2006 AMC 12B Problems/Problem 18
Contents
Problem
An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?
Solution
Let the starting point be 
. After 
steps we can only be in locations 
where 
. Additionally, each step changes the parity of exactly one coordinate. Hence after steps we can only be in locations where 
is even. It can easily be shown that each location that satisfies these two conditions is indeed reachable.
Once we pick 
, we have 
valid choices for 
, giving a total of 
possible positions.
Solution 2
moves results in a lot of possible endpoints, so we try small cases first.
If the object only makes 
move, it is obvious that there are only 4 possible points that the object can move to.
If the object makes 
moves, it can move to 
, 
, 
, 
, 
, 
, 
as well as 
, for a total of 
moves.
If the object makes 
moves, it can end up at 
, 
, 
, 
, 
, 
, 
, 
, 
, etc. for a total of 25.
At this point we can guess that for n moves, there are 
different endpoints. Thus, for 10 moves, there are 
endpoints, and the answer is 
.
See also
| 2006 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 17 |
Followed by Problem 19 |
| 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 12 Problems and Solutions | |
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