Math Forum / Mathematics / Undergraduate Math / April 2005

Do you mean congruent triangles? They aren't equilateral.

All possible squares?? What does that mean?

Too much to hope for without a logical statement of the problem.
Sorry.

--Lynn

You need to specify some ground rules.

Suppose one square is obtained from another by
rotation through 90 degrees; is it the same or different?
(Reading the colors clockwise from the right,
is BBYR the same as BYRB?)

Suppose one square is obtained from another by
reflection; is it the same or different?
(Is BBYR the same as BBRY?)

Don Coppersmith

Notwithstanding the poor statement of the problem, which others have succintly critqued, Let me give it a try.

First, I assume you meant to say "4 isoceles triangles", not "4 equilateral triangles".  Next, lets say that two triangles are colored red, one blue, and one yellow.  Finally, let us say that the only thing that distinguishes a square is its coloration and not its orientation.  Reflections might distinguish a square as long as it is not possible to achieve the same coloration with a rotation.

As for the "how many squares" questions there are two types of squares to consider. Let Type 1 be a square comprised of all four triangles, in an arrangement like that of the original section that produced the four triangles.  Let Type 2 be a square comprised of only two of the four triangles, form by adjoining their long edges.  Then there are 3 Type 1 squares.  They can be listed by calling out the color of the constituent triangles in counter-clockwise order:

(r,r,b,y), (r,r,y,b), (r,b,r,y)

Note that cyclic permuation are equivalent to a simple rotation of the square and hence, by the ground rule I filled in for the OP, do not produce unique squares.  Note that reflecting (r,r,b,y) can produce (r,r,y,b), but not rotation can transform one into the other.  So, by the ground rules I filled in for the OP, these two squares are unique.

There are 4 Type 2 squares.  They are listed by the color of the constituent triables:

(r,r), (r,b), (r,y), (b,y)

Again, cyclic permutations do not produce unique squares.

In all there are 7 unique squares that can be formed from the four triangle of three colors.

A rectangle can be formed from two Type 2 squares in such a way that where the squares touch they have the same colored edge, namely

(b,r)(r,y)

(I hope you can make sense of my shorthand.)

The only rectangle with all outer edges the same color is the Type 2 square (r,r).  Here I use the elementary fact that every square is a rectangle.

- MO