11*11 = 121 = 10*2 + 6*1 + 7*4 + 9*3 + 8*5
11*11 = 121 = 10*5 + 9*1 + 7*4 + 6*3 + 8*2
13*13 = 169 = 10*5 + 7*3 + 2*1 + 9*8 + 6*4
13*13 = 169 = 10*6 + 8*3 + 2*1 + 9*7 + 5*4
Here's a diagram showing the geometry involved.
Here's a sample run of tile.tc, a Tiny-C program that figures out all
possible solutions of the tiling problem.
The program tiles a square with 5 rectangles.
It contains a nested series of loops which exhaust all combinations of
the numbers 1 thru 10 which contain each number exactly once and satisfy
inequalities implied by geometry.
For example: n9 < minimum n5,n1
The innermost loop uses a nested if which uses a function which returns
true when its arguments satisfy a certain equality. The equalities are
implied by geometry.
For example: n5 = n9+n2
When all requirements are met, the 10 numbers are displayed.
n8 n7
+-----+-------------------------------------+
+ + +
+ + + n6
n1 + + +
+ +------------------------------+------+
+ + n10 + +
+ + + +
+ + n9 + +
+ + + +
+-----+------------------------------+ + n5
+ + +
+ + +
n2 + + +
+ + +
+------------------------------------+------+
n3 n4
L:\hdrive.3.16.20>tc tile.tc
*** TINY-C VERSION 1.0, COPYRIGHT 1977, T A GIBSON ***
This C version copyright 2017, T A Gibson
tile.tc - tct - 4/13/20
n1=10 n2=1 n3=6 n4=5 n5=8 n6=3 n7=9 n8=2 n9=7 n10=4
n1=10 n2=1 n3=9 n4=2 n5=8 n6=3 n7=6 n8=5 n9=7 n10=4
n1=10 n2=3 n3=7 n4=6 n5=4 n6=9 n7=8 n8=5 n9=1 n10=2
n1=10 n2=3 n3=8 n4=5 n5=4 n6=9 n7=7 n8=6 n9=1 n10=2
Here's a link to the source code:
http://primepuzzle.com/tc/tile.tc