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There are four possible kinds of mathematicians:
Pure and sane, Pure and insane,
Applied and sane, Applied and insane.
The first step is to determine what each kind of mathematician would say about themselves:
Pure and sane: "I am pure and sane."
Pure and insane: "I am applied and sane."
Applied and sane: "I am pure and insane."
Applied and insane: "I am applied and insane."
Note that the applied and insane mathematician is lying about a lie so is actually speaking the truth.
Next consider the first statement by A, "I am insane." Based on the four groups there are two that will say they are insane:
The applied and sane, The applied and insane.
What these two groups have in common is that both are applied, thus we can determine that A is applied.
By the same logic we can determine that B is sane, C is insane, and D is pure.
Now we know one of the two characteristics of each mathematician. Next jump to the last statement by D, "C is sane." We already know that D is pure so he actually believes that C is sane. However we also know that C is infact insane so D's belief must be incorrect. Since he is not deliberately telling a lie her belief is incorrect, making him insane. So D is pure and insane.
Let review what we know thus far:
A: applied, B: sane, C: insane,
D: pure and insane.
Next look at what B says about D, "D is insane." This is a true statement and since B is already sane we can conclude that he is also pure. Now we have:
A: applied, B: sane and pure,
C: insane, D: pure and insane.
Next consider what C says about B, "B is applied." This is not true since B is pure. We already know C is insane thus he must also be pure. If he were applied he would be lying about a lie, thus telling the truth. Now we have:
A: applied, B: sane and pure,
C: pure and insane, D: pure and insane.
Finally consider A's second statement, "C is pure." C is pure thus A is making a true statement. However A is applied, thus she thinks she is telling a lie. So A is lying about an incorrect belief, making a true statement. Since A's belief is incorrect he must be insane. Thus:
A: applied and insane, B: pure and sane, C: pure and insane,
D: pure and insane. |
