All Puzzles: Here
Puzzle: Mary baked a rectangular cake. Merlin secretly carved out a small rectangular piece, ate it and vanished! The remaining cake has to be split evenly between Mary’s two kids. How could this be done with only one cut through the cake?
Source: Heard a long time ago, in 1990s.
Solution: (by Yuri Robbers): There are two solutions to this problem. The first solution is perhaps a little bit silly, and it only works if the cake is vertically symmetrical. In that case it owuld be possible to halve the cake horizontally. The top half and the bottom half would then be identical.
A less silly solution is to cut along the line that runs through the center of the original cake, and through the center of the piece that is missing. If the centers happen to coincide, then any line through that joint center will do. Any rectangle is halved by any line through its center. Cutting through the center of the cake therefore halves the cake, while cutting through the center of the missing piece, halves the missing piece.
The second solution also works only if the cake is vertically symmetrical !
Let a the length of the cake, b the width of the cake, c the length of the rectangular missing piece, d the width of the rectangular missing piece, x the x coordinate of the bottom left of the missing piece! (assume (0,0) at the bottom left of the whole cake) and y the y coordinate of the bottom left of the missing piece. We are going to perform a single vertical cut at the horizontal coordinate k. In the general case, let E1 the part of the cake on the left, ignoring the missing piece and E2 the part of the cake on the right ignoring the missing piece. Also let, E3 the part of the missing piece on the left and E4 the part of the missing piece on the right. We want
E1-E3=E2-E4
which, according to the notation above implies
ab-(k-x)d=(a-k)b-(x+c-k)d
which, when solved wrt k gives:
ab-2xd-cd
k=————
2(b-d)
Notice that y plays no role since we perform a vertical cut!