Detailed Solution:

The hint is necessary, but comes into play later in the solution. It is very difficult to try and use the hint at the beginning. Instead, let’s examine the bottom horizontal tier of blocks. In order to get eight shaded squares in the bottom row, we’ll have to use a combination of three shaded in one block, three shaded in another, and two in the last block (since we have no stamp with four shaded squares). The only stamp to have three shaded squares on an edge is Stamp 4. Two copies of Stamp 4 leave one of two possible orientations of Stamp 2 for the remaining block. So now the question is, which column of blocks should not have a copy of Stamp 4? In the necessary orientation, the middle column cannot accommodate Stamp 4 (it would leave one square to be shaded in the rightmost column, which would require another copy of Stamp 4, but that leaves an unacceptable number of squares left to be shaded).

Now we need to determine the orientation of Stamp 2 in the remaining empty block in the bottom tier. In one position, there would still remain six squares to be shaded in one column. This would require two sets of three shaded squares each, which means using Stamp 2. Unfortunately, Stamp 2 has no shaded squares on one edge, thus not producing the correct combination of shaded squares. In the other orientation, two copies of Stamp 4 can cover the amount of squares left to be shaded. Note that the left column leaves one square to be shaded (as in step 1), which means using Stamp 4.

Now we need to determine which copy of Stamp 4 goes into the top two blocks of the middle vertical tier of blocks. We have already determined the orientations of both copies from the last step; we just need to figure out which one goes where. Actually, the wrong stamp can be eliminated from both blocks. Completing both deductions is redundant, but both are presented here.

Subtracting in the top horizontal tier of blocks, we find that only two copies of Stamp 2 will cover the remaining squares that need to be shaded. Using the hint, we can determine which orientation of Stamp 2 should be placed in which block. Note that the hint is necessary. Without it, we could not distinguish between the two possible placement combinations, since working vertically both orientations add up to the same number of shaded squares.

Working vertically in the left and right vertical tiers of blocks, we can determine the orientations of the needed copies of Stamp 2 in each remaining block.