Sometimes the mistake is not in counting badly, but in looking at a flat object when it actually has volume. A cake is not just a circle: it also has height.
The Cake in Eight Pieces
Riddle statement
You have a round cake.
What is the smallest number of straight cuts needed to divide it into 8 pieces?
Show solution
Solution
The answer is:
With two vertical cuts, as if cutting the cake into a cross shape when viewed from above, you can get 4 pieces.
But you still do not have 8.
The third cut does not have to be another vertical cut.
You can make a horizontal cut, parallel to the table, dividing the cake into an upper layer and a lower layer.
Then the 4 pieces on top and the 4 pieces below make:
pieces.
It cannot be done with fewer than 3 cuts, because with 1 cut you can get at most 2 pieces, and with 2 cuts you can get at most 4.
Answer: three cuts are enough.