# How do we solve this problem more elegantly with pencil pushing [closed]

100 Soldiers riddle

There are 100 soldiers. 85 lose a left leg, 80 lose a right leg, 75 lose a left arm, 70 lose a right arm. What is the minimum number of soldiers losing all 4 limbs?

Let $E$ be the number of soldiers losing it all. Then $E \geq\cdots\geq\cdots\geq\cdots\geq=10$.

None of the answers are like that. Well I kind of like pencil pushing method in Math. Makes me feel almighty.

it's normal algebra actually. I want an answer where we use the fact that n(A) = n(B) > = n(A U B) and then somehow got n(D)>=10. Can you do that? We can use other inequality too.

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How is that linear algebra? –  nik Mar 14 '12 at 11:02
I downvoted. The question is not clear. Please rewrite the question to make it more clear and self-contained (without referring to all answers in a different question). –  user2468 Mar 14 '12 at 16:49
it's normal algebra actually. I want an answer where we use the fact that n(A) = n(B) > = n(A U B) and then somehow got n(D)>=10. Can you do that? –  Jim Thio Mar 15 '12 at 9:02
Assuming you mean $\lvert A\rvert + \lvert B\rvert \ge \lvert A \cup B \rvert$ (note the plus instead of the equals sign), isn't the accepted answer by Eric Naslund precisely of the form you are looking for? –  Rahul Narain Mar 15 '12 at 9:16
see this link (it shows 3 overlapping sets that you may want to generalize to 4 sets): gmatclub.com/forum/formulae-for-3-overlapping-sets-69014.html –  Emmad Kareem Mar 15 '12 at 9:45