$2015$ $cm^2$ and perimeter $192$ $cm$ . A coin of diameter $2$ $cm$ is randomly tossed onto a floor laid with such rectangular tiles which tessellated in the following arrangement as shown (checkerboard arrangement).

A rectangular tile has area
Let
$P$
be the probability of the coin landing on
**
exactly
**
2 tiles (i.e. such that when it lands it covers part of exactly two tiles, as shown). Find
$2015P$
.

*
This problem is part of the set
2015 Countdown Problems
.
*

The answer is 184.

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Let $l$ and $h$ be the longer side and the shorter side of the tile respectively, and d be the diameter of the coin. $h=31$ and $l=65$ .

Notice that the blue areas correspond to where the centre of the coin needs to be within for the coin to land on exactly 2 tiles.

Hence

$P=\frac{\mbox{(total blue area)}}{\mbox{(total tile area)}}=\frac{(2(l-d)(d/2)+2(h-d)(d/2))}{lh}$

$=\frac{d(l+h-2d)}{lh}=\frac{(2(65+31-2×2))}{2015}=\frac{184}{2015}$ .