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?