Recreational Math

by Bernie Freidin © 1999-2000


These grids have the following property: take any four points in a rectangle formation {A, B, C, D} and they will satisfy the equation A2+D2 = B2+C2. So for example, in the first grid below, the points {63, 1224, 393, 1284} form a rectangle, and therefore 632+12842=12242+3932=1652625. I used to be very interested in finding large grids with this property - to this end I wrote some very complex programs to search for smaller grids and then "stitch them together" into larger grids. In the examples below, I have one 8 by 8 grid with 42 points on it (out of a total of 64), and one complete 6 by 4 grid with all 24 points - these were my master creations. I also found a complete grid which was 3 by 81, and I could have extended it even further, but the values were numbers hundreds of digits long.

Another recreational math interest of mine has been square-packing. Here are a few examples I did: example1, example2.

D=14, N=42


   8 ---- 63 ---- 216 -----+-------+----- 603 -- 1224 -----+
   |       |       |       |       |       |       |       |
   |       |       |       |       |       |       |       |
  172 --- 183 --- 276 --- 417 --- 609 --- 627 -- 1236 -- 2156
   |       |       |       |       |       |       |       |
   |       |       |       |       |       |       |       |
  388 --- 393 --- 444 --- 543 -----+----- 717 -- 1284 -----+
   |       |       |       |       |       |       |       |
   |       |       |       |       |       |       |       |
  712 -----+----- 744 --- 807 --- 921 --- 933 -- 1416 -- 2264
   |       |       |       |       |       |       |       |
   |       |       |       |       |       |       |       |
   +-------+---- 1272 -----+---- 1383 -----+---- 1752 -- 2488
   |       |       |       |       |       |       |       |
   |       |       |       |       |       |       |       |
 1448 -----+---- 1464 -- 1497 -----+-------+---- 1896 -----+
   |       |       |       |       |       |       |       |
   |       |       |       |       |       |       |       |
 1952 -- 1953 -----+-------+-------+---- 2043 -- 2304 -----+
   |       |       |       |       |       |       |       |
   |       |       |       |       |       |       |       |
   +-------+---- 3252 -----+---- 3297 -----+---- 3468 -- 3892


D=8, N=24


 2988 ---- 4356 ---- 5787 ---- 11164 --- 17046 --- 23948
   |         |         |         |         |         |
   |         |         |         |         |         |
   |         |         |         |         |         |
 5052 ---- 5964 ---- 7077 ---- 11884 --- 17526 --- 24292
   |         |         |         |         |         |
   |         |         |         |         |         |
   |         |         |         |         |         |
 12108 --- 12516 --- 13083 --- 16196 --- 20694 --- 26668
   |         |         |         |         |         |
   |         |         |         |         |         |
   |         |         |         |         |         |
 34812 --- 34956 --- 35163 --- 36436 --- 38646 --- 42148

This page © Bernie Freidin, 2000.
Last updated April 4th, 2000.