One of my favourite puzzles created by Barry Cipra was originally not a maths puzzle at all. The puzzle is based on a design by artist Solomon LeWitt. Sol LeWitt (after whom the puzzle is named) was a conceptual artist who often featured geometric and combinatorial themes to give a minimalist style to his works. His etching picture, titled Straight Lines in Four Directions and All Their Possible Combinations, is illustrated below in Figure 1, on the left.
|Figure 1. Left: the Sol LeWitt tiles. Right: an example of a red line connecting the edges through all the tiles and an example of a blue line that does not.|
To everyone except Barry this image simply showed 16 squares with lines drawn on them. However, Barry’s imagination was ignited when he noticed that some of the lines extend continuously from one side of the large square to another (red diagonal line in the right-hand of Figure 1), whilst others do not (blue horizontal line in the right-hand of Figure 1).
From this simple setting Barry asked the question:
"Is it possible to rearrange the tiles such that the resulting 4x4 grid has a pattern that allows all horizontal, vertical and diagonal lines to extend continuously across the grid, without interruption?"
Importantly, you are not allowed to rotate any of the pieces!
The simple answer is yes, you can produce such a pattern. In fact there are quite a few solutions to the problem! Have a go yourself. Cut out the squares and see you if can find one of them. Although finding one solution is satisfying, the more interesting investigation is finding a link between solutions.
Produce a few solutions and see if you can see some relation between them. Once you spot the link you will see how to produce many more solutions very easily. Not bad for a simple work of art!
Next time I will fill in the rest of the details, by presenting not only a solution but also furnishing you with the solution link that I am alluding to.