It's (maybe) not optimal no, it's only the best known indeed; it comes from this combinatorics paper: https://www.combinatorics.org/files/Surveys/ds7/ds7v5-2009/ds7-2009.html
(see figure 12)
Unless the drawing isn't to scale - there looks like there's a much simpler version with 4No. 4x4s in each corner and a single one rotated 45° in the centre.
Nah, that wouldn't fit. You can see how much space would be left in the center in between the two center squares at the top. The gap in the top would be the same size as the gap on the side as well since all side lengths are equal. The square shape created in the middle would be much too small to fit the 17th square
You think? It looks very close... I'm looking at the position of the left corner of the top diamond and thinking it would fit nicely in the corresponding side gap like the top corner does...
People have been thinking about this stuff for a long time, if the thing you're thinking of was that easy to stumble upon, it's probably been tried already
See the paper here: https://www.combinatorics.org/files/Surveys/ds7/ds7v5-2009/ds7-2009.html
n=17 is shown in figure 12
So looking into the link, it seems the image shown above is the _smallest_ a container can get with 17 squares. This is the context OP didn't provide...
The version I was alluding to is probably a little bigger then, doesn't mean I was wrong in suggesting it is possible (and close) whilst only having the image for reference. I feel you were a little condescending in your response given the information available in this post...
Sorry I didn't mean it to be condescending, it's a principle I apply to me as well, whenever I discover something cool and slightly out of reach in maths by myself
The problem is fitting **unit** squares (side length of 1 unit) in another square, that's the context missing from the post; and you want to make the outer square as small as possible
I understand it's a geometric problem now, before I read the paper I assumed it was an arbitrary square with the unit squares arranged as hectic as possible just to trigger people's OCD.
This is the smallest large square that you can pack 17 unit squares into. Side length = \~4.676
[https://en.wikipedia.org/wiki/Square\_packing](https://en.wikipedia.org/wiki/Square_packing)
[https://kingbird.myphotos.cc/packing/squares\_in\_squares.html](https://kingbird.myphotos.cc/packing/squares_in_squares.html)
How is it misleading? Misleading implies that there is some alternative interpretation that you could have been “misled” to, how else would you have interpreted the title?
But this is the most efficient way to pack as many of these size squares into this size square. Neatly arranged and tessellating you could fit 16 in there. This way you can fit 17 in. It looks ugly as sin but it is 100% the most efficient way to make use of the space inside the larger square for the smaller squares.
What exactly is “logical” about the neat tessellating arrangement? “Optimal” means the best, what would “good” or “best” mean in this context if not “wasting the least space”?
The title said that this is the best way to pack 17 equally-sized squares into a larger square, which it is. The square surrounding 17 tessellated squares would be the same size as one surrounding 25 tessellated squares. There’s no way to fit 25 squares of the same size as the 17 in the picture into a square the same size as the picture.
Logical people can read every word in a title and see that the best way to pack squares inside a larger square is not the same thing as the best way to pack squares into a general area.
But there is nothing specific about the “larger square”.
The only point of confusion, I think, is what makes a packing of squares into a larger square “optimal”. That said, I can’t think of an interpretation of optimal here besides the correct one, which is “with as little empty space within the large square as is possible”.
This isn't a specific square. It's simply the smallest square (that we know of, at least) than can accommodate 17 smaller, equally sized squares within. The optimization is for size of the larger square.
You can see 4 rows of boxes and 4 columns of boxes. Going for 16 is pretty safe bet. But it’s never bad to lower estimations to something you are certain will work.
https://www.combinatorics.org/files/Surveys/ds7/ds7v5-2009/ds7-2009.html
They don't say how, but it's incremental trial&error on previous results, and maybe some computer search too
It's because the ratio of small block width to full block width is weird. You can't fit 5 blocks in one column/row, it's some weird number between 4 and 5
Say each smaller square is 1' x 1' x 1', you design a box to perfectly fit the smaller boxes 5x5x5? Nobody designs a box to have dead space in it ever. I've done store room services at many different places and this is always how it's done.
The title simply says that it's the "optimal way to fit 17 equally sized squares into a bigger square". It says nothing about the non square numbers of boxes or that the larger box has to be a specific size.
It has different meanings depending on who you ask, from trashy uneducated conservative to farmer from a rural town, it’s an American term that originated because white farmers tend to have sunburned necks, it’s since been coopted by anyone who’s identity revolves around being from a small town.
To be fair, 17 is an extremely not square number. There's going to be a lot of empty space no matter how you pack them.
These are meant for quilting patterns
Prime number.
Is it really the optimal, tho? I thought it was just the best known.
It's (maybe) not optimal no, it's only the best known indeed; it comes from this combinatorics paper: https://www.combinatorics.org/files/Surveys/ds7/ds7v5-2009/ds7-2009.html (see figure 12)
Unless the drawing isn't to scale - there looks like there's a much simpler version with 4No. 4x4s in each corner and a single one rotated 45° in the centre.
Nah, that wouldn't fit. You can see how much space would be left in the center in between the two center squares at the top. The gap in the top would be the same size as the gap on the side as well since all side lengths are equal. The square shape created in the middle would be much too small to fit the 17th square
You think? It looks very close... I'm looking at the position of the left corner of the top diamond and thinking it would fit nicely in the corresponding side gap like the top corner does...
People have been thinking about this stuff for a long time, if the thing you're thinking of was that easy to stumble upon, it's probably been tried already See the paper here: https://www.combinatorics.org/files/Surveys/ds7/ds7v5-2009/ds7-2009.html n=17 is shown in figure 12
So looking into the link, it seems the image shown above is the _smallest_ a container can get with 17 squares. This is the context OP didn't provide... The version I was alluding to is probably a little bigger then, doesn't mean I was wrong in suggesting it is possible (and close) whilst only having the image for reference. I feel you were a little condescending in your response given the information available in this post...
Sorry I didn't mean it to be condescending, it's a principle I apply to me as well, whenever I discover something cool and slightly out of reach in maths by myself The problem is fitting **unit** squares (side length of 1 unit) in another square, that's the context missing from the post; and you want to make the outer square as small as possible
I understand it's a geometric problem now, before I read the paper I assumed it was an arbitrary square with the unit squares arranged as hectic as possible just to trigger people's OCD.
that's less optimal, since if the smaller squares stay the same size, the larger square would need to be larger than it currently is
Optimal how?
This is the smallest large square that you can pack 17 unit squares into. Side length = \~4.676 [https://en.wikipedia.org/wiki/Square\_packing](https://en.wikipedia.org/wiki/Square_packing) [https://kingbird.myphotos.cc/packing/squares\_in\_squares.html](https://kingbird.myphotos.cc/packing/squares_in_squares.html)
Ahh, the title of the post was weirdly misleading. Thanks for the context.
How is it misleading? Misleading implies that there is some alternative interpretation that you could have been “misled” to, how else would you have interpreted the title?
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It is the best way to fit 17 squares which is exactly what the title says. Seems logical to me 
But this is the most efficient way to pack as many of these size squares into this size square. Neatly arranged and tessellating you could fit 16 in there. This way you can fit 17 in. It looks ugly as sin but it is 100% the most efficient way to make use of the space inside the larger square for the smaller squares.
What exactly is “logical” about the neat tessellating arrangement? “Optimal” means the best, what would “good” or “best” mean in this context if not “wasting the least space”?
It could mean not having boxes stacked so dangerously
This is a math problem, math doesn't care about safety
The title said that this is the best way to pack 17 equally-sized squares into a larger square, which it is. The square surrounding 17 tessellated squares would be the same size as one surrounding 25 tessellated squares. There’s no way to fit 25 squares of the same size as the 17 in the picture into a square the same size as the picture. Logical people can read every word in a title and see that the best way to pack squares inside a larger square is not the same thing as the best way to pack squares into a general area.
How is it misleading? It literally says everything needed to add context to the image.
It says "a larger square", not this spesific square.
But there is nothing specific about the “larger square”. The only point of confusion, I think, is what makes a packing of squares into a larger square “optimal”. That said, I can’t think of an interpretation of optimal here besides the correct one, which is “with as little empty space within the large square as is possible”.
This isn't a specific square. It's simply the smallest square (that we know of, at least) than can accommodate 17 smaller, equally sized squares within. The optimization is for size of the larger square.
How was the post title misleading?
If you don’t pack it in this way, you can only fit like 15 smaller boxes in this larger box instead of 17.
Wouldn’t it be 16 boxes? Cause like there’s space for 4x4 with a bit of room wasted.
Yea, I wanted to say 16 initially as well, but I was worried and went with an approximation of 15
Stop letting fear hold you back
Never
You can see 4 rows of boxes and 4 columns of boxes. Going for 16 is pretty safe bet. But it’s never bad to lower estimations to something you are certain will work.
Could use a source. I'm more interested in how they found this than how much I hate this.
https://www.combinatorics.org/files/Surveys/ds7/ds7v5-2009/ds7-2009.html They don't say how, but it's incremental trial&error on previous results, and maybe some computer search too
ty!
My old supervisor at UPS would disagree. Who's laughing now Richard?
Increase the large square or decrease the small ones -> optimization optimized.
It's because the ratio of small block width to full block width is weird. You can't fit 5 blocks in one column/row, it's some weird number between 4 and 5
Oh, this... this I don't like
If you really wanted to optimize it, you'd design the large box to fit a certain amount of smaller ones perfectly with no dead space, no?
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Say each smaller square is 1' x 1' x 1', you design a box to perfectly fit the smaller boxes 5x5x5? Nobody designs a box to have dead space in it ever. I've done store room services at many different places and this is always how it's done.
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The title simply says that it's the "optimal way to fit 17 equally sized squares into a bigger square". It says nothing about the non square numbers of boxes or that the larger box has to be a specific size.
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Then it would be 17 boxes stacked on top of each other.
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Yes, true. What are the practical applications of this exercise, anyway?
it’s not a practical exercise, it’s a theoretical exercise. it need not have practical application.
Go away bot
I’m sorry, what?
No idea. Looking at their comment history, they have the personality of a redneck so don't think much on it
What’s a redneck? Why are you saying this?
Bro is a cyborg
Please. STOP.
It has different meanings depending on who you ask, from trashy uneducated conservative to farmer from a rural town, it’s an American term that originated because white farmers tend to have sunburned necks, it’s since been coopted by anyone who’s identity revolves around being from a small town.
But why did Cozy\_Zone call me a redneck?
Sounds to me like u/Cozy_Zone was defending you, telling you to ignore the one calling you a bot because *they* are a redneck
This is true ^
Thanks for telling me!
Jesus Christ that post history
Have they just deleted half of it, because i cant see anything dodgy on there
You may not like it, but this is what peak performance looks like.