# Dissect a square into 3:1 rectangles

I am being known for Geometrical and Topological Puzzles, So continuing with the trend here is another one.

Completely dissect a square into the lowest number of different sized rectangles with integer edges and a length to width ratio of 3 to 1.

EDIT:

Since people are having a hard time. I will add the solution here. Let your pointer do the work.

• Does the square have to be a particular size? – Beastly Gerbil Jan 2 '18 at 15:03
• @BeastlyGerbil No, one that gives the lowest number 3:1 rectangles. – prog_SAHIL Jan 2 '18 at 15:04
• This can be solved by inspection/intuition. – Jeff Zeitlin Jan 2 '18 at 15:42
• Re the edit: I don't see a solution there...? – Ankoganit Jan 4 '18 at 7:14
• @Ankoganit Look deep(deeper.) – prog_SAHIL Jan 4 '18 at 7:16

Here's the best I found so far with a square size of 96 as given by the image posted by OP as a solution. Twelve rectangles. To prove it is the smallest requires logic rather than my brute-force computer approach, since without some logical deductions I would have to search arbitrarily large squares with a huge list of sets of 11 or fewer rectangles which have the correct area. If the posted image which appeared to be a plain white square gave the answer then this is either superfluous or not optimal.

NB the smallest rectangle which is tiny has a '1' in it which divides it two neatly, don't mistake it for two small rectangles...

• I initially didn't see that small rectangle as 1x3 with a 1 in it, I saw it as two small rectangles too small to put a number in. Rotating the picture 90 degrees might make it slightly more presentable. – hvd Jan 6 '18 at 8:14
• Yeah I struggled with that. I'll just add a comment in the description... – theonetruepath Jan 6 '18 at 8:15

With a little research I found...

... a solution with 23 rectangles.

The numbers are the lengths of the short, horizontal, sides. Vertical sides are, of course, three times as long.

I don't know if this is optimal. I think there could be a solution with fewer rectangles that uses both orientations - i.e. 1x3 and 3x1 rectangles.

Credit: I adapted my image from a 1x2 rectangle dissection found at squaring.net

Method

I looked for a perfect square dissection tiling a 1x2 rectangle, then added another square on the end to make a 1x3 rectangle and stretched this to a square, thus stretching all the original sub-squares to be 1x3 rectangles.

• This is very near to the optimal solution but not it! – prog_SAHIL Jan 3 '18 at 7:51

A rectangle of dimensions $k\times3k$ has area $3k^2$. So first we need to solve the following sub-problem:

Find distinct natural numbers $a_1,a_2,\dots,a_n$ and $N$ such that $3(a_1^2+a_2^2+\dots+a_n^2)=N^2$.

Every square is congruent to either 0 or 1 modulo 3. So in order that $a_1^2+a_2^2+\dots+a_n^2$ is a multiple of 3, the number of $a_n$ which are not multiples of 3 must itself be a multiple of 3.

Every square is congruent to either 0, 1, or 4 modulo 8. So in order that $3(a_1^2+a_2^2+\dots+a_n^2)$ is a square modulo 8, the number of odd $a_n$ must be congruent to 0, 3, or 4 modulo 8.

I found one solution: $n=4,a_1=1,a_2=3,a_3=4,a_4=7$, giving $(1\times3)+(3\times9)+(4\times12)+(7\times21)=15^2$. Unfortunately, we can't fit a $7\times21$ rectangle into a $15\times15$ square, so this won't solve the main problem.

• I think it's relevant to specify that N is a multiple of 3 since N^2 is one. This can help. – Alix Eisenhardt Jan 2 '18 at 16:12
• I wrote some code to find sets of N kx3k rectangles that sum to a square S*S. Turns out that it's 'open ended' in the sense that you need to search arbitrarily large k even for small N. So this is never going to be trivial. If I limit it to k=75 it makes good progress, nothing below N=8. I also started my tiling program searching small S with all k that will fit, it's searching S=62 now, there's nothing smaller. – theonetruepath Jan 3 '18 at 7:33
• I fixed a bug in my program that finds sets of rectangles. There are lots of rectangle sets to consider, so my tiling program is not just wasting CPU cycles. And there are plenty of sets with even as few as 4 rectangles, although you need quite a bit more than four to tile a square. – theonetruepath Jan 4 '18 at 6:24

It's not too hard to find a (most likely suboptimal) solution with

65

rectangles, by combining several solutions to the

problem. Unfortunately, the resulting square has an edge size of

856240

meaning it won't be easy to draw a pixel-perfect image of it. However, we can draw each part separately and then combine them using image magic. The result is here (each rectangle contains two numbers - the size within each part and the size within the whole solution):

The smallest rectangle is blue-1x1 (6160x18480) and the largest rectangle is blue-80x80 (492800x1478400)

To see why there aren't duplicate rectangles across parts, we note that

the edge sizes are almost mutually coprime, meaning that the only the full squares would scale up to the same size. The "almost" part refers to the first two squares having a GCD of 2, which allows an additional collision - 56x56 from the first part would scale up to the same size as 55x55 from the second part. Neither exists in their respective pairs.

The parts are:

112A AJD 1978 from http://squaring.net/sq/ss/spss/o21/spsso21.pdf
110A AJD 1978 from http://squaring.net/sq/ss/spss/o22/spsso22.pdf
139A AJD 1990 from http://squaring.net/sq/ss/spss/o22/spsso22.pdf
The integer in name refers to the edge size. The number of rectangles is in the URL.