On the DVD release's commentary, writer Terrance Dicks claims that he wrote the First Doctor's "solution" to the checkerboard puzzle, pretty much, off the top of his head., and that he has been asked to explain it, even though he wrote it purposefully with no answer. (The novelization even makes an aside that the Doctor never bothered to explain what the solution was.) At first glance, it seems impossible that he could have used the mathematical formula for Pi, the circumferance of a circle divided by its diameter in Euclidean geometry, to solve a problem based on a square. However, Pi can also be a constant expressed in relativity, something the Doctor should know a lot about. In that way, it can also be defined using an infinite series such as Pi/4 = 1  1/3 + 1/5  1/7 + 1/9  ... In general relativity, space and spacetime are expressed as nonEuclidean geometries. In a nonEuclidean geometry, circumference divided by diameter can be more or less than Pi. NonEuclidean geometries are used in physics, so the ratio is very nearly Pi over small distances i.e. any finite value divided by infinity is so small it might as well be 0, so it is called 0. Thus, over small distances, there are relatively unnoticeable differences in the measurements for Pi. The Doctor could well have been working in a nonEuclidean geometry for the puzzle's solution, but, Pi would still be roughly the same value, as he states, 3.14159265. Dicks just stumbled into this by blind luck, most likely, and it still doesn't explain what the puzzle solution is, though.
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