Twenty Quadrilaterals from Nine Dots
Date: 04/04/99 at 20:44:20 From: Mistie Subject: 20 quadrilaterals I have to come up with 20 quadrilaterals using 9 dots, and they can't be congruent. HELP!
Date: 04/05/99 at 12:19:42 From: Doctor Peterson Subject: Re: 20 quadrilaterals Hi, Mistie. I'll assume the nine dots you have to use for vertices are arranged in a square, and I'll label them like this: 1 2 3 4 5 6 7 8 9 Let's see if we can find an orderly way to list quadrilaterals. We'll want to avoid duplicates (congruent quads); and we can make a rule that we'll always list the vertices in a clockwise order. If the quadrilateral contains a corner (1, 3, 7, or 9), we can put it at 1, so I don't have to try any quads starting at 3, 7, or 9. If it doesn't contain any corners, it consists only of 4 out of the 5 vertices 2, 4, 5, 6, and 8, so we'll just have to pick one to leave out. So our list of quads with no corner dots will look like this: 2 6 8 4 (leaving out the center) 2 5 8 4 (leaving out an edge) Now we can work on different possibilities for quads containing 1. We might have either 1, 2, 3, or 4 corner dots in it; that gives these possibilities: 1 ? ? ? (only one corner) 1 3 ? ? (2 adjacent corners) 1 9 ? ? (2 opposite corners) 1 3 9 ? (3 corners) 1 3 9 7 (4 corners) What possibilities are there for the 3-corner case? The fourth vertex can be any of 2, 4, 5, and 8, but some of those will not make quads (because two vertices are collinear), and others will be the same shape. Only one shape, 1 3 9 8 is really a quad! Keep thinking in this way, and you should be able to make a list. So far I've got 4 actual quads, and the beginnings of many more. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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