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In the Easter of 2003, when my wife was expecting our first child, she and my mother-in-law were crocheting a variegated bedsheet from spare woolen threads. Because it looked too random to me, I decided to design some colour patterns for the smallest possible number of distinct colours.
The original sheet comprised squares of three colours each. I wanted to create a bedsheet consisting of all different squares fulfilling some simple rule. First, I required the colours in a square to be distinct. Since the order of colours in the square matters, one can choose three colours from a palette of n in n×(n-1)×(n-2) ways. For n=6, we get 120 different squares, which can be arranged in a 12×10 matrix.
The next natural step was to see if the number of colours can be reduced further. I noticed that 120=5×4×3×2×1. This is a strong hint that five colours might be enough to generate 120 distinct squares. In fact, from that equation it is obvious that there are 120 distinct squares, each consisting of 4 (or 5) colours out of the 5-colour palette. It took some fiddling to get a nice pattern out of the squares.
In the Easter of 2008, my wife wanted to knit cushion covers from the wool that was left over from the bedspreads. From four colours, we can obtain 24=4×3×2×1 different 3-colour (or 4-colour) squares, or 4×3 squares on both sides of a cushion.
The generating programs, written in plain C, output the images in NetPBM PPM format. Feel free to examine or download the C source code of the generator of the 6-colour bedsheet, the generator of the 5-colour bedsheet and the generator of the 4-colour cushion cover.