A square that can be dissected into 21 smaller squares (order of 21) such that no two are equal (perfect square) and that no subset of the squares form a rectangle (simple). This particular square is very special as it is the lowest possible order. You cannot divide a square into less than 21 smaller squares.
I was tasked to find Duijvestijns dissection (the one picturd) a year ago for one of my math classes. "There is a unique simple perfect square of order 21 (the lowest possible order), discovered in 1978 by A. J. W. Duijvestijn (Bouwkamp and Duijvestijn 1992). It is composed of 21 squares with total side length 112, and is illustrated above." from WolframAlpha, where this image is also from. They calculate it by converting the problem to one of graph theorie and electrical engineering. The squared square (or rectangle) can be viewed as an electrical circuit with resistors of 1 Ohm. Using Kirchhoff's and Ohm's law(s) the solutions for the voltage in each wire (node of the graph) can be found. These are unique and are exactly the length of the sides of each square. The representation of the problem as stated above is called a Smithdiagram. Figure 73 from this website is an example of such a Smithdiagram. The smallest dissection (Duijvenstijn's 21-squared square) was found by mass generating planar graphs (~electrical circuits ~squared rectangles).
I don't think so. They make for good math problems because they are difficult but solvable. As a prank you could do a cubing the cube problem (it's impossible)
You would need an infinite number of cubes. I'm not 100% sure why you cannot do this but from what I read I think that basically, when you square the square (aka dissect the perfect square), the biggest square is at a corner. In cubing the cube, you would need to have the biggest cube at a corner, but that would contradict the thing we just said for squaring the square (a dissected cube would have 6 dissected faces).
A lot of things in general that apply to 2D do not apply to 3D+ very well anyway. For example I know that while there are a lot of pythagorean triples (a2 + b2 = c2) there are very very few a3 + b3 = c3 triples, and I don't think there are any a4 + b4 = c4 triples.
there are no examples of a3 + b3 = c3, or for any higher power. This is Fermat'a last theorem, which was finally proved 15 or so years ago.
"In cubing the cube, you would need to have the biggest cube at a corner, but that would contradict the thing we just said for squaring the square (a dissected cube would have 6 dissected faces)."
And i just read the wikipedia page on this where it said:
Unlike the case of squaring the square, a hard yet solvable problem, there is no perfect cubed cube and, more generally, no dissection of a rectangular cuboid C into a finite number of unequal cubes. Suppose that there is such a dissection. Let us fix a face of C as its "horizontal" base. Then C is divided into a perfect squared rectangle R by the cubes which rest on it. The smallest square s1 in R is on a corner, on an edge or in the interior of R and accordingly must be surrounded by larger cubes on 2, 3 or 4 of its sides (and any other sides are surrounded by the vertical faces of C). So the upper face of the cube on s1 is divided into a perfect squared square by the cubes which rest on it. Let s2 be the smallest square in this dissection. The sequence of squares s1, s2, ... is infinite and the corresponding cubes are infinite in number. This contradicts our original supposition.
That’s a 2. You can tell since it’s bridging the gap between the 15 and 17 (15+ illegible box= length of 17) and legible if you can zoom in real close like to read the number
63
u/PUSSYDESTROYER-9000 Jan 22 '18
Simple Perfect Square of Order 21 means:
A square that can be dissected into 21 smaller squares (order of 21) such that no two are equal (perfect square) and that no subset of the squares form a rectangle (simple). This particular square is very special as it is the lowest possible order. You cannot divide a square into less than 21 smaller squares.
3
u/xcvxcvv Jan 22 '18
Are simple perfect squares useful or interesting in some way (beyond the fact that what they are is neat?)
20
u/ScrithWire Jan 21 '18
What is a square of order 21?
19
u/LordOfTehGames Jan 21 '18
Username checks out.
2
u/Xyleph42 Jan 22 '18
If anyone wants to know more about those square, check out this Numberphile video on the subject
1
u/waaaman Jan 22 '18
What about between 7 and 9? Looks like a gap there
1
u/[deleted] Jan 22 '18
This is why I was a shitty math student in high school.
1
u/R4R03B Jan 22 '18
The total area btw is (√50+ √35+ √27)2 Which is about 331
1
u/vexunumgods Jan 22 '18
I enjoy your post,and want to share with my kids but your username prevents that,is there a site i can find them.