Algorithm to divide a chocolate bar in equal parts

Question

A random thought popped into my head (when I was sharing a chocolate bar of course!). I was wondering if there is a generic algorithm to solve this problem.

The problem

Answer 1

It seems to me that you're looking for numbers that are evenly dividable by all numbers between 1 and n inclusive. That's called the least common multiple of 1, ..., n. A square containing the least common multiple of 1, ..., n squares would by definition be evenly dividable into pieces of size 1, ..., n. You're looking for a maximum of n splits, which adds additional complexity to the problem which may or may not be possible.

Your example for n = 4 is the LCM(4,3,2,1) which is 12. LCM(5,4,3,2,1) is 60. LCM(6,5,4,3,2,1) is also 60.

They can always be laid out as 1xLCM(n,...,1) rectangles, and always be dividable into 1,...,n even piles in n-1 or fewer divisions.

For example, when n = 4, LCM(4,3,2,1) = 12. The rectangle is

############

And can be divided as follows:

1: ############     // 0 cuts
2: ###### ######    // 1 cut
3: #### #### ####   // 2 cuts
4: ### ### ### ###  // 3 cuts (3 being n-1)