Hey this is Presh Talwalkar, here is an email I received. I really like your YouTube videos and I wanted to send you one of my favorite ### riddle

s. It's a bit difficult in my opinion, but the solution is very cool. Alice and Bob are playing a game with a ## chessboard

This is an 8x8 grid Alice starts by placing a ### knight

on the board, then they take turns moving the ### knight

to a new square it wasn't on before standard chest rules are valid. The ### knight

can only move in an L-shape, which is two squares one direction and one square to the side.

This is one possible place the ### knight

could go, and these are all other places the ### knight

could go from the original square could go. Every time the ### knight

moves it has to move in this l-shape move the ### knight

to a new square loses the game who wins if both players play optimally and what is the winning strategy so I admit that this ### problem

left me perplexed and I couldn't figure it out. I want to thank Sebastian for suggesting this issue and To send me the solution you can find out try this issue and when you are ready keep watching the video for a solution Find.

Before I get to the solution, I want to mention that this ### problem

is an example of game theory and a #### math

ematical field Where you #### solve

games of strategic interaction Alice's best move depends on what she thinks Bob will do, and Bob's best move will depend on what alice has done and could do situations and actually find solutions to some games in this game we can prove that bob can always win no matter how alice plays how can we do that? We're going to use an interesting concept called the graph coloring proof. Start with our ## chessboard

and break it down into eight distinct four-by-two regions, that is, one, two, three, four, five, six, and seven eight.

So what's so important about these, let's analyze one of them individually. Suppose the ### knight

is on one of these squares where the ### knight

could go while staying in the same 4x2 region. This is the key to the whole ### problem

. From a square in a given 4 x 2 region, a ### knight

has only one legal move to stay in the same 4 by 2 region, so there is only one possible place from that square that the ### knight

could go while it is remains in this 4x2 region, so we will code these two squares with the same color, so from each of these squares ### knight

can only move to the other position this applies to every single square in this 4 by 2 region, for example from the bottom right square there is only one possible spot where the ### knight

could go from there and vice versa so we're going to color these two the same color we can do for the remaining squares in this four by two region we mark two squares of the same color if the ### knight

can now move between them what is this coloring good at, it's applied to each and every one of these four by two regions so know of a g In a four by two region n we know that there is only one legal move to stay in the same four-by-two region.

How does this help Bob win the game? Alice starts the game by placing the ### knight

in a four by two region Here Bob will move the ### knight

to the only other legal square in the same 4x2 region. The key is that Bob took the only other legal square in this 4x2 region. This forces Alice to move the ### knight

to a new 4x2 region on her next move, she can't stay in the same 4x2 region because those two squares have already been visited, so let's say Alice now moves here where Bob is fine should move, he will then move the ### knight

to the only other legal square in the four by two region he applies the same color pattern to that region.

In this case, the ### knight

is on a yellow square, so Bob will move to the other yellow square, as long as Bob continues this strategy, he always has a move and he forces Alice to find a new 4x2 region every single move Alice has to keep finding new squares in different 4x2 regions and eventually she won't find any the game has to end in 65 moves or less because there are 64 squares in total so bob can always win this game and like magic we #### solve

d this ### problem

and showed it that Bob can always win. did you figure it out Thank you for watching this video.

Please subscribe to my channel and make videos about #### math

. You can see me on my blog You can check out my books listed in the video description and you can support me on Patreon if you have a #### math

topic suggestion you can email me at presh mindyourdecisions.com and you can meet me on social media at either mindyourdecisions or preshtalwalkar

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