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# Spin the bottle

## Question

6 friends are sitting in a circle whom you have been observing playing the game called 'spin the bottle'. They invite you to play as well. Being a good sport, you agree to play. You happen to notice however that it is HIlga's turn and you would rather not kiss Hilga. You have also noticed that the bottle when spun with the left hand has a tendency to stop on a person an even distance away (e.g. the second, fourth, sixth person from her) and when spun with the right hand will land on a multiple of 3 distance away from HIlga. Currently the bottle is pointing directly across from Hilga but if it were to land on herself she would immediately spin it again from that position, switching hands in the process. You have no idea which hand she will use first nor do you know which way rotation she will place on the bottle [clockwise or counter-clockwise].

Where should you sit?

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## 3 answers to this question

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I'll make some simplifying assumptions. I'll first assume that, if you don't sit across from Hilga, the person who is sitting across from Hilga will spin the bottle with either hand with equal probability.

I'll then assume that, if spun with the left hand, the bottle will stop on any of the three people an even distance away (pointing toward either of the two people sitting next to Hilga, or pointing toward the person who was spinning) with equal probabilities of 1/3. So odds of getting smacked if you're in those positions are [odds that the bottle is spun with the left hand] x [odds that you're the unlucky one out of the three it might land on] = 1/2 x 1/3 = 1/6.

I'll then assume that, if spun with the right hand, the bottle will stop on either the person spinning or the person across from them with equal probability. So 1/2 of the time the person sitting across from Hilga will get kissed, and 1/2 of the time the bottle will point to Hilga and she'll spin with her left hand. Then the bottle will point to one of the two people sitting two seats away from Hilga, or to Hilga again, with probability 1/3. So odds that you get kissed if you're sitting two seats away from Hilga are [odds that the bottle is first spun with the right hand] x [odds that it points to Hilga after the first spin] x [odds that she spins it and it points to you] = 1/2 x 1/2 x 1/3 = 1/12.

But wait, there's also a possibility that after those two spins, the bottle ends up pointing to Hilga yet again and she keeps spinning. With the third spin (with the right hand) the bottle lands on Hilga or the person opposite her with 1/2 probability, on the fourth spin (if needed, with the left hand) she'll hit either of the people two seats away from her or herself yet again, and there's a potential for an infinite series. To circumvent having to calculate it, I will state (without proof) that if you sit in one of the spots two seats away from Hilga, you'll be hit with probability no greater than 1/2 if she starts spinning. So your probability of being hit, even accounting for this infinite series, is no greater than [odds that the bottle is first spun with the right hand] x [odds that it points to Hilga after the first spin] x [odds that you eventually get hit by her spinning] = 1/2 x 1/2 x [something less than 1/2] = something less than 1/8.

Now if you sit directly across from Hilga, you'll get to choose whether to spin with the left hand or the right hand. If you spin with the left you'll get kissed with probability 1/3, and if you spin with the right then you'll get kissed with probability 1/2 (actually slightly greater than 1/2, but that's a moot point). So you definitely don't want to sit across from Hilga.

Most favorable odds are in a seat adjacent to the person who's about to spin. Unless of course you happen to know that they're likely to spin with their right hand.

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position 5

Edited by bonanova
spoiler

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Position five relative to?

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