Difference between revisions of "2020 AMC 10B Problems/Problem 23"
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Let (+) denote counterclockwise/starting orientation and (-) denote clockwise orientation. | Let (+) denote counterclockwise/starting orientation and (-) denote clockwise orientation. | ||
Let 1,2,3, and 4 denote which quadrant A is in. | Let 1,2,3, and 4 denote which quadrant A is in. | ||
| + | |||
Realize that from any odd quadrant and any orientation, the 4 transformations result in some permutation of <math>(2+, 2-, 4+, 4-)</math>. | Realize that from any odd quadrant and any orientation, the 4 transformations result in some permutation of <math>(2+, 2-, 4+, 4-)</math>. | ||
| + | |||
The same goes that from any even quadrant and any orientation, the 4 transformations result in some permutation of <math>(1+, 1-, 3+, 3-)</math>. | The same goes that from any even quadrant and any orientation, the 4 transformations result in some permutation of <math>(1+, 1-, 3+, 3-)</math>. | ||
| − | We start our first 19 moves by doing whatever we want, 4 choices each time. Since 19 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 <math>(1+)</math>, and we must use that transformation. | + | |
| + | We start our first 19 moves by doing whatever we want, 4 choices each time. Since 19 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 <math>(1+)</math>, and we must use that transformation. | ||
| + | |||
Thus <math>4^{19}=\boxed{(C) 2^{38}}</math> | Thus <math>4^{19}=\boxed{(C) 2^{38}}</math> | ||
Revision as of 00:17, 8 February 2020
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 squares 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.)
Solution
Let (+) denote counterclockwise/starting orientation and (-) denote clockwise orientation. Let 1,2,3, and 4 denote which quadrant A is in.
Realize that from any odd quadrant and any orientation, the 4 transformations result in some permutation of 
.
The same goes that from any even quadrant and any orientation, the 4 transformations result in some permutation of 
.
We start our first 19 moves by doing whatever we want, 4 choices each time. Since 19 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 
See Also
| 2020 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 22 |
Followed by Problem 24 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
