Who won the race?

Question

4 players enter a race. The judge is not at the finish line to judge the ending. When the judge shows up to award the prize, these are the statements by 4 players

Alpha - I won the race

Beta - I came in last

Gamma - I did not finish last

Theta - I was neither first nor last

The judge goes to Alpha to hand the prize, when a member from the audience who has been watching the race all along shouts - "Exactly one of these 4 is lying". To whom should judge award the prize?

Answer 1

Here's the situation if no one lied. A ⚪ means "this player may come at this place":

Player	1st	2nd	3rd	4th
Alpha	⚪
Beta				⚪
Gamma	⚪	⚪	⚪
Theta		⚪	⚪

Now let's check who lied.

If it's Alpha, then Gamma must have won and Beta must have come last.

Player	1st	2nd	3rd	4th
Alpha		⚪	⚪	⚪
Beta				⚪
Gamma	⚪	⚪	⚪
Theta		⚪	⚪

If it's Beta, no one comes last. This is not a valid race.

Player	1st	2nd	3rd	4th
Alpha	⚪
Beta	⚪	⚪	⚪
Gamma	⚪	⚪	⚪
Theta		⚪	⚪

If it's Gamma, both Beta and Gamma comes last. This is not a valid race either.

Player	1st	2nd	3rd	4th
Alpha	⚪
Beta				⚪
Gamma				⚪
Theta		⚪	⚪

If it's Theta, Alpha must have won and Beta must have come last. There's no place for Theta.

Player	1st	2nd	3rd	4th
Alpha	⚪
Beta				⚪
Gamma	⚪	⚪	⚪
Theta	⚪			⚪

Conclusion: Alpha lied and the real winner is Gamma.

Answer 2

Let's look for statements leading to another falsehood if they're false. That would lead to a contradiction:

- If Gamma is lying, so is Beta, making Gamma a truth teller. Gamma must have wound up at the 1st, 2nd or 3rd place.

- If Theta is lying, so is either Alpha or Beta, making Theta a truth teller. Theta must have wound up at the 2nd or 3rd place.

Exactly one of Alpha and Beta is lying, so the 1st and 4th places can't be reserved for them (which would imply an even number of lies). The only way to prevent that is Gamma winning the race.