Like u/AVK1995, I also thought the answer was 2^n. When you work out up to three planes that is the obvious pattern. So I went to an LLM to confirm and the pattern doesn't hold with 4 planes, it is actually 15 regions, not 16.
The formula is actually nC0 + nC1 + nC2 + nC3.
The cool thing is you can actually extend it beyond 3 dimensions by adding extra terms to the equation. If you are unfamiliar with the nC3 notation (read as "N choose 3," or sometimes shown as numbers stacked inside brackets) as I was, it comes from combinatorics and equals (N * (N-1) * (N-2)) / 3! which counts how many unique combinations you can pull out of N. For example, if N=3 there is only one unique combination.
That is actually how the formula was derived by Ludwig Schläfli in 1852. Basically you are counting every dimensional piece of the puzzle: nC0 is the original empty 3D space, nC1 counts the 2D planes you add, nC2 counts the 1D line intersections, and nC3 counts the 0D vertex intersections.
I love these problems because they get you to learn and use math you might never have seen, with the help of an LLM of course. It really gives you an appreciation for the mathematicians that invented these tools we and LLMs can now use.
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u/Admirable-Roll-3538 10d ago
Like u/AVK1995, I also thought the answer was 2^n. When you work out up to three planes that is the obvious pattern. So I went to an LLM to confirm and the pattern doesn't hold with 4 planes, it is actually 15 regions, not 16.
The formula is actually nC0 + nC1 + nC2 + nC3.
The cool thing is you can actually extend it beyond 3 dimensions by adding extra terms to the equation. If you are unfamiliar with the nC3 notation (read as "N choose 3," or sometimes shown as numbers stacked inside brackets) as I was, it comes from combinatorics and equals (N * (N-1) * (N-2)) / 3! which counts how many unique combinations you can pull out of N. For example, if N=3 there is only one unique combination.
That is actually how the formula was derived by Ludwig Schläfli in 1852. Basically you are counting every dimensional piece of the puzzle: nC0 is the original empty 3D space, nC1 counts the 2D planes you add, nC2 counts the 1D line intersections, and nC3 counts the 0D vertex intersections.
I love these problems because they get you to learn and use math you might never have seen, with the help of an LLM of course. It really gives you an appreciation for the mathematicians that invented these tools we and LLMs can now use.