Now, we are looking at an old puzzle of more than a century. Old, it's called 15-puzzle. And let me explain the rules. So you have a small rectangular board, and there are 15 small pieces, small squares. And what you can do, you can move them. There is one empty space here. And if you have an empty space, you can move a piece to this space. So here are the game rules. So you can move pieces and the goal is to reach some configuration. So for example, you start with the initial configuration as you have seen. And you want to get the same configuration with one change, 14 and 15 are interchanged. So the question is, of course, whether it's possible or not? And if you want to do this, maybe it's more convenient to start with the configuration you want and then to get back the initial configuration. Of course, it's just like you can go back and forth if you can get a new configuration from the first one. Then you can go back and then return to the standard position from a new one, so it's equivalent question. And try this, if you succeed, if you can get exchange to pieces 14 and 15, we will pay you some prize, not much, but reasonable. So would you try? Let me tell you a story of this game. It's not very clear actually. It's clear that it's invented in 19th century and there is some complicated history, I looked in Wikipedia and you can look yourselves, but it's not very clear who invented it the first time. But what is clear, that it was popular at some time. And indeed like in our case, there were cash prizes announced in the newspaper, so a lot of people were trying to do this. And so the question is how the organizer, why decided that they could do this, why they take the risk? Because, of course, if many people will ask for a prize, they will not have enough money to pay them. So why they take this problem not seriously? Because they have a proof. At that moment before they started this publication, there was a proof that such a thing is not possible. And so okay, maybe you believe that our proof is incorrect, so you still can try to get your prize. But at least it's not that easy. And for now, our challenge is to reinvent this proof. So we want to understand what is the proof of possibility. And in this way, we will develop some small part of group theory about permutation group specificity but forget about this mathematical words, we just want to solve the puzzle.