Adding together the area of each square will give the area of the rectangle: 1056 square centimetres.
Using the factors of 1056, this gives the possible dimensions of: 1 x 1056, 2 x 528, 3 x 352, 4 x 264, 6 x 176, 8 x 132 ... 48 x 22, 44 x 24, 33 x 32.
I thought the most likely of these was 33 x 32 and listed each combination of getting 33cm or 32cm from the lengths given. It was clear that the 1cm square had to be on the interior of the rectangle as nothing else could fit in the space which would be left if it was on the exterior. I therefore crossed out all of the combinations involving 1cm. For 33cm this left 18 +15, 18 + 8 + 7, 14 + 10 + 9 and 14 + 4 + 7 + 8. For 32cm I had 18 + 14, 15 + 10 + 7 and 15 + 9 + 8.
Any numbers appearing in both lists were likely to be on the corners of the rectangle, so I cut out the squares and tried 18 + 15 and 18 + 14 for two of the sides. It was clear that the 1cm square would be needed to fill in the gap between the 8cm square and 9cm square so I tried this in the other corner and the other pieces filled the remaining gaps.