Arbuzoid riddle

Question

Hello again!Here's another riddle, which I hope will be met better than the previous one.There's a planet inhabited by arbuzoids (water-melon-ers). Those creatures are found in three colors: red, green and blue. There are 13 red arbuzoids, 15 blue ones, and 17 green. When two differently coloured arbuzoids meet, they both change to the third color.The question is, can it ever happen that all of them assume same color?Anton

Mr Wordy · Accepted Answer

Hi Anton,

I wouldn't exactly call this a "riddle"!

The answer is no.

I was intrigued by the word "arbuzoid", by the way. I'm guessing this derives from the Russian word for watermelon. Is that right?

Ant_222 · Answer

Hello, Mr. Wordy.
You're right about both the word's origin (arbuz == water melon) and the answer to the riddle. But let me ask you why do you think it technically does not qualify as a riddle, and what your solution is. Some people that I have asked it tended to answer "no" after a thought experiment wherein they tried every way to have them all take the same color and failed. But that doesn't prove there's no solution...

Mr Wordy · Answer

To me, a "riddle", in the sense of a puzzle posed for people to solve for fun, or as a mental challenge, is something like "When it rains I do X, when the sun shines I do Y. I have Z legs. What am I?" (It could diverge a little from this exact form, but you get the general idea.) "Riddle" can also be used in a more general sense, as in "Russia is a riddle wrapped in a mystery inside an enigma". However, I would never call a mathematical or logical puzzle such as yours a "riddle".

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This is how I did it. Assign the number 0 to the red arbuzoid, 1 to the blue, and 2 to the green. The sum of all the arbuzoids taken modulo 3 is invariant under the allowed transformations. With 13 red arbuzoids, 15 blue and 17 green, the sum, modulo 3, is 1. However, if all 45 arbuzoids were of the same colour, it would be 0. Therefore it's impossible to get from one to the other just by performing the allowed transformations.

Ant_222 · Answer

Mr WordyThis is how I did it

Hey, that's a fun solution! Never thought in that direction... How did it occur to you? That makes a third solution in my repository, one whereof is mine

Mr WordyHowever, I would never call a mathematical or logical puzzle such as yours a "riddle".

Hope it's not too much off-topic...

Mr Wordy · Answer

Ant_222Never thought in that direction... How did it occur to you?

The first thing I thought of was to try to find an invariant -- in this case, some number that could be calculated from the numbers of red, green and blue arbuzoids that wouldn't change when they switched colours in the permitted ways. Then I obviously hoped that the invariant for the starting state would be different from that of all of the states that we were trying to prove are impossible. At this stage I should have straight away thought of modular arithmetic, but actually I didn't. It was obvious that ordinary addition wouldn't work, so I tried multiplication, and found that assigning the red, green and blue arbuzoids each of the three (complex) cube roots of unity and then multiplying them up would work. Then it was obvious that, because these multiplications are just equivalent to rotations in the complex plane, I was effectively just doing addition modulo 3 (kind of like "clock arithmetic" but with only three hours in the day).

Ant_222 · Answer

No comment...

Arbuzoid riddle

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