Question

PLEASE ANSWERRR!

Three men are suspected of being double agents. When questioned by the authorities, they speak as follows:

Jackson: “Max is a mole.”
Max: “Gregory is a mole.”
Gregory: “Max is lying.”

Assuming that moles like, other agents tell the truth, and there is just on mole among the three, determine:
Who is the mole?
If, on the other hand there are two moles present, who are they?

Answer 1

There are two ways to look at this:

If someone is a mole, negate their proposition:

Jackson(mole) "Max is not a mole"
Max(mole) "Gregory is not a mole"
Gregory(mole) "Max is not lying"

Looking at it this way their are two options, either Max is a mole, or Jackson and Gregory are both moles.

To find out for certainty, we must go at a maximum with no contradictions, meaning one agent and two moles:

Jackson(mole) "Max is a mole"
Max(mole) "Gregory is a mole"
Gregory(agent) "Max is lying"

This is a contradiction because if Gregory is telling the truth, and Jackson is lying, then Gregory and Max are not moles

Jackson(mole) "Max is a mole"
Max(agent) "Gregory is a mole"
Gregory(mole) "Max is lying"

This works because if Jackson and Gregory are both moles, then Max is telling the truth and is not a mole, and Max states Gregory is a mole which is again true

Jackson(agent) "Max is a mole"
Max(mole) "Gregory is a mole"
Gregory(mole) "Max is lying"

This is also a contradiction since Max says Gregory is a mole, and in this case he is, Max would be telling the truth, but to our knowledge the moles are lying.

Therefore Jackson and Gregory are moles

Answer 2

Consider all possibilities.

There is only one mole:

if Jackson is truthful,
Max is mole, and Gregory must also be true. It lines up since Gregory says that Max is lying.

if Max is truthful,
Gregory is a mole, and Jackson must also be true. so then Max would be the mole, but we established that's not possible. This option is not logical.

if Gregory is truthful,
Max is lying, and Jackson must also be truthful. So this option also leads you to believe Max is the mole.

To summarize, Max is the mole.

for two moles:

if Jackson is truthful,
both max and Gregory must be moles. So Max is definitely a mole, and according to Gregory (who is lying), Max is lying. so we can say max is NOT lying when he says Gregory is a mole. But if Max is not lying, then he is not a mole, so this logic is wrong.

if Max is truthful,
both Jackson and Gregory are false. So Max is not a mole and is not lying. So Gregory is definitely a mole according to Max, and since Jackson is lying when he says Max is the mole, that makes Jackson the other mole. This it's the best argument!

If Gregory is truthful,
both Jackson and Max are lying. Max is definitely a mole then, and Jackson is technically correct, even though he is a mole and moles lie. so this logic is not correct.

to summarize, Jackson and Gregory are the moles.