A $2\times 10$ rectangle needs to be tiled with tiles of size $1\times 2$ as shown in figure. The number of ways it can be done is $K$, then $K-82$ is euqal to 

Let a_n denote the number of ways in which

a 2 × n rectangle be filled with tiles of size

1 × 2. Clearly, a₁=1 and a₂ = 2. For $\mathrm{n}\ge 3$

we will build the titles one tile at a time. If the first tile is vertical, then we have 2 × (n-1) rectangle that can be tiles in a_n-1 ways.

If the first tile is horizontal, then the next tile is also horizontal and we are left with $2\times (\mathrm{n}-2)$rectangle and this can be tiled in ${\mathrm{a}}_{\mathrm{n}-2}$ways. With the initial conditions${\mathrm{a}}_1=1\text{ and }{\mathrm{a}}_2=2$ this yields the sequence 1, 2, 3, 5, 8, 13, 21, 34, 55, 89

$\therefore {\mathrm{a}}_{10}=89$