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tiramichu@sh.itjust.works ⁨2⁩ ⁨days⁩ ago

These categories of geometric problem are ridiculously difficult to find the definitive perfect solution for, which is exactly why people have been grinding on them for decades, and mathematicians can’t say any more than “it’s the best one found so far”

For this particular problem the diagram isn’t answering “the most efficient way to pack some particular square” but “the smallest square that can fit 17 unit-sized (1x1) squares inside it.” - with the answer here being 4.675 unit length per side.

Trivially for 16 squares they would fit inside a grid of 4x4 perfectly, with four squares on each row, nice and tidy. To fit just one more square we could size up to 5x5, and it would remain nice and tidy, but there is then obviously a lot of empty space, which suggests the solution is in-between. But if the solution is in between, then some squares must start going slanted to enable reduction in size, as it is only by doing this we can utilise the unfilled gaps and start poking corners in there.

So, we can’t answer what the optimal solution is going to look like, but we can certainly demonstrate that it’s going to be very ugly and messy.

Another similar (but less ugly) geometric problem is the moving sofa problem which has again seen small iterations over a long period of time.

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