Problem:
Square in the coordinate plane has vertices at the points , and . Consider the following four transformations:
, a rotation of counterclockwise around the origin;
, a rotation of clockwise around the origin;
, a reflection across the -axis; and
, a reflection across the -axis.
Each of these transformations maps the square onto itself, but the positions of the labeled vertices will change. For example, applying and then would send the vertex at to and would send the vertex at to itself. How many sequences of transformations chosen from will send all of the labeled vertices back to their original positions? (For example, is one sequence of transformations that will send the vertices back to their original positions.)
Answer Choices:
A.
B.
C.
D.
E. 2^
Solution:
Let denote counterclockwise/starting orientation and denote clockwise orientation. Let , and 4 denote which quadrant is in. Realize that from any odd quadrant and any orientation, the transformations result in some permutation of .
The same goes that from any even quadrant and any orientation, the transformations result in some permutation of . We start our first moves by doing whatever we want, choices each time. Since is odd, we must end up on an even quadrant. As said above, we know that exactly one of the four transformations will give us , and we must use that transformation.
Thus, the answer is
Notice that any pair of transformations either swaps the and -coordinates, negates the and -coordinates, swaps and negates the and coordinates, or leaves the original unchanged. Furthermore, notice that for each of these results, if we apply another pair of transformations, one of these results will happen again, and with equal probability. Therefore, no matter what state we are in after we apply the first pairs of transformations, there is a chance the last pair of transformations will return the figure to its original position.
Therefore, the answer is
The problems on this page are the property of the MAA's American Mathematics Competitions