Billiards handicap consistency: A can give B 20 points, A can give C 32 points, and B can give C 15 points. Assuming consistent skill ratios, how many points make the game?

Introduction / Context:
In handicap problems, “A gives B k in N” means when A scores N points, B scores N − k points in the same time. Consistency across pairs lets us solve for N (the game size).

Given Data / Assumptions:

• A gives B 20 in N ⇒ A:B = N : (N − 20)
• A gives C 32 in N ⇒ A:C = N : (N − 32)
• B gives C 15 in N ⇒ B:C = N : (N − 15)

Concept / Approach:
When A scores N, B scores N − 20. In that time, C should score (N − 20)*(N − 15)/N by the B:C ratio. But also, by A:C, C must be N − 32. Equate to solve N.

Step-by-Step Solution:

Verification / Alternative check:
Plugging N = 100 satisfies all three handicaps.

Why Other Options Are Wrong:
They violate at least one of the three given pairwise handicaps.

Common Pitfalls:
Adding point differences linearly; the correct method uses proportional scoring in equal time.

Final Answer:
100