Did you solve it? The colourful truth about elves | Mathematics

Earlier today I set you the following puzzle:

Four elves Glarald, Mnementh, Virthana and Tinsel are each wearing tunics of a different colour. At least one of these elves is a liar. (A liar is someone who only says statements that are untrue). During break at elf school, the following conversation is overheard:

  • Glarald: Mnementh wears green. (1)

  • Virthana: The elf in green is a liar. (2)

  • Tinsel: I wear blue. (3)

  • Glarald: I wear yellow. (4)

  • Mnementh: I’m in pink. (5)

  • Virthana: The elf in the red tunic beat Tinsel at the 2020 elf curling championship. I do not play curling. (6)

  • Tinsel: One of us is in yellow. (7)

  • Mnementh: Only one liar is among us. (8)

  • Virthana: I do not wear green. (9)

  • Tinsel: I was beaten by the elf in red at the 2020 elf curling championship. (10)

Who beat the elf in the blue tunic at the 2020 elf curling championship?

(You need to assume that elves who are not liars always tell the truth.)

Solution: Mnementh.

Workings: First of all we need to work out which elf is wearing which colour, how many elves are liars and who they are.

STEP 1 Let’s say Virthana is a liar. Then from (2), the elf in green is not a liar, and from (9), Virthana wears green. But if Virthana wears green, (2) is now true, thus contradicting the statement that Virthana is a liar. We cannot allow contradictions, thus Virthana is NOT a liar. We conclude that Virthana does not wear green, but that one of the elves does.

STEP 2 Virthana tells the truth. Thus from (6) and (10) we can deduce that Tinsel also tells the truth. From (3) Tinsel is in blue, and from (7) one elf is in yellow. We conclude that the four colours of elf tunic are yellow, blue, red and green.

STEP 3 We know no elf is in pink, thus from (5) Mnementh is a liar, which means that (8) is a lie. We know that at least one elf is a liar, and that two elves are speaking the truth, thus there must be two liars: Mnementh and Glarald.

STEP 4: Tinsel is in Blue. Virthana doesn’t wear green, and from (1) neither does Mnementh, so the green elf must be Glarald. Virthana cannot be red, from(6) and thus Virthana is yellow, making Mnementh the elf in red. The elf in red is the elf that beat Tinsel (in blue), and thus Mnementh is the elf that beat the elf in blue at the 2020 elf curling championship.

I hope you enjoyed the puzzle. Congrats again to Loïs B, the 12-year-old boy from Luxembourg, who came up with it.

I’ll be back in 2 weeks.

Photograph: Faber & Faber

My latest book is The Language Lovers Puzzle Book.

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

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