Some interesting retro problems can be obtained simply by asking you to reconstruct a partially broken position, or to complete an incomplete position. Of course your aim is to obtain a legal position, possibly satisfying some additional stipulation.
A very nice explicit example of a broken board is the following:
B. Lurje and N. Plaksin
Special Prize Schachmaty w SSSR 1985
13+14. Arrange the two parts into a legal "Mate in 1" problem
Here you have eight possible arrangements into an 8x8 board with white square h1. Only one is a legal mate in 1.
Other stipulations falling into the reconstruction category are