1. Try to design a method to divide a square into 8 smaller squares without repetition or omission (the sizes of the smaller squares can be different); also, how to divide a square into 31 smaller squares under the same requirements?
Three, 1. It is easy to divide a square into $4^{2}=16$ smaller squares, then combine 9 of them located at one corner into one square, resulting in a total of $16-9+1=8$ squares.

Divide into 16 smaller squares, then further divide any 5 of them into 4 smaller squares each, resulting in a total of $16-5+5 \times 4=31$ squares.