Tiling squares with distinct Number-Rectangles
Hello Math-Fun,
Say every integer > 9 can produce an h x w
rectangle (height first, then width).
We would then have:
10 = 1 x 0 (no rectangle, nothing that we will use here)
11 = 1 x 1 (the unit cell)
12 = 1 x 2 (see below)
13 = 1 x 3 (see below)
...
21 = 2 x 1 (see below)
31 = 3 x 1 (see below)
...
100 = 10 x 0 (no rectangle, nothing that we will use here)
101 = 10 x 1 (no other choice, we don't accept any
height or width starting with zero)
...
111 = 1 x 11 or 11 x 1
112 = 1 x 12 or 11 x 2
...
2023 = 20 x 23 or 202 x 3
...
We will use "NR" for the integers of the above left
column (NR stands for Number-Rectangle)
Now we want two things:
1) to tile an n x n square with distinct shapes of NRs
2) the sum of the NRs involved in the tiling is itself
a square.
Examples
+---+
| | the 11-NR
+---+
+---+---+
| | | the 12-NR
+---+---+
+---+---+---+
| | | | the 13-NR
+---+---+---+
+---+
| |
+---+ the 21-NR
| |
+---+
+---+
| |
+---+
| | the 31-NR
+---+
| |
+---+
+---+---+---+
| 11| |
+---+ +
| | 32 | a sound 3 x 3 square as 11 + 21 + 32 = 64 = 8^2
| 21| +
| | |
+---+---+---+
+---+---+---+
| 12 | 11|
+-------+---+ the above square, tilted here 90 degrees
| | clockwise, is NOT a sound 3 x 3 square
+ 23 + (as 12 + 11 + 23 = 46 is NOT a square)
| |
+---+---+---+
| | the 11-NR
+---+
+---+---+
| | | the 12-NR
+---+---+
+---+---+---+
| | | | the 13-NR
+---+---+---+
+---+
| |
+---+ the 21-NR
| |
+---+
+---+
| |
+---+
| | the 31-NR
+---+
| |
+---+
+---+---+---+
| 11| |
+---+ +
| | 32 | a sound 3 x 3 square as 11 + 21 + 32 = 64 = 8^2
| 21| +
| | |
+---+---+---+
+---+---+---+
| 12 | 11|
+-------+---+ the above square, tilted here 90 degrees
| | clockwise, is NOT a sound 3 x 3 square
+ 23 + (as 12 + 11 + 23 = 46 is NOT a square)
| |
+---+---+---+
+---+---+---+
| 13 |
+---+---+---+
| | another sound 3
x 3 square as 13 + 23 = 36 = 6^2
+ 23 + (found
by Scott Shannon)
| |
+---+---+---+
+---+---+---+---+
| 11| 12 | |
+---+---+---+ +
|
13 | |
+---+---+---+ 41| a sound 4 x 4 square as 11 + 12 + 13 + 23 + 41 = 100 = 10^2
|
| |
+
23 + +
|
| |
+---+---+---+---+
Question
Are there more such sound n x n squares?
Best,
É.
Maximilian H. was quick to answer, as usual:
> Eric,
are you sure the painting on your blog page is not turned by 90° ?
;-)
Yes there are many such "sound squares".
I think it's quite a challenge to write a program for the general case,
but considering just decompositions in 3 rectangles is very easy.
I get this list:
3 x 2 + 1 x 1 + 2 x 1 (= 8^2),
1 x 5 + 4 x 1 + 4 x 4 (= 10^2),
1 x 5 + 4 x 2 + 4 x 3 (= 10^2),
1 x 7 + 6 x 1 + 6 x 6 (= 12^2),
1 x 7 + 6 x 2 + 6 x 5 (= 12^2),
1 x 7 + 6 x 3 + 6 x 4 (= 12^2),
8 x 7 + 1 x 1 + 7 x 1 (= 13^2),
8 x 7 + 2 x 1 + 6 x 1 (= 13^2),
8 x 7 + 3 x 1 + 5 x 1 (= 13^2),
9 x 2 + 1 x 7 + 8 x 7 (= 14^2),
9 x 2 + 2 x 7 + 7 x 7 (= 14^2),
9 x 2 + 3 x 7 + 6 x 7 (= 14^2),
9 x 2 + 4 x 7 + 5 x 7 (= 14^2),
12 x 8 + 1 x 4 + 11 x 4 (= 16^2),
12 x 8 + 2 x 4 + 10 x 4 (= 16^2),
12 x 8 + 3 x 4 + 9 x 4 (= 16^2),
12 x 8 + 4 x 4 + 8 x 4 (= 16^2),
12 x 8 + 5 x 4 + 7 x 4 (= 16^2),
13 x 12 + 1 x 1 + 12 x 1 (= 38^2),
13 x 12 + 2 x 1 + 11 x 1 (= 38^2),
13 x 12 + 3 x 1 + 10 x 1 (= 38^2),
13 x 12 + 4 x 1 + 9 x 1 (= 38^2),
13 x 12 + 5 x 1 + 8 x 1 (= 38^2),
13 x 12 + 6 x 1 + 7 x 1 (= 38^2),
8 x 15 + 7 x 1 + 7 x 14 (= 40^2),
8 x 15 + 7 x 2 + 7 x 13 (= 40^2),
8 x 15 + 7 x 3 + 7 x 12 (= 40^2),
8 x 15 + 7 x 4 + 7 x 11 (= 40^2),
8 x 15 + 7 x 5 + 7 x 10 (= 40^2),
15 x 6 + 1 x 9 + 14 x 9 (= 18^2),
15 x 6 + 2 x 9 + 13 x 9 (= 18^2),
15 x 6 + 3 x 9 + 12 x 9 (= 18^2),
15 x 6 + 4 x 9 + 11 x 9 (= 18^2),
15 x 6 + 5 x 9 + 10 x 9 (= 18^2),
15 x 6 + 6 x 9 + 9 x 9 (= 18^2),
15 x 6 + 7 x 9 + 8 x 9 (= 18^2),
etc.
Best wishes,
-M.
(PARI)
{dec3(n)=my( cc(x,y)=eval(Str(x,y)), p(a,b,c,d,e,f,g)=
printf("%d x %d + %d x %d + %d x %d (= %d^2), ",a,b,c,d,e,f,g));
for(x=1,n-1, my( HW=cc(n-x,n), WH=cc(n,n-x), s ); for( a=1, (n-1)\2,
issquare( HW + cc(x,a) + cc(x,n-a), &s ) && p( n-x,n, x,a, x,n-a, s);
issquare( WH + cc(a,x) + cc(n-a,x), &s ) && p( n,n-x, a,x, n-a,x, s)))}
for(x=1,n-1, my( HW=cc(n-x,n), WH=cc(n,n-x), s ); for( a=1, (n-1)\2,
issquare( HW + cc(x,a) + cc(x,n-a), &s ) && p( n-x,n, x,a, x,n-a, s);
issquare( WH + cc(a,x) + cc(n-a,x), &s ) && p( n,n-x, a,x, n-a,x, s)))}
+---+---+---+---+---+
| 15 |
+---+---+---+---+---+
| | |
+ + +
| | |
+ 41| 44 + sum = 15 + 41 + 44 = 100 = 10^2
| | | + + + | | | +---+---+---+---+---+ +---+---+---+---+---+ | 15 | +---+---+---+---+---+ | | | + + + | | | + 42 + 43 + sum = 15 + 42 + 43 = 100 = 10^2 | | | + + + | | | +---+---+---+---+---+ +---+---+---+---+---+---+---+ | 17 | +---+---+---+---+---+---+---+ | | | + + + | | | + + + | | | + 61| 66 + sum = 17 + 61 + 66 = 144 = 12^2 | | | + + + | | | + + + | | | +---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+ | 17 | +---+---+---+---+---+---+---+ | | | + + + | | | + + + | | | + 62 | 65 + sum = 17 + 62 + 65 = 144 = 12^2 | | | + + + | | | + + + | | | +---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+ | 17 | +---+---+---+---+---+---+---+ | | | + + + | | | + + + | | | + 63 | 64 + sum = 17 + 63 + 64 = 144 = 12^2 | | | + + + | | | + + + | | | +---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+---+ | | 11| + +---+ | | | + + + | | | + + + | | | + 87 + 71| sum = 87 + 11 + 71 = 169 = 13^2 | | | + + + | | | + + + | | | + + + | | | +---+---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+---+ | | | + + 21| | | | + +---+ | | | + + + | | | + + + | 87 | 61| sum = 87 + 21 + 61 = 169 = 13^2
| 15 |
+---+---+---+---+---+
| | |
+ + +
| | |
+ 41| 44 + sum = 15 + 41 + 44 = 100 = 10^2
| | | + + + | | | +---+---+---+---+---+ +---+---+---+---+---+ | 15 | +---+---+---+---+---+ | | | + + + | | | + 42 + 43 + sum = 15 + 42 + 43 = 100 = 10^2 | | | + + + | | | +---+---+---+---+---+ +---+---+---+---+---+---+---+ | 17 | +---+---+---+---+---+---+---+ | | | + + + | | | + + + | | | + 61| 66 + sum = 17 + 61 + 66 = 144 = 12^2 | | | + + + | | | + + + | | | +---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+ | 17 | +---+---+---+---+---+---+---+ | | | + + + | | | + + + | | | + 62 | 65 + sum = 17 + 62 + 65 = 144 = 12^2 | | | + + + | | | + + + | | | +---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+ | 17 | +---+---+---+---+---+---+---+ | | | + + + | | | + + + | | | + 63 | 64 + sum = 17 + 63 + 64 = 144 = 12^2 | | | + + + | | | + + + | | | +---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+---+ | | 11| + +---+ | | | + + + | | | + + + | | | + 87 + 71| sum = 87 + 11 + 71 = 169 = 13^2 | | | + + + | | | + + + | | | + + + | | | +---+---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+---+ | | | + + 21| | | | + +---+ | | | + + + | | | + + + | 87 | 61| sum = 87 + 21 + 61 = 169 = 13^2
+ + +
| | |
+ + +
| | |
+ + +
| | |
+---+---+---+---+---+---+---+---+
+---+---+---+---+---+---+---+---+
| | |
+ + +
| | 31|
+ + +
| | |
+ +---+
| | |
+ + |
| 87 | 51| sum = 87 + 31 + 41 = 169 = 13^2
+ + +
| | |
+ + +
| | |
+ + +
| | |
+---+---+---+---+---+---+---+---+
+---+---+---+---+---+---+---+---+---+
| | |
+ + +
| | |
+ + +
| | |
+ + +
| | |
+ + +
| 92 | 87 | sum = 92 + 87 + 17 = 196 = 14^2
+ + +
| | |
+ + +
| | |
+ + +
| | |
+ +---+---+---+---+---+---+---+
| | 17 |
+---+---+---+---+---+---+---+---+---+
+---+---+---+---+---+---+---+---+---+
| | |
+ + +
| | |
+ + +
| | |
+ + +
| | |
+ + +
| 92 | 77 | sum = 92 + 77 + 27 = 196 = 14^2
+ + +
| | |
+ + +
| | |
+ +---+---+---+---+---+---+---+
| | |
+ + 27 +
| | |
+---+---+---+---+---+---+---+---+---+
+---+---+---+---+---+---+---+---+---+
| | |
+ + +
| | |
+ + +
| | |
+ + +
| | |
+ + +
| 92 | 67 | sum = 92 + 67 + 37 = 196 = 14^2
+ + +
| | |
+ +---+---+---+---+---+---+---+
| | |
+ + +
| | 37 |
+ + +
| | |
+---+---+---+---+---+---+---+---+---+
+---+---+---+---+---+---+---+---+---+
| | |
+ + +
| | |
+ + +
| | |
+ + +
| | |
+ + +
| 92 | 57 | sum = 92 + 57 + 47 = 196 = 14^2
+ +---+---+---+---+---+---+---+
| | |
+ + +
| | |
+ + 47 +
| | |
+ + +
| | |
+---+---+---+---+---+---+---+---+---+
+---+---+---+---+---+---+---+---+---+---+---+---+
| | |
+ + +
| | |
+ + +
| | |
+ + +
| | |
+ + +
| | |
+ + +
| | |
+ 128 + 114 +
| | |
+ + +
| | |
+ + +
| | |
+ + +
| | |
+ + +
| | |
+ +---+---+---+---+
| | 14 |
+---+---+---+---+---+---+---+---+---+---+---+---+
sum = 128 + 114 + 14 = 256 = 16^2
Next sums (and according 12 x 12 squares):
128 + 104 + 24 = 256 = 16^2
128 + 94 + 34 = 256 = 16^2
128 + 84 + 44 = 256 = 16^2
128 + 74 + 54 = 256 = 16^2
I think it's quite a challenge to write a program for the general case,
RépondreSupprimerbut considering just decompositions in 3 rectangles is very easy.
I get this list:
3 x 2 + 1 x 1 + 2 x 1 (= 8^2),
1 x 5 + 4 x 1 + 4 x 4 (= 10^2), 1 x 5 + 4 x 2 + 4 x 3 (= 10^2),
1 x 7 + 6 x 1 + 6 x 6 (= 12^2), 1 x 7 + 6 x 2 + 6 x 5 (= 12^2), 1 x 7 + 6 x 3 + 6 x 4 (= 12^2),
8 x 7 + 1 x 1 + 7 x 1 (= 13^2), 8 x 7 + 2 x 1 + 6 x 1 (= 13^2), 8 x 7 + 3 x 1 + 5 x 1 (= 13^2), 8 x 7 + 4 x 1 + 4 x 1 (= 13^2),
9 x 2 + 1 x 7 + 8 x 7 (= 14^2), 9 x 2 + 2 x 7 + 7 x 7 (= 14^2), 9 x 2 + 3 x 7 + 6 x 7 (= 14^2), 9 x 2 + 4 x 7 + 5 x 7 (= 14^2),
12 x 8 + 1 x 4 + 11 x 4 (= 16^2), 12 x 8 + 2 x 4 + 10 x 4 (= 16^2), 12 x 8 + 3 x 4 + 9 x 4 (= 16^2),
12 x 8 + 4 x 4 + 8 x 4 (= 16^2), 12 x 8 + 5 x 4 + 7 x 4 (= 16^2), 12 x 8 + 6 x 4 + 6 x 4 (= 16^2), ...
Pasting this PARI code into their online interpreter you can get more:
{dec3(n)=my( cc(x,y)=eval(Str(x,y)), p(a,b,c,d,e,f,g)=
printf("%d x %d + %d x %d + %d x %d (= %d^2), ",a,b,c,d,e,f,g));
for(x=1,n-1, my( HW=cc(n-x,n), WH=cc(n,n-x), s ); for( a=1, n\2,
issquare( HW + cc(x,a) + cc(x,n-a), &s ) && p(n-x,n, x,a, x,n-a, s);
issquare( WH + cc(a,x) + cc(n-a,x), &s ) && p( n,n-x, a,x, n-a,x, s)))}
/* then do: */ for(n=2,12,dec3(n))
oops, above program lists duplicates, replace n\2 by (n-1)\2 or n\/2-1 to fix this bug.
RépondreSupprimerNote, n=15 is the smallest size with a "horizontal" and distinct "vertical" partition, and distinct square sums.