Octopus Problems

I’ve translated two problems from the 2009 Moscow Math Olympiad. In both of them our characters are genetically engineered octopuses. The ones with an even number of arms always tell the truth; the ones with an odd number of arms always lie. In the first problem (for sixth graders) four octopuses had a chat:

  • “I have 8 arms,” the green octopus bragged to the blue one. “You have only 6!”
  • “It is I who has 8 arms,” countered the blue octopus. “You have only 7!”
  • “The blue one really has 8 arms,” the red octopus said, confirming the blue one’s claim. He went on to boast, “I have 9 arms!”
  • “None of you have 8 arms,” interjected the striped octopus. “Only I have 8 arms!”

Who has exactly 8 arms?

Not only do octopuses lie or tell the truth according to the parity of the number of their arms, it turns out that the underwater world is so discriminatory that only octopuses with six, seven or eight arms are allowed to serve Neptune. In the next problem (for seventh graders), four octopuses who worked as guards at Neptune’s palace were conversing:

  • The blue one said, “All together we have 28 arms.”
  • The green one said, “All together we have 27 arms.”
  • The yellow one said, “All together we have 26 arms.”
  • The red one said, “All together we have 25 arms.”

How many arms does each of them have?

My students enjoyed the octopuses, so I decided to invent some octopus problems of my own. In the first problem, the guards from the night shift at Neptune’s palace were bored, and they started to argue:

  • The magenta one said, “All together we have 31 arms.”
  • The cyan one said, “No, we do not.”
  • The brown one said, “The beige one has six arms.”
  • The beige one said, “You, brown, are lying.”

Who is lying and who is telling the truth?

In the next problem the last shift of guards at the palace has nothing better to do than count their arms:

  • The pink one said, “Gray and I have 15 arms together.”
  • The gray one said, “Lavender and I have 14 arms together.”
  • The lavender one said, “Turquoise and I have 14 arms together.”
  • The turquoise one said, “Pink and I have 15 arms together.”

What number of arms does each one have?

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10 Comments

  1. DeepButi:

    1. either the blue or the striped, but not both.
    2. green 6; yellow, red, blue 7 each
    … tbc

  2. DeepButi:

    3. magenta right => 7+8+8+8 & cyan lying. Only one could lie, but brown & beige cannot both be true. So magenta is lying, cyan right.
    brown cannot be true because beige with 6 would be lying. So brown lying, beige true (and beige has 8 arms :) ).

  3. DeepButi:

    4. Pink 7, Green 7, Lavender 6, Turquoise 8
    Green cannot be true (8=> no solution for Pink; 6=> no solution for Tuquoise), so G=7 and the others are easy.

  4. Sue VanHattum:

    These are lovely! (The last one was way harder, for me.) I’ll have to try to make one up.

  5. Noelle:

    1. Only the striped one can be true, since the red octopus must be lying when he says that the blue octopus is telling the truth.

    Red says he has nine arms. If he had an even number, he would be truthful of that and not claim he has an odd number of arms, therefor he does have an odd number of arms, but not nine of them, and what he said about blue (that blue has 8 arms) cannot be true either. Since blue lied about how many arms it had, it must have an odd number of arms. And the green is already lying. So only the striped one told the truth (I have eight arms, and none of you do.)

    I didn’t do the others.

  6. Bill:

    I disagree with DeepButi’s solution for #4.

    All liars have 7 arms. If Gray is a 7-armed liar, then its statement is false, so Lavender cannot also have 7 arms, and is therefore a truth teller. But if Gray is an even-armed truthteller, then its statement is true and Lavender must also be an even-armed truthteller. Lavender must be a truthteller either way! Based on Lavender’s true statement, Turquoise must also be an even-armed truth-teller, making Pink a 7-armed liar.

    So Pink has 7 arms, Gray has 8 arms, Lavender has 6 arms, and Turquoise has 8 arms.

  7. DeepButi:

    Noelle: absolutely right, I missed this one :(

    Bill: Pink 7 + Grey 8 = 15 … so Pink with 7 arms would be telling the truth …

    By the way, I love all of them, I used to do the old ones on J&S (a french magazine) with lots of true/lie tellers and other special characters (fool: sometimes true/sometimes lier, galactic perroquet: he was true/lier depending on the previous sentence of the problem), really nice ones.

  8. Bill:

    DeepButi, you are right!

  9. vadim:

    A related pun:

    In front of a door to a cellar full of gold, there stand three guards. One of them tells only truth, the second one tells only lies and the third one smashes the face of anyone who dares to ask stupid questions.

    Thanks for nice riddles! :)

  10. cmoh:

    Brute force approach: Made a truth table for p,l,t=(T,T,T), (T,T,F), (T,F,T), …