Henry Ernest Dudeney/Modern Puzzles/91 - Counting the Wounded/Solution
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Modern Puzzles by Henry Ernest Dudeney: $91$
- Counting the Wounded
- When visiting with a friend one of our hospitals for wounded soldiers, I was informed that
- exactly two-thirds of the men had lost an eye,
- three-fourths had lost an arm,
- and four-fifths had lost a leg.
- "Then," I remarked to my friend, "it follows that at least twenty-six of the men must have lost all three -- an eye, an arm, and a leg."
- That being so, can you say exactly how many men were in the hospital?
Solution
$120$ men are in the hospital.
Proof
We know that:
- exactly two-thirds of the men had lost an eye.
We also know that:
- three-fourths had lost an arm.
By Inclusion-Exclusion Principle, at least:
- $\dfrac 2 3 + \dfrac 3 4 - 1 = \dfrac 5 {12}$
of the men had lost both an eye and an arm.
Similarly, since we know that:
- four-fifths had lost a leg
by the Inclusion-Exclusion Principle again, at least:
- $\dfrac 5 {12} + \dfrac 4 5 - 1 = \dfrac {13} {60}$
of the men had lost all three.
Dudeney equates this to $26$.
Hence there are exactly:
- $26 \div \dfrac {13} {60} = 120$
men in this hospital.
$\blacksquare$
Sources
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $91$. -- Counting the Wounded
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $174$. Counting the Wounded