Alice, Bob, Charlie and Diane are playing when someone breaks a vase.

They all know who did it and each makes these statements:

Alice: “It was Bob”

Bob: “It was Diane”

Charlie: “It was not me”

Diane: “What Bob says is wrong”

a) Explain why at least one of the four must be lying.

b) Explain why at least one of the four must be telling the truth.

c)) Let’s now assume that exactly one of them is lying.

Who is lying? Who broke the vase?

d) Let’s now assume that exactly one of them is telling the truth.

Who is telling the truth? Who broke the vase?

e) Let’s now assume exactly two of the statements are true and two false.

f) Hence, show that if we don’t know how many of the statements are true,

then any one of the four could have broken the vase?

a) A contradicts B (or, B or D are lying), implies “at least one must be lying”

b) IF all lie, then B contradicts D, implies “at least one is telling the truth”

c) A or B are lying. IF A lies and B truth, then D is lying: contradiction.

Bob is lying and Bob broke the vase.

d) B or D truth, then A and C are lying, and B is lying.

Diane is telling the truth and Charlie broke the vase.

e) B or D truth. IF B tell truth, then D broke vase and C is telling truth OR

IF D true, then B lying, IF A truth, then C truth: contradiction, so A lying, C truth,

then Alice broke vase.

f) The scenarios include cases where each of the four broke the vase.