Henry Ernest Dudeney/Puzzles and Curious Problems/306 - A Puzzle in Billiards/Solution
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Puzzles and Curious Problems by Henry Ernest Dudeney: $306$
- A Puzzle in Billiards
- 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.
Solution
- $82$ points.
Proof
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 $82$.
$\blacksquare$
Sources
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $306$. -- A Puzzle in Billiards