So, here's an outline of, of how to show that 1 over 2 to the n, minus up is positive. Let's just look at the left going first move. If left moves first, I'd claim that the best move for left is to play to this star. What's left is then 1 over 2 to the n plus star. Now, now it's right's turn, what can right do? Right can, can move from star to zero. Remember that the two, the only option in star is for left to move to zero or for right to move to zero. In which case, what's left is 1 over 2 to the n, left just chops the blue edge, and now there's nothing left, and so left wins. The other possibility is for right to cut off one of these red edges, and that just delays things. So, right cuts off one of those edge, edges the left then moves from star to 0, and what's left is right cuts off another red edge, blue cuts off the blue edge, and now left wins. So, so that's the argument when left goes first. When left goes second, there's a little bit more stuff to do, but the argument is very similar. And again, if left goes second, left wins. Put this all together and that says that 1 over 2 to the n minus star is positive. Which says, with what we did previously, that up is bigger than zero and less than 1 over 2 to the n for any n. And there's no number that does that. So, up is not a number. Okay. That ends week 3. I'll post a little quiz and I'll, I'll talk through it for the last part of week 3. Okay. So, the quiz for week 3 is, I have a whole bunch of games. You have to decide in each game whether it's positive, negative, equal to zero or fuzzy with zero. It can't be two of these, it has to be exactly one of these. The first game is these hack and bush with three stems like this. And second one, hack and bush with, with one green stem and one blue stem. Game number three is 2 copies of one green stem and one red stem. And number four is a little more abstract, go back and look at the definitions of up and star and figure out whether up plus star is positive, negative, zero or fuzzy with zero. Work this out and we'll post the solutions by the time you see this. Take care.