Henry Ernest Dudeney/Puzzles and Curious Problems/306 - A Puzzle in Billiards/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $306$

Alfred Addlestone can give Benjamin Bounce $20$ points in $100$, and beat him;

Bounce can give Charlie Cruikshank $25$ points in $100$, and beat him.

Now, how many points can Addlestone give Cruikshank in order to beat him in a game of $200$ up?

Of course we assume that the players play constantly with the same relative skill.

$82$ points.

Let $A$, $B$ and $C$ denote Addlestone, Bounce and Cruikshank respectively

From the definition of the problem, $A$ can score $100$ while $B$ can score only $79$.

Similarly, $B$ can score $100$ while $C$ can score only $74$.

Multiply $79$ by $74$, double, and divide by $100$, and you get $116.92$.

So $C$ can score $117$ (there are no fractional points) while $A$ can make $200$.

Therefore $A$ can give $C$ $82$ points and win.

There is a case for saying that $A$ can give $C$ $83$ points and still win -- it depends on how fractional points are interpreted.

However, it is certain that $A$ can win having given $C$ those $82$.

$\blacksquare$