Math in chessboard
There is a chessboard. Put the numbers 1 to arbitrarily in those squares. Then there must exist 2 adjacent squares (sharing a common vertex or side) the difference of the numbers put in those 2 squares is at least . Find .
The answer is 17.
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1 solution
Consider the motion of an ant on a chess board which can move a unit square to the right or to the front. Say it is at (1,1) & it needs to go to (x,y)(assuming x>y).The minimum no. of moves in which it can do this is first to go to (1,1) to (y,y) and then to (x,y).It will take (y+x-y-1) ways or (y-1) moves.Now difference between two squares at most = n^2-1.There exists an algorithm to reach any point in <=(n-1) moves.So by PHP it's easy to conlude ther exists atleast 2 adjacent squares with difference >=(n^2-1)/(n-1)=(n+1).(Equality holds if all the no.s in a row are arranged in an ap of common difference n).So ANSWER = n+1=17.