Frog is in the origin of decartes coordinate system. Every second frog jumpes horizontally or vertically in some of the $4$ adjacent points which coordinates are integers. Find number of different points in which frog can be found in $2015$ seconds.
Problem
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Tags: combinatorics
adamz
25.05.2015 22:58
The sum of the coordinates is an odd number less than or equal to 2015 as we can use 2 jumps to stay in the same position.
Therefore the answer is $2016^2$
neverlose
25.05.2015 23:11
mihajlon wrote: Frog is in the origin of decartes coordinate system. Every second frog jumpes horizontally or vertically in some of the $4$ adjacent points which coordinates are integers. Find number of different points in which frog can be found in $2015$ seconds. Let the frog is situated in coordinate $(x,y)$ then the frog can be found after $2015$ second in coordinate $(x_{0},y_{0})$ only if $|x - x_{0}| + |y - y_{0}| \leq 2015$ and $|x - x_{0}| + |y - y_{0}| \equiv 1$$(mod$ $2)$.So by counting all the ways and exclude the ways when $|x - x_{0}| + |y - y_{0}| \equiv 0$ $(mod$ $2)$ we get that the answer is $2016^2$