## A Tangled Tale May 31, 2007

Posted by Peter in Exam 1/P.

As is now common knowledge, Lewis Carroll was not merely a writer of children’s tales, but an amateur mathematician. He was fond of puzzles of a logical nature, and in his work A Tangled Tale, he posed a question that is particularly relevant to the concepts of probability theory tested on Exam 1/P:

“Sad — but very curious when you come to look at it arithmetically,” was her aunt’s less romantic reply. “Some of them have lost an arm in their country’s service, some a leg, some an ear, some an eye — ”

“And some, perhaps, all!” Clara murmured dreamily….

“Say that 70 per cent have lost an eye — 75 per cent an ear — 80 per cent an arm — 85 per cent a leg — that’ll do it beautifully. Now, my dear, what percentage, at least, must have lost all four?”

Being the writer that he was, Carroll posed the question in the setting of a conversation between a young girl, Clara, and her apparently unsympathetic aunt, who I wonder must have had Asperger’s syndrome, and perhaps would have made an excellent actuary. But the question is this: Given that 70% of veterans have lost an eye, 75% an ear, 80% an arm, and 85% a leg, what is the minimum percentage of veterans that have lost all four appendages? The solution, as I furnished to the individual who directed my attention to this question, is as follows:

If no less than 70% of the soldiers lost one eye, then no more than 30% of the soldiers did not lose one eye. Similarly, no more than 25% of the soldiers did not lose one ear; no more than 20% of the soldiers did not lose one hand; and no more than 15% of the soldiers did not lose one leg.

We see that the minimum possible percentage of soldiers who lost each of these parts is attained when the maximum percentages of each who did not lose at least one part are mutually disjoint. This is because if we maximize the number of soldiers who retained at least one body part, we minimize the number of soldiers who lost all such parts. To this end, the maximum number of soldiers retaining at least one body part occurs if no soldier has more than one surviving body part; that is, the 30%, 25%, 20%, and 15% of soldiers who retained an eye, ear, hand, and leg, respectively, are assumed to have lost all other parts.

Consequently, the total percentage of soldiers who have retained at least one body part is the maximum 30+25+20+15 = 90%. Therefore, we are guaranteed that at least 10% of the soldiers have lost all such body parts.