Depends on patience I suppose.
With n pieces, there are n! arrangements they can be in. If they are identical polygons, they can cover the field of they are all rectangles (two ways to place them, long side up or down) triangles (three angles), squares (four), or hexagons (six), unless one gets really exotic. I'll assume that the whale puzzle of yours consists of a 3 x 3 grid of squares, so there are 9! x 4^9 = 95,126,814,720 different ways to put it together, four of which form the picture of a whale (facing up, down, left, right). If you are really handy do go through those positions at a rate of one per second, that's about three thousand years.
The 6,144 positions of a 2x2 grid would take less than two hours at the same pace.. still doable. So I'd say.. a 2x3 grid is the smallest that I would consider "virtually unsolvable". Working at that 24/7 at a rate of one arrangement per second would take roughly one month.