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The Four Mathematicians (Posted on 2003-04-12) Difficulty: 3 of 5
Four mathematicians have the following conversation:

A: I am insane.
B: I am pure.
C: I am applied.
D: I am sane.
A: C is pure.
B: D is insane.
C: B is applied.
D: C is sane.

You are also given that:

Pure mathematicians tell the truth about their beliefs.
Applied mathematicians lie about their beliefs.
Sane mathematicians beliefs are correct.
Insane mathematicians beliefs are incorrect.

Describe the four mathematicians.

See The Solution Submitted by Ravi Raja    
Rating: 3.0000 (6 votes)

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Solution First figure out... | Comment 1 of 14
First figure out what each of the four types of mathematicians would say about themselves (Use of "he" is made as a grammatical convenience):
pure and insane: says he's applied, says he's sane
pure and sane: says he's pure, says he's sane
applied and insane: says he's applied, says he's insane
applied and sane: says he's pure, says he's insane

Since A says he's insane he must be applied.
Since B says he's pure he must be sane.
Since C says he's applied he must be insane.
Since D says he's sane he must be pure.

Therefore D is saying a falsehood when he says C is sane. Since D is pure, D must be insane.

So B is telling the truth that D is insane, and since B is sane, B must be pure.

So C is saying what is false when he says that B is applied, and since C is insane, C must be pure.

So A is telling the truth when he says C is pure, and since he's applied he must be insane.

Final result:
A: applied & insane
B: pure & sane
C: pure & insane
D: pure & insane
  Posted by Charlie on 2003-04-12 04:40:10
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