To start with, for the first quilt, we worked out the area of the rectangle which was each of the squares' side lengths, squared and added together. This came to 1056. We then worked out the factors of 1056 to find the dimensions of the rectangle and the only possible lengths were 22cm and 48cm, 24cm and 44cm and 32cm and 33cm. This was because each length had to be over 18cm. We then eliminated 22 and 48 as it wouldn't work, this was because if you put the smaller squares on the outside lengths then you wouldn't have the right sized squares to fill in all the gaps. For example, if you put the 4 next to the 18 to make 22 then you wouldn't have any more small squares to fill the space. We also found this to be true for 24 and 44. Therefore, we knew the dimensions had to be 33 and 32 and also that the larger squares had to be on the outside. We then made the squares and used this knowledge to make half of the rectangle and then filled in the gaps with the remaining shapes.
For the second quilt, we did the same process and found out the area of the rectangle was 3055. We then found out the only possible dimensions were 47 and 65 as they were the only factors both over 25. We also knew that the larger squares had to be on the outside otherwise there would be gaps. We then, once again, made the squares and formed half of the rectangle and then used the remaining squares to complete it.
We have attached pictures to show our solutions.