A particle is allowed to move in the $\mathbb{Z}\times \mathbb{Z}$ grid by choosing any of the two jumps:
1) Move two units to right and one unit up
2) Move two units up and one unit to right.
Let $P=(30,63)$ and $Q=(100,100)$, if the particle starts at origin then?
a) $P$ is reachable but not $Q$.
b) $Q$ is reachable but not $P$.
c) Both $P$ and $Q$ are reachable.
d) Neither $P$ nor $Q$ is reachable.
I could make out that the moves given are a subset of that of a knight's in chess.
I think that it'd never be able to reach $(100,100)$ but I'm not sure of the reason. It has got to do something with the move of the knight but I cannot figure out what.
I don't have a very good idea about chess, so I'd be glad if someone could answer elaborately.
Best Answer
Both possible moves increase the sum of coordinates by $3$, so any reachable position must have the sum of coordinates a multiple of $3$. This means $Q$ is not reachable, since its sum of coordinates is $200$, not divisible by $3$.
$P$ is also not reachable – to reach an $x$-coordinate of $30$ it must make at most $30$ jumps, which means that the highest $y$-coordinate it could reach is $2×30=60<63$.