Piece of cake
Probability
Level
3
What is the maximum number of pieces into which a 3-dimensional cube can be cut by exactly 45 planes?
Image credit: Pinterest.
The answer is 15226.
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1 solution
We are looking for the maximum number of regions into which a 3-dimensional cube can be partitioned by exactly n planes. This is referred to as the cake number and it is defined as:
C n = ( 3 n ) + ( 2 n ) + ( 1 n ) + ( 0 n ) = 1 / 6 ( n 3 + 5 n + 6 )
It is called the cake number due to its relation to the cake cutting problem: how to fairly divide a circle into n equal area pieces using cuts in its plane. One method of proving that a fair cake cutting is always possible relies on the Frobenius-Konig theorem . This theorem states that the permanent of an n x n matrix with all entries either 0 or 1 is 0 iff the matrix contains an r x s submatrix of 0s with r + s = n + 1 .