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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 

    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   
Associated Topics:
High School Geometry
High School Puzzles
High School Triangles and Other Polygons

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