burton (07/28/82)
One of the neat discoveries I made in early high school was that the difference between any two consecutive perfect squares was always an odd number. When I showed this to my teacher, she encouraged me to prove it; it was an obvious proof, but it developed in me a taste for proving 'obvious' math relations. You can work the proof out for yourself, but one of the neat relations that pops out of it is that to get the next perfect square, you multiply the square root of the current perfect square by 2, add 1, then add it to the current perfect square; i.e., if n is the current perfect square, then n +2*sqrt(n) + 1 = next square in the series Again, nothing spectacular, but really neat for a junior high schooler! Doug Burton ihps3!inuxc!burton
grunwald (07/30/82)
#R:inuxc:-26500:uiucdcs:10300003:000:1101 uiucdcs!grunwald Jul 30 03:30:00 1982 Actually, this is one of my favorite problems to show people (mainly people who are not really turned on by math) the concepts of logical types and mappings or isomorphisms between two logically equivilent problems. To show you what I mean, let me present two representations of the sum: 1 = 1 1+3 = 4 1+3+5 = 9 1+3+5+7 = 16 Now, look at it this way: area of 1 = 1 area of 33 13 = 4 area of 555 335 135 = 9 area of 7777 5557 3357 1357 = 16 Note that there are five fives, seven sevens etc -- they form the new "corner" of the square. The digits that I use to construct the squares are just there to show that there are 5 5's 7 7's, etc. Showing the problem in this light make the proof much easier to see because it maps the problem to a more visual medium. I found this in Gregory Batesons "Mind and Nature", where he talks at great length about the problem of confusing "logical types" (not like computer logical types -- it's a super set of the same concept) and how this causes lots of problems with people understanding the world. A very good book on human understanding.