In the plane, the points with integer coordinates are called lattice points. Suppose a flea in the plane jumps from one lattice point to another. Each jump is one unit to the right, one unit to the left, one unit up, or one unit down. If the flea starts at the origin and makes exactly 10 jumps, how many lattice points could possibly be the final landing place of the flea?

Question

In the plane, the points with integer coordinates are called lattice points. Suppose a flea in the plane jumps from one lattice point to another. Each jump is one unit to the right, one unit to the left, one unit up, or one unit down. If the flea starts at the origin and makes exactly 10 jumps, how many lattice points could possibly be the final landing place of the flea?

Answer 1

To solve this problem, we need to consider the possible combinations of 10 jumps the flea can make. Since each jump can be one unit to the right, one unit to the left, one unit up, or one unit down, we can view each jump as a movement in the x or y direction.

First, let's consider the x-coordinate of the landing point. The flea can make any number of right or left movements in its 10 jumps. If we denote a right movement as +1 and a left movement as -1, the x-coordinate will be the sum of these 10 numbers.

Mathematically, let's consider the sum of 10 integers: sum = x1 + x2 + x3 + ... + x10.

For the x-coordinate to be an integer, we need the sum to be divisible by 2. This is because if the flea makes an odd number of right or left movements, the x-coordinate won't be an integer since odd + odd = even, and even + odd = odd.

Next, let's consider the y-coordinate of the landing point. As before, we can denote an up movement as +1 and a down movement as -1. The y-coordinate will also be the sum of 10 numbers: sum = y1 + y2 + y3 + ... + y10.

Similarly, for the y-coordinate to be an integer, we need the sum to be divisible by 2.

Now, let's combine these conditions. The x-coordinate must be divisible by 2, and the y-coordinate must also be divisible by 2. This means the sum of the 10 numbers must be an even number for both the x and y coordinates.

Here's how we can find the number of lattice points:

1. Count the number of even numbers from 0 to 10 (both inclusive). There are six even numbers: 0, 2, 4, 6, 8, 10.

2. For each even number, count the number of ways to choose a combination of right (+1) and left (-1) movements that sums up to that even number. This can be done using combinations or other counting techniques.

3. Multiply the number of even numbers (6) by the number of combinations for each even number. This will give you the total number of lattice points the flea could possibly land on.

I hope this explanation helps you understand the concept behind solving this problem!

262144

To solve this problem, we need to consider the possible combinations of 10 jumps the flea can make. Since each jump can be one unit to the right, one unit to the left, one unit up, or one unit down, we can view each jump as a movement in the x or y direction.