# 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?

I want answer where

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.

-
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 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
show 5 more comments

## closed as off-topic by Lord_Farin, azimut, Dominic Michaelis, Nicholas R. Peterson, Tobias KildetoftOct 9 '13 at 7:39

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Lord_Farin, azimut, Dominic Michaelis, Nicholas R. Peterson, Tobias Kildetoft
If this question can be reworded to fit the rules in the help center, please edit the question.