Say, you are given a cake which you must share with 7 others. So, you must cut the cake into 8 pieces. But, you are only allowed to make 3 straight cuts. You cannot move pieces of the cake after the first cut.
Obviously, this doesn't work in the plane. So you need a 3D solution.
A 3D solution is simple. Cut the cake with 2 perpendicular cuts through the center, then make a horizontal cut at half the height of the cake. It is not fair regarding topping, but you have 8 pieces.
Florian's answer mentions that it's 'obvious' that the cuts don't work in the plane, but I figure it's worth a short proof that you in fact can't cut a (convex!) 2d cake into eight pieces with three slices.
Firstly, since the cake is convex we may as well say that it's infinitely large and just look at cutting the plane into pieces; this won't affect the maximum number of pieces we can get. Now, since each slice is a line, any two of them intersect in at most one point. If all three of them intersect in the same point then they partition the plane into at most six regions. Likewise, if any two slices don't intersect, then they partition the plane into at most six regions. Otherwise, pick two of the slices: they divide the plane into four quadrants. Now, the third slice intersects the other two in one point each, and those two intersection points border a common quadrant. Whichever quadrant is opposite to that one can't be cut by the third slice, because either getting 'in' or 'out' of it would require another intersection with one of the first two slices definiing the quadrants; this means that the three slices partition the plane into at most 4 (the quadrants from the first two slices) + 3 (the new regions defined by the third slice) =7 pieces.
And why convex? Well, cutting this nonconvex 'cake' into eight pieces with just three slices is left as an exercise to the reader:
cut the cake in an even X making 4 slices. Then stack the slices on top of each other and cut them down the center making 8 equal pieces with equal icing
Cut 4 pieces at 45 degree angles - same as diving a wristwatch from 10:30-1:30 , 1:30-4:30, 4:30-7:30 and 7:30-10:30. Each piece is equal to 3 hours on a watch. Take the piece with 3 o'clock portion and align this with the 12 o'clock position. Similarly, align the 9 o'clock portion with 6 o'clock. Make sure all the center portions are aligned in one straight line which will pass through 3, 12, the cake center, 6 and 9. A single cut along this line will give 8 exact pieces equal to 1:30 hrs on a clock. This does not require stacking which can destroy the cake icing etc. Enjoy the perfect 8 equal pieces.
Just for the fun, assume the cake is only 2 dimensional, so that the other answers don't work. A solution is still possible. Cut the cake one line through the center and now bend it to form a cone. Be careful not to move the pieces, as that is not allowed, merely bend them. Some parts will overlap especially on the bottom of the cone. Now cut the cone in a straight horizontal line about at half height. Unbend it and you have a circular cut but you only did a straight cut. Now just do a second cut through the center and you have 8 pieces and all with topping. Enjoy!
(the solution still works for a 3 dimensional cake. After all it works with paper and paper is 3 dimensional. Cake is a bit (a lot) thicker, so it might be a bit more difficult but it works in theory)
cutting the cake in an even X making 4 slices. then cutting a concentric circle making them 8 pieces.
It is simple.First of all find 360 degree divided by 8. So we get 45 degrees.Now, in our cake concept, cake is the circle with 360 degrees and as we cut into 8 equal parts, each cut part should have 45 degrees. Now how to do this is: Cut the cake directly into half.Take 45 degrees from an end of the cake to right side.Cut the cake right at this new position, in full to further half. Repeat this step once again. We get 8 equal parts of piece cakes.