To solve this problem, we need to focus on a checkerboard's black squares, where a checker can move diagonally. The goal is to place one red checker and one black checker such that the red checker can jump over the black checker.

Step 1: Black Squares on a Checkerboard

A standard $8 \times 8$ checkerboard contains 32 black squares (since black and white squares alternate). These squares are arranged in such a way that checkers can move diagonally on them.

Step 2: Understanding a Valid Jump

A checker jumps from one black square over another in a diagonal path. For this to happen:

• The red checker must start on one black square.
• The black checker must be positioned diagonally adjacent to the red checker.
• There must be a vacant black square behind the black checker in the same diagonal direction for the red checker to land after the jump.

Step 3: Pairs of Adjacent Black Squares

On the checkerboard, the adjacent diagonal pairs of black squares can be identified. Every black square is part of two diagonal lines: one sloping upward from left to right and one sloping upward from right to left.

Let's identify how many such "jumpable" pairs of black squares exist:

• In the long diagonals (those stretching across the board), there are 6 pairs on the longest diagonals.
• The diagonals closer to the edges are shorter, with 5, 4, 3, and 2 pairs.
• Overall, this gives us a count for each type of diagonal (both directions combined):
  • 2 diagonals of length 6: $2 \times 6 = 12$ pairs
  • 4 diagonals of length 5: $4 \times 5 = 20$ pairs
  • 4 diagonals of length 4: $4 \times 4 = 16$ pairs
  • 4 diagonals of length 3: $4 \times 3 = 12$ pairs
  • 2 diagonals of length 2: $2 \times 2 = 4$ pairs

Step 4: Total Number of Pairs

Summing these values gives the total number of pairs of black squares where a jump can occur:

$12 + 20 + 16 + 12 + 4 = 64 \text{ pairs}.$

Step 5: Arrangement of Checkers

In each of these 64 pairs, there are two possible ways to arrange the red and black checker: either the red checker is placed on the first square of the pair, or the red checker is placed on the second square of the pair.

Thus, for each of the 64 pairs, there are 2 possible configurations of checkers. This gives:

$64 \times 2 = 128 \text{ ways}.$

Final Answer:

There are 128 ways to place a red and a black checker on two black squares so that the red checker can jump over the black checker.

Would you like any further details, or do you have any questions?

Follow-up Questions:

1. What are the properties of a checkerboard in terms of color distribution?
2. How does the diagonal movement of checkers influence game strategies?
3. Can you calculate the number of ways if the checkerboard is larger than $8 \times 8$?
4. What is the total number of squares on a checkerboard?
5. How do you determine the number of adjacent black squares in a diagonal?

Tip:

Always visualize problems involving checkerboards by marking important squares, such as diagonals, which often hold the key to solving the problem.