Answer: 6

Setting up.

Let the table top be $10'$ along $x$ and $6'$ along $y$. A blue rectangle of $2.5' \times 8'$ that shares two sides with two edges of the table must sit in a corner, with its $8'$ side along the long edge and its $2.5'$ side along the short edge.

There are $4$ corners, so each blue rectangle occupies one of $4$ corner positions:

• Bottom-left (BL), Bottom-right (BR), Top-left (TL), Top-right (TR)

Counting placements.

We choose $2$ of the $4$ corners for the two blue rectangles:

$\binom{4}{2} = 6$

Since the table is fixed to the ground, rotations and reflections give visually distinct patterns. The $6$ pairs are:

1. BL + BR (blue band along bottom edge)
2. TL + TR (blue band along top edge)
3. BL + TL (two stripes on the left)
4. BR + TR (two stripes on the right)
5. BL + TR (one diagonal)
6. BR + TL (other diagonal)

Each gives a different observable pattern from above.