# The joke office

Alice and Bob are supposed to get form A at the joke office.

Forms A, B, C, D and E are each at exactly one of the following five terminals. At one Terminal is only one form.

Terminal 1: Here there are form B.

Terminal 2: Here there are form C or E.

Terminal 3: Here there are form D.

Terminal 4: Here there are form A, C or E.

Terminal 5: Here there are form A.

Bob is ​​at a loss when he reads the signs at the counters.

Alice remembers what the janitor is said: "Exactly one statement on the terminals is wrong, the four other statements are true."

Question:

At which terminal is form A available?

• Is switch the right word, or were you looking for "counter" or "terminal" ? – Falco Jul 9 '19 at 11:20
• Yes, terminal is better. – Matti Jul 9 '19 at 11:32
• You should specify that at each terminal there is exactly one form, otherwise there are alternate solutions, for example with B at 1, C and E at 2, D at 3 and A at 5. – Magma Jul 9 '19 at 14:03

Notation: (P,Q,R,S,T) denotes that form P is in Switch 1, Q in Switch 2, etc.

If Switch 1 is wrong, then

(?,C/E,D,C/E,A), only choice left for ? is B -> contradiction

If Switch 2 is wrong, then

(B,?,D,C/E,A), only choices left for ? are C and E -> contradiction

If Switch 3 is wrong, then

(B,C/E,?,C/E,A), only choice left for ? is D -> contradiction

If Switch 4 is wrong, then

(B,C/E,D,?,A), only choices left for ? are C and E -> contradiction

If Switch 5 is wrong, then

(B,C/E,D,A/C/E,?), one scenario can be (B,C/E,D,A,C/E) -> possible

Form A: Switch 4; Wrong Sign: Switch 5

Form A is available at

Switch 4.

Assume

Switch 1 was incorrect. This means that form B must be somewhere else. But there is no other sign which references form B, so that would mean more than one sign was incorrect (contradiction!). If we assume Switch 3 was incorrect, we’d get the same result. So one of Switches 2, 4, or 5 are wrong. It can’t be 4, because then B or D would have to be there too and those have already been placed. Assuming that Switch 2 was wrong would put A at Switch 2, but then Switch 5 would also be wrong. Therefore Switch 5 must be wrong, so all of the others are right, and since Switch 4 is the only place with mention to A, it must be there.

• you sniped me with only the answer, but i have the full explanation first. let the judge decide! good game! – Omega Krypton Jul 8 '19 at 12:24

Let us assume for contradiction sake Switch 5 was correct. If Switch 1 and 3 are correct only C and E are left for Switches 2 and 4 which would thus be correct, too. Similarly if Switches 2 and 4 are correct only B and D are left for Switch 1 and 3 and either both or none of them is correct. Thus, there is no way that Switch 5 is correct and exactly one of the remaining switches incorrect. This implies that Switch 5 must be the incorrect one. The only Switch remaining which could possibly contain Form A is Switch 4.

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Terminal 4 is wrong as it must contain either Forms C or E, thusly Bob should go to Terminal 5 to retrieve Form A