Suppose you have four squares of stained glass, all of different colours, and you wish to make a 2x2 square stained glass window. How many different windows are possible?
(Note that any pattern may be rotated 180°, flipped vertically, or flipped horizontally. You should
count all the possible resulting patterns as the same window.)
To find the number of different windows possible, we need to consider symmetry and rotation.
Let's break down the problem:
Step 1: Count the number of ways we can arrange the squares without considering symmetry or rotation.
Since we have four squares of different colors, we have four choices for the first square, three choices for the second square, two choices for the third square, and only one choice for the last square. Therefore, there are 4 * 3 * 2 * 1 = 24 possible arrangements.
Step 2: Account for the symmetry and rotations.
For a 2x2 square stained glass window, let's consider the different types of symmetries and rotations:
a) No symmetry or rotation
In this case, there is only one possibility.
b) 180° rotation
To calculate the number of possibilities here, we need to divide the number of arrangements we counted in step 1 by 2 since each arrangement has a symmetric counterpart when rotated 180°.
24 / 2 = 12 possibilities.
c) Vertical flipping
Similarly, when we flip the arrangement vertically, we get another symmetric arrangement. So we need to divide the result from step 2b by 2.
12 / 2 = 6 possibilities.
d) Horizontal flipping
Just like the vertical flipping, when we flip the arrangement horizontally, we get another symmetric arrangement. So we divide the result from step 2c by 2.
6 / 2 = 3 possibilities.
Step 3: Add up all the possibilities.
Now we add up the results from step 2a, step 2b, step 2c, and step 2d:
1 + 12 + 6 + 3 = 22
Therefore, there are 22 different possible stained glass windows for a 2x2 square using four squares of different colors, accounting for the given symmetry and rotations.