Devil and the Hero on the Chessboard

Given that the devil will always win in a 3x3 board (which can be inferred very easily), let us consider a 4x4 board. Whatever be the first move of the hero, the devil can take its first move in such a way that the devil and the hero are now in a 3x3 grid and hence the devil will win.

Similarly it can be seen with any of the NxN boards where the problem would reduce to N-1 x N-1 boards and eventually to a 3x3 board and hence the devil would always win. This can be proved inductively.

The human is 14 (straight-adjacent) tiles away from the devil.

If the human is on the wall, he can at best increase the distance by 1 in a round, but the devil will always be able to decrease the distance by 2 (by moving diagonally). This means if the human stays on the wall, the devil wins. If he moves away from the wall, he decreases the distance, and cannot re-increase this distance as the devil will at least maintain the same speed as the human.

If the human is away from the wall, he can either: travel directly away from the devil to maintain his distance, or the devil will be able to decrease the distance faster than the human increases it. every time he moves directly away, he moves closer to a wall, meaning he will become trapped as stated before.

The dragon needs to move to the right until the hero is directly below the dragon. This is guaranteed to happen either after the dragon's or the hero's move. Then the dragon should move down each move (or down diagonally to match the hero's horizontal moves) until it reaches the hero.

Win by smothering.

On any finite chessboard, with any starting positions of the hero and devil:

(1) If the devil ever shares a column or row with the hero, the devil can force a win.

(2) The devil can always move to where it shares a column or row with the hero.

By (1) and (2): the devil can force a win.

Proof of (1). Let us just take the row case. (The column case holds by symmetry.) The devil and hero share a row and it is the hero’s turn. Define "rank" as the shortest distance measured in squares from the devil's column to the hero's column. By the rules for movement, barring edge constraints, the hero can change rank by -1, 0, or 1. At the same time the hero can also change to an adjacent row (either by moving diagonally, in connection with changing the rank; or by moving sideways [in relation to the hero/devil axis] when not changing the rank). After any of these possible moves the Devil can always reduce the rank by 1 and restore (if necessary) row alignment with the hero. So row alignment is restored with either the same or lower rank distance. The hero cannot however continue increasing rank indefinitely (backing away) becase s/he will reach a board edge. From this point on, after every move by the hero, the devil can play such that rank will decrement by one and the devil will win.

Proof of (2) Just taking the row case, define "alignment offset" as the distance between the devil's row and the hero's row. By the rules for movement, when the devil moves it can always reduce the alignment offset by 1 (with or without an accompanying diagonal component in the move, for the sake of optimization for example). The hero can change the alignment offset by -1,0,1 but is similarly bounded by an edge, as described in proof of (1). When the hero no longer can evade sideways, the devil will be able to reduce the alignment offset to zero, i.e. enter the same row as the hero.

The devil can reach the same or adjacent row of the hero. Then it moves always to the same row while closing the columns.

Lets now start with the time when the devil and the hero are in a straight line and its hero's turn to move.Also without loss of generalization the hero is above the devil in the same column(otherwise you can rotate the board and proceed with the following arguments).

1. Now at maximum the number of moves the hero can make is 8 . Call them N, NE, E, SE, S, SW, W, NW (based on the directions).
2. If the hero moves E or W, the devil moves NE or NW resp.
3. If the hero moves SE S or SW then devil moves NE, N or NE resp.
4. If the hero moves NE, N or NW, then the devil copies the moves of the hero.

Now we can see that the devil will always be in the same column as the hero and also the vertical distance them must be same or reduce.The distance will be same if and only is the hero moves as case 4. But then he can make only finite many moves as case 4 and hence the devil will meet the hero.

We now need to prove that the devil can always reach the same column as the hero. This is easy to prove as the devil can keep on going right till he is in the same column as the hero. If the devil by chance is in the same column as the hero and it's his turn, then he can move in the same column to reduce the problem to above case

The devil will always win. The devil just needs to move toward the hero, and at some point the devil will be able to move directly in front of the hero. Then, the hero will be forced to move away from the devil. Regardless of the move the hero takes from the three possible moves to get away from the devil, the devil will be able to take the previous position again, forcing the hero to move back again. Eventually, the hero will reach an edge of the board and the hero is trapped!

If devil wins in a 3x3 then it always win in square

In the first move, the hero moves to a black square. Then the devil moves to a black square. For every size NxN board, the devil will win. The only way the hero can win is if he starts on a different color square than the devil. This happens on an NxM board where either N or M is even and the other is odd.

The absolute distance (by Pythagoras if need be) between Devil and Hero, can often decrease but NEVER has to increase. Therefore the Devil will steadily approach the Hero.

The simple answer for this is the hero only win if he avoids the devil forever which means the game will continue forever and sometime the devil must be able to catch him