Squared Squares - Numberphile

Key Takeaways

so this thing is actually the logo of Trinity math Society Trinity College in Cambridge math society when they have lectures about mathematics and they go out socializing to the pub well their society has this as their logo is a square made of other squares but all the squares are different sizes uh so you see this one here with 50 well that means it's a 50 unit by 50 unit Square uh this one here with the 35 is 35 by 35 all these squares are different sizes yet they fit together to make one big Square which is 112 by 112 So it's called a squared square why have they picked this as their logo for their society cuz they're nerds well yeah and that and because it was a trinity math students that solved this this one is actually the smallest possible squared square so this is something that feels like it would be done by the ancient Greeks or something this feels like an old problem how can you make a square from other squares of different sizes that feels like a classic problem this didn't really turn up until the 20th century which surprises me they are integer squares we're not using the same square twice or if you do they're called imperfect so you can use that but I'd rather they're all different sizes there were four students at the Trinity math society that really solved this problem they were called Arthur Stone Cedric Smith Roland Brooks and Bill Tutt the first three were math students Bill Tutt was actually a chemistry student but he was in the math Society because he liked math so you don't have to be a mathematician to be in the math society and I think it was Arthur Stone who brought this problem in who heard of this problem before the only thing that had been done before was there were a couple of rectangles that had been found made out of squares not quite the same thing here they are just two there were just two rectangles that been found one of them uses nine squares the other uses 10 squares rectangles they are easier than making squares but only two had been found so far so the students started to find more their methods were a little bit ad hoc so they were trying to find these rectangles they were hoping that they would get lucky and find a square made out of squares just by chance but it wasn't working they kept trying they found rectangles but they just couldn't find a square they thought right we're going to have to get more systematic so when they started finding these rectangles made out of squares they thought okay let's do this and they started to extend the horizontal lines like this you see this horizontal line toot toot toot toot toot right once you've extended the lines we're going to turn this rectangle into a network and what you do is you connect two horizontal lines if the square are touching so this one here 25 is touching these two horizontal lines that's the the top of 25 and that's the bottom of 25 so I'm just going to connect them with a line I'm actually going to draw an arrow here here to show top to bottom and markets that's the 25 Square so I'm just going to do the whole thing now with the networks so 16 that's the top of 16 that's the bottom of 16 so let's connect those with a line and I'll mark it with 16 that's the 16 line and then I'll keep going so they turned it into to a network and when they did that they recognized it they went that seems familiar and why they recognized it was because it reminded them of an electrical network if you connect this to a battery and if these were wires in a network and each wire has one unit resistance uh and you connect it to a battery which has the same potential as the height of the rectangle put all these numbers in these are the currents flows through that electrical Network so suddenly they realized that this was connected to electrical networks and they could use all these maths that was already there they could borrow it completely to solve the squared squared problem uh in particular these are called kirkhoff laws a couple of kirkhoff laws are the current flowing into a point should equal the current flowing out of a point so let's look at the current flowing into here 36 is flowing in two is flowing in so we got 38 in total flowing in and flowing out we've got 33 and a five so it they are equal another thing we can learn from kirkov laws if you've got a circuit like this circuit here a circuit should make zero I'll show you what I mean uh so from going from here we've got a nine we've got a seven which makes 16 and then I go in the opposite direction here so this is a minus 16 so Al together the whole circuit is a z that's crazy yeah so all this math suddenly you could just borrow and and they could solve the squared squared problem a couple of things they solved on the way uh they showed that the smallest squared rectangle uh just uses nine squares uh and there's two of them actually uh that's one of them that's the smallest squared rectangle meaning it just uses nine squares and the other one was that one that was done before them remember there were two that was done previously one of those is a small squared rectangle they showed that there are no cubed cubes and that's kind of disappointing you can't make a cube out of cubes of different sizes so using these networks they could find what made valid squared rectangles and what are not valid squared without having to draw out the squared rectangles so they started to build up a catalog one of their ideas was if they could find a squared rectangle and another squared rectangle that had the same dimensions they could do it like this a squared rectangle like that and another one which had the same dimensions but using completely different squares inside so they're mutually exclusive but they're the same size then you could join them together like this this would be a square up here this would be a square here and it would make a squared squared automatically or if you can't do that wouldn't have duplicates in it though so you want to avoid duplicates so you want two squared rectangles which have no square in common that's what they were looking for or maybe they could have one square in common if it was a corner Square because you could do this you could have a rectangle like this another one that's meant to be exactly the same Dimension that justes share one corner and that would make a squared square as well so that was their plan and it worked H and they finally found one the first squared square uh it was massive the first one they found it used 55 squares uh and then they started to improve they wanted to find smaller ones uh they found one that used 38 uh Bill Tutt found one that used 26 and these were getting better and the question was what's the smallest squared square now that is something they struggled with because it became too hard a problem and this was a problem that was then left open for another 38 years until the 1970s until computers came along and in using computers they could use the students method that they had done they could use those methods with a computer and began searching for squared squares and they found the smallest uses 21 squares it's definitely the smallest it's Unique there's only one squared square that uses 21 and this is it there are none that use fewer squares uh and then you can look at many that uses more than 21 squares but those students all of them became mathematicians uh well one of them became a civil servant but he was still a maths fan I think that was Roland Brooks uh but the other three became mathematicians including Bill Tutt who started off as a chemistry student he swapped over to maths he then became a Bletchley Park codebreaker in World War II uh very important job and became a proper mathematician after the war and he was a Pioneer in graph Theory which is the maths of networks and he CR crits this problem as his training in graph Theory Okay so we've seen this one which is the smallest squared square the dimensions of it is 112 units by 112 units there is actually smaller squares if we're just talking about Dimensions but they use more squares and they're not unique either there are three known squares that are of that size so there smaller Dimensions but they use more squares one thing I found that was interesting theoretically we can do this uh 1 2 + 2 2 + 3 2 + 4 S imagine consecutive squares right going up to 24 s is equal to a big number I'm not even sure what it is well I can tell you what it is it's 70 SAR which means that it is theoretically possible to make a square which is 70 by 70 made out of consecutive squares one 2 3 4 but it can't be done it's so tempting the maths looks like it can be done but geometrically they don't fit together it's so disappointing I know it be like a magic squared square perfect super perfect and it's so disappointing it doesn't exist ah or does it no it doesn't oh another favorite squared square well it's this one I say's favorite because this one was actually found before the Trinity students found theirs they were scooped I mean everyone else is like it's not a very nice square is it don't what should we tell Matt he seems so excited about it well you know what I am excited about it this is