From: utzoo!decvax!pur-ee!uiucdcs!grunwald Newsgroups: net.math Title: Re: Math Discoveries as a child - (nf) Article-I.D.: uiucdcs.293 Posted: Fri Jul 30 04:26:22 1982 Received: Sat Jul 31 02:37:35 1982 #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.