2020 AMC 10B Problems/Problem 13
Andy the Ant lives on a coordinate plane and is currently at facing east (that is, in the positive -direction). Andy moves unit and then turns left. From there, Andy moves units (north) and then turns left. He then moves units (west) and again turns left. Andy continues his progress, increasing his distance each time by unit and always turning left. What is the location of the point at which Andy makes the th left turn?
Solution 1 (Associative Property)
Andy makes a total of moves: horizontal ( left and right) and vertical ( up and down).
The -coordinate of Andy's final position is The -coordinate of Andy's final position is Together, we have
Solution 2 (Pattern)
You can find that every four moves both coordinates decrease by Therefore, both coordinates need to decrease by two times. You subtract, giving you the answer of
Solution 3 (Pattern)
Let's first mark the first few points Andy will arrive at:
Starting Point ( move):
In the move Andy lands on , in the move, Andy lands on .
There is a pattern, for every moves (starting from the move), Andy will arrive on a coordinate in the form of . From this we can deduce:
We have , for which and . So, on the move Andy is at .
Because the problem asks for the move, , on the move, Andy will be on .
2015 AMC 10B Problem 24 https://artofproblemsolving.com/wiki/index.php/2015_AMC_10B_Problems/Problem_24
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