Homophobe!

Why are there gaps on either side of the upper-right square? Seems like shoving those closed (like the OP image) would allow a little more twist on the center squares.

What? No. The divisibility of the side lengths have nothing to do with this.

The problem is what's the smallest square that can contain 17 identical squares. If there were 16 squares it would be simply 4x4.

Fun fact: Bees actually make round holes, the hexagon shape forms as the wax dries.

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 "what is 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 the container up to 5x5, and it would remain nice and tidy, but there is then obviously a lot of empty space, which suggests the solution must be in-between. But if the solution is in between, then some squares must start going slanted to enable the outer square to reduce in size, as it is only by doing this we can utilise unfilled gaps to save space by poking the corners of other squares into them.

So, we can't answer what the optimal solution exactly is, or prove none is better than this, but we can certainly demonstrate that the solution is 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.

It's not necessarily the most efficient, but it's the best guess we have. This is largely done by trial and error. There is no hard proof or surefire way to calculate optimal arrangements; this is just the best that anyone's come up with so far.

It's sort of like chess. Using computers, we can analyze moves and games at a very advanced level, but we still haven't "solved" chess, and we can't determine whether a game or move is perfect in general. There's no formula to solve it without exhaustively searching through every possible move, which would take more time than the universe has existed, even with our most powerful computers.

Perhaps someday, someone will figure out a way to prove this mathematically.

Any other configurations results in a larger enclosed square. This is the most optimal way to pack 17 squares that we’ve found

That's not more efficient because the big square is bigger

We actually haven't found a universal packing algorithm, so it's on a case-by-case basis. This is the best we've found so far for this case (17 squares in a square).

It's the best we've found so far

It is confirmed. I don't understand it very well, but I think this video is pretty decent at explaining it.

https://youtu.be/RQH5HBkVtgM

The proof is done with raw numbers and geometry so doing it with physical objects would be worse, even the MS paint is a bad way to present it but it's easier on the eyes than just numbers.

Mathematicians would be very excited if you could find a better way to pack them such that they can be bigger.

So it's not like there is no way to improve it. It's just that we haven't found it yet.

Proof via "just look at it"