Okay. Welcome back. Still week six, wow. One more week to go. So, here we're in impartial games, reversible moves and we'll have a few examples and MEX. But I think that it's going to be two pieces. So, let's take a look at this. So, let's talk about impartial games. Impartial game is where like Nim, both left and right have the same possible moves. Not like five which gives five moves to the left and no moves to right. Or minus five, which gives five moves to right but none to left. Five and minus five are partisan games. They're games where it depends on whether you're left or right. Whereas, Nim is impartial. In general, a game is called impartial if every left move is also a right move, and every right move is a left move. And every move in every position is also an impartial game. So, let's look at an example. Okay. Now, when we write down the impartial games, we don't have to say what the left moves are and the right moves are. All we have to do is say what the moves are. This is because every left move is a right move and every right move is a left move. So, let's look at the game where the moves are to Nim heap for size zero, size one, size two, size five or size seven. Okay. So, my claim is, is the same as this. So, so think of this as a Nim heap for size three, that you can add two coins to or you can add let's see three, three, four coins to. So, so start with start with a Nim heap for size three, and either you remove one or more coins or you add two coins or you add four coins. That's this game G. Now if you add two coins, I'll just take them away and put them in my pocket. And if you add four coins, I'll just take the, take, take the four coins and put them in my pocket. So the, the move to star five, to five coins, which you get by adding two coins and the move to star seven, which you get by adding four coins, is irreversible. The other player just takes them away and pockets them. And if they're like something really exotic like a, like a rare year of the Morgan Dollars or something, it might be worth something, it might be worth something. In, say, grade 65 or so, that would be worth a lot of money. So , so, if you add coins, the other player can just take them away and you're back to where you started. What this says is, this game here is the same as this game here. But this game here is the same as just the Nim heap of size 3. Which you can then move from Nim heap of size 3 or through a Nim heap of size 2, 1, or 0. And we note here that 3 is the, what's recalled the minimal excluded number out of 1, 2 and 3. That is, you look at 1, 2 and 3 and look at the first natural number that's missing. And in general, what happens, so let's take a look at some examples, of, of Mex. And we'll leave that for you to compute for next time. And then we'll have some, go back to examples of games. So the minimal excluded of 1, 3, 2, 1, 17. Let's see. One is there, two is there. I'm sorry. 0 is there, 1 is there, 2 is there, 3 is there, 4 isn't. So it's 4. The minimal excluded of this is, well, 0 is not there. So, it's 0. And the minimal excluded of this is 5, if I did this correctly. And so here's the minimal excluded for you to get. Now, it occurs to me that I didn't do the problem from last time. And so let me just leave up some numbers here. If we take a Nim heap of size 11, 3, and 10. 11, 13, and 10. Write them out in binary. There's one here, go up to the one here. Circle that. 101, take 101 Nim sum 100. We get what do we get? We get 001. So, the winning move is here. Just keep the 11 the same. Take the 13. And change it to a 1 and take the 10 and this is the same. So that's, that's the solution, I believe to the to the problem at the end of the last module. And there we have it, end of another module.