(puzzle by Henry Dudeney):
When visiting with a friend one of our hospitals for wounded soldiers, I was informed that exactly twothirds of the men had lost an eye, threefourths had lost an arm, and fourfifths had lost a leg.
”Then,’ I remarked to my friend, ‘it follows that at least twentysix 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? It is a very simple calculation, but I have no doubt it will perplex a good many readers.
By basic divisibility, the total number of men = 60m, and say m=1, so a total of 60, of whom 26 have all 3 wounds, which is sufficient because 2*60/3>26, 3*60/4>26 and 4*60/5>26. But this does not explain why the number must be 26, as claimed by Dudeney.
To deal with this we have to consider the necessary overlap between the three fractions.
2/3+3/4 overlap to the extent of (2/31/4) = 5/12
4/5+5/12 overlap to the extent of (5/121/5) = 13/60.
So the necessary overlap for m=1 is 13, not 26.
It follows that m=2, when the necessary overlap is 26, so there are 120 wounded men in the hospital.

Posted by broll
on 20230503 23:15:59 