more. Imagine we want to find two perfect squares starting with this number which are different. Prove that there is another number n, and n has a different first digit and still n square also starts with 31415. So we need to prove another thing, another n. What can we do? Actually, we can do the same. Take the square root but in a different way. So can you see what to do? Yeah, here it is. So let's put decimal point in a different place. So let's put the decimal point here. Then the square root is different. Actually, we divide, more or less, by square root of 10. But the first digits is now completely different. But we can use the same trick. We can try something a bit bigger than this. So it will be natural to try 5605, but we see that it's too big because here we have 6 instead of 5. So we need more digits, actually. We can try this, replace 2 by 1. And then, with this, we get what we wanted. So here is the desired sequence. And now we have two perfect squares, two n which start with different digits and still their squares are okay, and this is from the previous slide and this is what we get now. Problem solved.