Why Not Five Colors, All Touching?
Date: 04/30/2003 at 03:58:01 From: Joanna Subject: Five colors, all touching? My 4th-grade class was instructed to use 5 colors for the problem I'm about to describe. With 5 colors we were supposed to create a situation in which all colors were touching each other color simultaneously. When two of the colors were touching at points or corners, that didn't count as "touching." For example, you could not evenly slice up a pentagon - assign each 'piece' a color, and therefore say you've solved the problem. So when I say that each color has to touch all the other colors: - Blue, Green, Yellow, Orange, Red - B has to touch G,Y,O,R, and G must be touching B,Y,O,R, etc. It must be 5 "blobs" or whatever shape you need to achieve this. It can't be more than 5, though. I am able to create a situation in which 4 of the colors are touching the other colors. It comes down to one final color being isolated, or surrounded, by the other colors and therefore not allowing one final color to touch it to complete the promblem. The work is totally visual. I have gone through several sheets of paper sketching out different scenarios and I can't conjure up the solution. One color always remains cut off.
Date: 04/30/2003 at 20:29:18 From: Doctor Ian Subject: Re: Five colors, all touching? Hi Joanna, This is easier to think about if we use graphs, rather than shapes. Suppose we make a shape like +---+---+ | a | b | +-------+ | c | +-------+ We can represent each region as a node in a graph. An arc between two regions means that there is a border between the regions: a-------b \ / \ / \ / c Now, suppose we add a fourth region, which touches all the others. It might look like this, +--------+ | d | +---+---+ | | a | b | | +-------+ | | c |--+ +-------+ or it might look like something else, but on our graph, we really have only two choices for where to put d, so that it can connect to all the other regions. It can go inside, a-----------------b \ . . / \ . . / \ d / \ . / \ / \ . / \ / \ / c or it can go outside, . . . . . . d . . . . . . a-------b . \ / . \ / . \ / . c . . And in fact, these are really the same case: one in the middle, connected to the other three. (Just switch the labels 'b' and 'd', and one graph turns into the other.) Now, when we try to add a fifth region, what will happen in our graph? If it goes outside the triangle, a-----------------b \ . . / \ . . / \ d / \ . / \ / e \ . / \ / \ / c then we can't get to the region that's surrounded without crossing an existing border (i.e., breaking up one of our regions). But if we go inside the triangle, a-----------------b \ . e . / \ . . / \ d / \ . / \ / \ . / \ / \ / c then we've surrounded the new region with three regions, and cut off the fourth. What you're always going to find is that the fourth region you add cuts off one of the others. So it's not surprising that you haven't been able to find the arrangement you're looking for. What would be surprising is if you _did_ find it! See the Dr. Math archives (and you can find more examples using the Dr. Math searcher at http://mathforum.org/library/drmath/mathgrepform.html to look for the keywords four color theorem ): Four-Color Theorem http://mathforum.org/library/drmath/view/57256.html The Four Color Map problem http://mathforum.org/library/drmath/view/52466.html Four Color Map Problem http://mathforum.org/library/drmath/view/57231.html - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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