Starting from the point (0,0), a grasshopper makes a series of leaps on the coordinate plane. The grasshopper's first leap takes her to (3,4). After a move which adds (x,y) to the grasshopper's coordinates, her next move adds either (x,y-1) or (x-1,y) to her coordinates. Thus her second move adds either (3,3) or (2,4) to her position, landing her at (6,7) or (5,8) accordingly.
If the grasshopper's x and y coordinates never decrease, and if she ends at the point (a,a) where a is as large as possible, then what is the value of a?
Give an explanation with the answer. Thanks!!!! :)
14
To find the value of "a," let's analyze the pattern of the grasshopper's moves.
The grasshopper starts at (0,0) and makes her first leap to (3,4). This means she moves three units to the right (x + 3) and four units up (y + 4).
In the second move, the grasshopper has two options:
1. Move (x, y-1): If she chooses this move, she moves three units to the right and three units up (3, 3), resulting in (0 + 3, 4 + 3) = (3, 7).
2. Move (x-1, y): If she chooses this move, she moves two units to the right and four units up (2, 4), resulting in (3 + 2, 4 + 4) = (5, 8).
To maximize "a," the grasshopper needs to choose the move that results in the largest x and y coordinates.
As we observe, the second move, where she adds (2, 4) to her coordinates, results in larger values compared to the first move. Therefore, the grasshopper chooses the second move and ends up at (5, 8).
Since she ends at (a, a) and a = 8, the value of "a" is 8.
Hence, the value of "a" is 8 in this scenario.
To determine the value of a, we need to follow the grasshopper's leaps and find a pattern. Let's analyze the first few moves:
1. First leap: (3,4)
2. Second leap options: (3,3) or (2,4)
We can see that each leap adds either (x, y-1) or (x-1, y) to the grasshopper's position.
3. Third leap options for (3,3): (3,2) or (2,3)
4. Third leap options for (2,4): (2,3) or (1,4)
At this point, we can observe that the available options for the third leap are the same for both possible positions after the second leap. It seems that regardless of which option is chosen, the subsequent options remain the same.
Continuing this pattern:
5. Fourth leap options for (3,2): (3,1) or (2,2)
Fourth leap options for (2,3): (2,2) or (1,3)
Fourth leap options for (1,4): (1,3) or (0,4)
Again, we notice that the available options for the fourth leap are the same across all possible positions after the third leap.
By observing this pattern, we can form a general rule: For each leap, the grasshopper's options will always be either (x, y-1) or (x-1, y), regardless of the previous position.
Now, let's determine the options for the final leap, which must result in the grasshopper's coordinates ending at (a, a), where a is as large as possible.
If the grasshopper ends at (a, a), then the previous position before the final leap must be either (a, a-1) or (a-1, a).
For the first case, (a, a-1):
- The penultimate position must have been (a, (a-1)-1) = (a, a-2).
- The previous position would have been (a, (a-2)-1) = (a, a-3).
- This pattern continues until we reach (a, 0) as the starting point.
For the second case, (a-1, a):
- The penultimate position must have been ((a-1)-1, a) = (a-2, a).
- The previous position would have been ((a-2)-1, a) = (a-3, a).
- This pattern continues until we reach (0, a) as the starting point.
Comparing both cases, we can see that in both scenarios, the highest value a can take is max(a, 0), where max(a, 0) denotes the maximum value between a and 0.
Therefore, the value of a is the maximum value between the starting x-coordinate (0) and the starting y-coordinate (0), which is max(0, 0) = 0.
Hence, the value of a is 0.