For the uninitiated: this is the current most-efficient method found of packing 17 unit squares inside another square. You may not like it, but this is what peak efficiency looks like.
(Of course, 16 squares has a packing coefficient of 4, compared to this arrangement’s 4.675, so this is just what peak efficiency looks like for 17 squares)
Isn’t this only true if the outer square’s size is not an integer multiple of the inner square’s size? Meaning, if you have to do this to your waffle iron, you simply chose the dimensions poorly.
The optimisation objective is to fit n smaller squares (in this case, n=17) into the larger square, whilst minimising the size of the outer square. So that means that in this problem, the dimensions of the outer square isn’t a thing that we’re choosing the dimensions of, but rather discovering its dimensions (given the objective of "minimise the dimensions of the outer square whilst fitting 17 smaller squares inside it)
But you can fit 25 squares into the same space. This isn’t efficiency, it’s just wasted space and bad planning.
You raised the packing coefficient by ⅝ to squeeze one extra square in with all that wasted space, so don’t argue that 25 squares has a packing coefficient of 5. Another ⅜ will get you an extra 8 squares, and no wasted space.
Precisely. That’s why I wrote the parenthetical about the greater efficiency of 16 as a perfect square. As the other commenter pointed out, this is a meme. This is only the most efficient packing method for 17 squares. It’s the packing efficiency equivalent of the spinal tap “this one goes to 11” quote.
Some autistics thrive on chaos, some thrive on order. I’m not the “pack a prime number into a square” kind of autistic, I’m the “why would you want to do that” kind
Basically just to see if they can. We can think of the problem from multiple angles. The general problem is: “if we have a larger square with side length of a, what’s the maximum number of smaller squares (with side length of b) that we can fit into that larger square?”. If we have a larger square with side length of 4, then we can fit 16 squares in. If the larger square had a side length of 5, then we can fit 25 squares in. So this means that if we want a neat packing solution, and we can control how large the outer square is (in relation to the inner squares), then we want each side of the larger square to be a whole number multiple of the smaller square’s side length.
But what if that isn’t our goal? The fact that packing 25 squares into a 5x5 square is an optimal packing solution with no spare space means that it will be impossible to fit 25 smaller squares into a square that’s less than 5x5 large. But what about if we do have awkward constraints, and the number of smaller squares we have to pack isn’t a square number? The fact that this weird packing solution in the OP has 17 squares isn’t because 17 is prime, but rather that 17 is 1 more than 16 (it’s just that 17 happens to be prime).
This is a long way of saying that because packing 16 squares into a square is easy, the natural next question is “how large does the larger square need to be to be able to pack 17 squares into it?”. If this were a problem in real life where I had to pack 17 squares into a physical box, most people would just get a box that’s at least 5x5 large, and put extra packing material into all the spare space. But asking this question in terms of “what’s the smallest possible box we could use to pack 17 squares in?” is basically just an interesting puzzle, precisely because it’s a bit nonsensical to try to pack 17 squares into the larger square. We know for certain we need a box that’s larger than 4x4, and we also know that we can do it in a 5x5 box (with a heckton of spare space), so that gives us an upper and lower bound for the problem — but what’s the smallest we could use, hypothetically?
As a fellow autistic person, I relate to your confusion. But I’d actually wager that there were a non-zero number of autistic people who were involved in this research. It sort of feels like “extreme sports” for autistic people — doing something that’s objectively baffling, precisely because it feels so unnatural and wrong
Okay, but none of that applies to waffles. They said they wanted more squares for syrup, but they actually got more unused space on the waffle surface.
I guess I’m not the “figure out how to fit a prime number into a square” kind of autistic, I’m the “why would you want to do that” kind of autistic.
To me, square numbers are beautiful because of how harmoniously they can be arranged, and prime numbers are beautiful because of how unique and impossible to neatly arrange they are. Trying to treat one like the other feels like an itch that can’t be scratched…
I mean, the actual answer is severalfold: “sometimes, when you need to fill a space, you don’t end up with simple compound numbers of identical packages” is one, but really, it’s a problem in mathematics which, were we to have a general solution to find the most efficient method of packing n objects with identical properties into the smallest area, we would be able to more effectively predict natural structures, including predicting things like protein folding, which is a huge area of medical research. Simple, seemingly inapplicable cases can often be generalised to more specific cases, and that’s how you get the entire field of applied math, as well as most of scientific and engineering modeling
Even when it can’t be generalized, you still often learn something by trying. You may invent a new way to look at a set of problems that no one’s done before, or you may find a solution to something totally unrelated. There’s a lot to learn even when it looks like you’ll gain nothing.
(this is the part where you tack on a silly harmless lie at the end, like - “this specific packing optimization improvement was actually discovered accidentally, through a small mini-game introduced into Candy Crush in 2013. Players discovered the novel improvement, hundreds of individual times, within the first several minutes of launch. Scholars pursuing novel packing algorithms even colloquially call this event ‘The Crushening’”)
That candy crush story is, as the commenter said, a lie. I don’t know why they would suggest that adding on a lie is in any way good, since we know that this packing was discovered in the late 1990s. It’s on the wikipedia article for square packing (with sources) but I don’t feel like looking it up again.
For 25 squares of size 1x1 you’d need a square of size 5x5. The square into which 17 1x1 squares fit is smaller than 5x5, so you can’t fit 25 squares into it.
You raised the packing coefficient by ⅝ to squeeze one extra square in with all that wasted space, so don’t argue that 25 squares has a packing coefficient of 5. Another ⅜ will get you an extra 8 squares, and no wasted space.
You raised the packing coefficient by ⅝ to squeeze one extra square in with all that wasted space, so don’t argue that 25 squares has a packing coefficient of 5. Another ⅜ will get you an extra 8 squares, and no wasted space.
For the uninitiated: this is the current most-efficient method found of packing 17 unit squares inside another square. You may not like it, but this is what peak efficiency looks like.
(Of course, 16 squares has a packing coefficient of 4, compared to this arrangement’s 4.675, so this is just what peak efficiency looks like for 17 squares)
Isn’t this only true if the outer square’s size is not an integer multiple of the inner square’s size? Meaning, if you have to do this to your waffle iron, you simply chose the dimensions poorly.
The optimisation objective is to fit n smaller squares (in this case, n=17) into the larger square, whilst minimising the size of the outer square. So that means that in this problem, the dimensions of the outer square isn’t a thing that we’re choosing the dimensions of, but rather discovering its dimensions (given the objective of "minimise the dimensions of the outer square whilst fitting 17 smaller squares inside it)
Or maybe you just want waffles with 17 squares in them.
But you can fit 25 squares into the same space. This isn’t efficiency, it’s just wasted space and bad planning.
You raised the packing coefficient by ⅝ to squeeze one extra square in with all that wasted space, so don’t argue that 25 squares has a packing coefficient of 5. Another ⅜ will get you an extra 8 squares, and no wasted space.
Precisely. That’s why I wrote the parenthetical about the greater efficiency of 16 as a perfect square. As the other commenter pointed out, this is a meme. This is only the most efficient packing method for 17 squares. It’s the packing efficiency equivalent of the spinal tap “this one goes to 11” quote.
My autistic ass can’t comprehend why anyone would want to arrange a prime number in a square pattern…
???
It’s pretty common for people with autism to prefer things to be efficient and logical.
Some autistics thrive on chaos, some thrive on order. I’m not the “pack a prime number into a square” kind of autistic, I’m the “why would you want to do that” kind
LOL’ed, but also
Basically just to see if they can. We can think of the problem from multiple angles. The general problem is: “if we have a larger square with side length of a, what’s the maximum number of smaller squares (with side length of b) that we can fit into that larger square?”. If we have a larger square with side length of 4, then we can fit 16 squares in. If the larger square had a side length of 5, then we can fit 25 squares in. So this means that if we want a neat packing solution, and we can control how large the outer square is (in relation to the inner squares), then we want each side of the larger square to be a whole number multiple of the smaller square’s side length.
But what if that isn’t our goal? The fact that packing 25 squares into a 5x5 square is an optimal packing solution with no spare space means that it will be impossible to fit 25 smaller squares into a square that’s less than 5x5 large. But what about if we do have awkward constraints, and the number of smaller squares we have to pack isn’t a square number? The fact that this weird packing solution in the OP has 17 squares isn’t because 17 is prime, but rather that 17 is 1 more than 16 (it’s just that 17 happens to be prime).
This is a long way of saying that because packing 16 squares into a square is easy, the natural next question is “how large does the larger square need to be to be able to pack 17 squares into it?”. If this were a problem in real life where I had to pack 17 squares into a physical box, most people would just get a box that’s at least 5x5 large, and put extra packing material into all the spare space. But asking this question in terms of “what’s the smallest possible box we could use to pack 17 squares in?” is basically just an interesting puzzle, precisely because it’s a bit nonsensical to try to pack 17 squares into the larger square. We know for certain we need a box that’s larger than 4x4, and we also know that we can do it in a 5x5 box (with a heckton of spare space), so that gives us an upper and lower bound for the problem — but what’s the smallest we could use, hypothetically?
As a fellow autistic person, I relate to your confusion. But I’d actually wager that there were a non-zero number of autistic people who were involved in this research. It sort of feels like “extreme sports” for autistic people — doing something that’s objectively baffling, precisely because it feels so unnatural and wrong
Okay, but none of that applies to waffles. They said they wanted more squares for syrup, but they actually got more unused space on the waffle surface.
I guess I’m not the “figure out how to fit a prime number into a square” kind of autistic, I’m the “why would you want to do that” kind of autistic.
To me, square numbers are beautiful because of how harmoniously they can be arranged, and prime numbers are beautiful because of how unique and impossible to neatly arrange they are. Trying to treat one like the other feels like an itch that can’t be scratched…
It’s not just primes.
https://en.wikipedia.org/wiki/Square_packing
But it’s especially primes, cause they can’t even fit in a rectangle unless it’s 1×
I mean, the actual answer is severalfold: “sometimes, when you need to fill a space, you don’t end up with simple compound numbers of identical packages” is one, but really, it’s a problem in mathematics which, were we to have a general solution to find the most efficient method of packing n objects with identical properties into the smallest area, we would be able to more effectively predict natural structures, including predicting things like protein folding, which is a huge area of medical research. Simple, seemingly inapplicable cases can often be generalised to more specific cases, and that’s how you get the entire field of applied math, as well as most of scientific and engineering modeling
Even when it can’t be generalized, you still often learn something by trying. You may invent a new way to look at a set of problems that no one’s done before, or you may find a solution to something totally unrelated. There’s a lot to learn even when it looks like you’ll gain nothing.
(this is the part where you tack on a silly harmless lie at the end, like - “this specific packing optimization improvement was actually discovered accidentally, through a small mini-game introduced into Candy Crush in 2013. Players discovered the novel improvement, hundreds of individual times, within the first several minutes of launch. Scholars pursuing novel packing algorithms even colloquially call this event ‘The Crushening’”)
Are you sure the story is real? I can find anything that points to it, so a link would help a lot
That candy crush story is, as the commenter said, a lie. I don’t know why they would suggest that adding on a lie is in any way good, since we know that this packing was discovered in the late 1990s. It’s on the wikipedia article for square packing (with sources) but I don’t feel like looking it up again.
I didn’t even understand the point that it was a lie, and not the original comment had a lie, reading skill issue
For the lolz? Little harmless fun? Not everyone’s cup of tea, admittedly.
I’m not sure where I came across it, but it’s out there somewhere. You can do it!
Mathematicians try this with every number
For 25 squares of size 1x1 you’d need a square of size 5x5. The square into which 17 1x1 squares fit is smaller than 5x5, so you can’t fit 25 squares into it.
Do I need to tap the sign?
You can’t fit 25 squares into a square 4.675x bigger unless you make them smaller. Yes, that will increase the volume available for syrup.
Literally already addressed that, but go off
Yeah, it’s not at all an optimal waffle. It’s more a cool math meme waffle. ;3
– Frost
Thank you I was very lost lmao
Does coefficient in this context mean the length of the side of the big square?
Exactly. It is the length of the side of the bigger square, relative to the sides of the smaller identical squares.