Fariha’s Game
Fariha invented a game!
In that game, Fariha kept 5 plates on a circular dining table!
The plates were marked as 1,2,3,4,5.
Fariha had an apple.
On the first move, she put that apple to the clockwise neighboring plate no 2 from plate no 1.
On the second move, she moved the apple from no 2 table to no 4 table.
So on her kth move she actually moved the apple for k positions!
And after some moves she reached the plate no 1 again.
Now her friend Nabila put some more plates on the table.
Now could Fariha find out if it was possible for her to reach plate no 1 again after any number of moves!!!
the apple is moving in this way : (1+2+3+4+….)
so if the sum of (1+2+3+4+5+…) is divisible by the number of plates then the apple will reach the plate no 1
1 then 2 then 3 then 4 then 5 etc (how moves go)
so position of apple will be like,1,1+2,1+2+3,1+2+3+4,as every move goes (position from first plate)
so if we divide total by number of plates remainder will tell the position of apple at end .since sum can be formalised into n(n+1)/2 where n is last value of series .so the number of plates must be factor of n(n+1)/2 if we want it to end in first place.
so if number of plates is k then if i choose n=k square.(k^2) so it will be.k^2(k^2 +1)/2 should be divisible by k and it is divisible by k so it will always happen no matter how many plates it is always possible
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