There are 11 beans, one weighs different than the others. How would you figure out in 3 turns?

Question

yeah so there's 11 beans right. and you only get to weigh them 3 times on the scale. and how do you figure out WHICH ONE doesn't weigh the same as every other bean?

Answer 1

The tricky part of this question is that it doesn't tell you whether the odd bean is heavier or lighter. At first I didn't think this could be done in 3 turns, but I thought about it more and I'm pretty sure the following works:

Start by separating the 11 beans into four groups A B C D ... where A B and C all have 3 beans each ... and group D only has two beans. Here's how we use our three turns.

1. Compare A and B (our first turn). If they are the same, then the odd bean is in C or D ... go to step 3 for our second turn.

2. If A and B are different in our first turn, then pick the heavier group (let's say it's B). Compare B and C (our second turn).

2a. If B and C are the same then we know the odd bean is in group A and it is *lighter*. Then it's trivial to find the odd bean in one more turn (weigh two of them ... if one is lighter then that's your winner, if not, then the third one is the winner).

2b. If B is heavier than C (it can't be lighter), then we know the odd bean is in B and it is *heavier*. Then it's trivial to find the odd bean in one more turn (weigh two of them ... if one is heavier then that's your winner, if not, then the third one is the winner).

3. If A and B are the same in our first turn (so neither contains the odd bean) then compare B and C (this is our second turn).

3a. If C is lighter then we know it contains a lighter bean and it's trivial to find it in one more turn (see 2a).

3b. If C is heavier then we know it contains a heavier bean and it's trivial to find it in one more turn (see 2b).

3c. If B and C are also the same weight, then we know the odd bean is in D (the group of two). Pick one from group D, and any bean from the other groups (since they are all normal beans and weigh the same). Weigh them. If the D-bean is heavier or lighter, we know it is the odd bean. If they are the same then the other D-bean (the one we didn't weigh) is the odd bean.

Bonus observation: Note that in this very last case, we have succesfully located the odd bean, but we don't know if it is heavier or lighter than the others, since we have never weighed it.

(Phew.)