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You’re given a base width W of an isosceles triangle. Your task is to determine the maximum height H of an isosceles triangle that can be formed such that the height and base are integers.

Example:

Clarifying Questions

  1. Is W guaranteed to be a positive integer?
    • Yes, W is guaranteed to be a positive integer.
  2. Does the base width W always refer to the full width of the isosceles triangle at its bottom?
    • Yes, W is the full width of the base of the isosceles triangle.
  3. Are there any constraints on the range of W?
    • The problem does not specify, but typically W will be within a reasonable range of values.
  4. Is the requirement to have an integer height and base?
    • Yes, both the height and the base should be integers.

Strategy

To form an isosceles triangle with a given base width W and the maximum height H, observe that the height is constrained by how evenly the height can be divided by 2 until it can’t be anymore.

In essence, for each increment in height, you require two additional units in width to maintain an isosceles triangle. This keeps happening until the base width can no longer accommodate an additional 2 units per increment step.

The mathematical relationship for an isosceles triangle where the height is maximal given a base width W can be approximated as: [ H = \left\lfloor \frac{W}{2} \right\rfloor ]

This is because each step in height adds 1 unit on each side, thus increasing the width by 2 units at each step.

Code

Here is the Python function to calculate the maximum height H based on the given base width W:

def max_height_of_triangle(W):
    # The largest integer `H` such that 2*H fits within the width W
    return W // 2

# Example Usage
W = 8
print(max_height_of_triangle(W))  # Output: 4

Time Complexity

The time complexity of this function is (O(1)) since it involves just a single division operation and flooring the result which are constant time operations.

The space complexity is also (O(1)), as we are only using a fixed amount of additional space regardless of the input size.

By providing this function, we can efficiently determine the maximum height for any given base width of an isosceles triangle.

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