[Friday Riddler] Troll v Dwarves

[Mike]

[Mike]

For the Express question about the baby, I don't have the math to do a definitive calculation, but by crunching it on a spreadsheet, I get that her "couch time" is approaching 50% as time progresses.

[Mike]

For the dwarves, is there a way to save more than half of them? I mean, there must be, because half is super obvious. Each odd number dwarf simply names the color of the even numbered dwarf in front of him. All even numbered dwarves will live, and the odd numbered dwarves each get a 50/50 shot.

But can we do better than that? Can we guarantee safety for six?

[poorpete]

[Mike]

The dwarves are killing me here.

There are two questions here: 1. What strategy guarantees the safety of the most dwarves? 2. What strategy is most likely to save the most dwarves?

For question #1, I can't guarantee the safety of more than five dwarves at this point.

Strategy A: Odd numbered dwarves call out the color of the dwarf ahead of them. Even numbered dwarves simply repeat what the last guy said. Guaranteed 5 dwarves saved, and 7.5 dwarves are saved on average.

Strategy B: Dwarf #1 calls out the majority color he sees in front of him. Everyone else simply repeats what #1 said. No matter the configuration, you are guaranteed to save at least 5 dwarves, and 6.23 dwarves are saved on average. Okay... that's inferior.

Strategy C: All dwarves, on their own turn, call out the majority color they see in front of them. If the number is tied, then instead call out the same color that was called last. I think this one still guarantees 5, but I haven't figured out an easy way to calculate the average number saved for all cases.

[Tahlvin]

I suppose cheating is out of the question? Because if you could vary the cadence of your answer, or something similar, you could convey multiple information in one word. So the first dwarf says the color of the dwarf in front on him, and has a 50/50 chance. The second dwarf repeats that answer, but varies the cadence of his answer based on the color on the dwarf in front of him, which conveys to that dwarf what color he needs to say, and so on. That would guarantee that you lose one dwarf in a worst case scenario, and none in a best case scenario. If you've go time to plan for something like Mike's even/odd scenario, then plan for something like this that would guarantee more dwarfs survive.

So if I answer "WHITE", that indicates the dwarf in front of me should answer WHITE as well. If I answer "WHIIIIITE" (or put different stress on a different syllable of the word, etc.), then that indicates the dwarf in front of me should answer BLACK. If I say "BLACK", then the next dwarf also says BLACK; if I say "BLAAAACK", then the next dwarf answers with "WHITE".

[Mike]

I'm assuming that is cheating and outside the scope of the puzzle. If that were the case, then we could just convey all the info we want by breathing in Morse Code.

But it certainly is a good answer.

[Mike]

Physics Girl has the answer today to the dwarf riddle:



The answer is super-simple. If the first dwarf says white, that means he sees an odd number of white dots. If he says black, that means he sees an even number of white dots.

From that, all other dwarves will survive.

[Zen]

Ah... I see... So the first dwarf needs to pick the color based on a code to tell them what the balance of black and white dots they see is, which will tell the next tallest one what color dot they have, cascading down the line... So they should be able to save all but the tallest, with a 50% chance of saving everyone...

[Tahlvin]

They all just need to keep track of how many people behind them have called white, so one knows whether an even or odd number of white dots in front of one, plus what's already been called, indicates whether oneself has a white or black dot. It works as long as nobody makes a mistake, in which case that person and everyone in front of him is screwed.

And I liked how the guy in the video offered the same cheat as me, about varying the pitch of the answer.

[Phoebe]

I just want to say how much I enjoy this and am looking forward to another round. Unfortunately the problems above were known to me before (in different terms) but they are fun problems.

[Zen]

I would also like to add that it is pleasant to read a story where being short is advantageous... Just a personal thing... Not that I have a chip on my shoulder about my (declining) height or anything...

[Tahlvin]

[Zen]

[poorpete]

Chat about todays riddler here or in a new thread?

https://fivethirtyeight.com/tag/the-riddler/

[Mike]

I say new thread with a couple words to identify it.