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### Twenty Quadrilaterals from Nine Dots

```
Date: 04/04/99 at 20:44:20
From: Mistie

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

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

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

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