the angel and the devil if you had an infinite chess board in your cupboard who would you want to play like many difficult things the rules are super simple one of the players is the angel and at each turn the angel flies to some square on the board before the game begins we would agree on how far the angel is able to travel and we call that number of squares its power for instance the angel of power one can travel just one square in any direction essentially the king in chess an angel of power two could travel anywhere within this region NUMBER_TOKEN of power NUMBER_TOKEN to any of these squares and so on in comes the second character the devil whenever the devil lands on a square it burns that square making it inaccessible to the angel for the remainder of the game much like the angel on every turn the devil can move to another square but unlike the angel is able to travel as far as it wishes and those are basically the rules the devil then wins the game if it eventually traps the angel and the devil has a waiting strategy if it is always able to win a useful way to think about this is the devil having a set of instructions such that regardless of what the angel does the devil can look up the current layout in its large instructions book and deliver the perfect reply which will keep the angel contained in some large predetermined region at all times the angel has a winning strategy if the devil doesn't meaning it can always evade the devil so which character here has the upper hand well when john conway introduced this problem and when he then wrote an article about it some years later he published a collection of counter-strategies to ones that might seem appealing initially i will briefly talk about the first and simplest one idea you might have is i'll have my angel fly some amount of north at every move conway calls this angel the fool and that's because the fool loses to the devil let's take a fool of power one here's what the devil will do it'll picture all the squares the fool might land on in the forthcoming terms and it plans to build a wall on the rope just after the devil also takes note of the half distance between the fool and the future wall every time the four moves the devil burns one out of four squares above when the fool reaches the halfway point the devil restricts its interest to the new smaller region now accessible to the fool and concentrates its efforts on a shorter barrier with every move the devil burns squares that half are now taken and we repeat twice more blocking the fool of power one conway's first example is to show that this can be scaled to be fools of any power using the exact same tactic of progressively filling in a wall of suitable width far away and every time the fool has its distance from there the devil restricts its efforts to a wall half that size
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