12 balls: The solution.

At least, I think it's a solution. Please have a good, long look at it, figure out what it's talking about and then tell me if I've got anything wrong.

If you haven't had a go yourself, don't look at the image, just go here for the original problem, and here for a clue. It wouldn't be any fun if you went straight for the solution, would it!


The solution
Warning! This is the solution and contains spoilers!

O.k, that's not very easy to read, is it? Might be a good idea to click on it and view it full-size.

Just in case you're not sure what's going on, here's a walk-through of just one of the possibilities to give you an idea:

1. First up, you need to name your balls. I've gone for the rather unimaginative (but not too confusing) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 & 12. You could go for A-L if you wanted to, or you could think of 12 different names, but then you might get too attached.
2. Weigh* any four balls against any other four. I've gone for 1, 2, 3 & 4 against 5, 6, 7 & 8. Three things can happen here:
   i) The two sets of balls balance;
   ii) The two sets don't balance, with the first set rising and the second falling;
   iii) The two sets don't balance, with the first set falling and the second rising.
   Let's pretend they're balanced:
3. If the two sets are balanced, you know that balls 1 - 8 are fine, and the odd one out is either 9, 10, 11 or 12. To find out which one, first you need to weigh** two of the suspects against one suspect and one fine ball. In my example, I've gone with 9 & 10 against 11 & 1. Again, three things can happen:
   i) The two sets balance;
   ii) The two sets don't balance, with the first set rising;
   iii) The two sets don't balance, with the first set rising.
   Let's pretend they're unbalanced, and that the first set (9 & 10) rise.
4. If 9 & 10 rise, then one of two things is true:
   i) Either 9 or 10 is too light;
   ii) 11 is too heavy.
   To work out which it is, weigh*** the balls in the first set against each other (e.g. 9 against 10). Once more, three things can happen, and the solution depends on which:
   i) The two balls are balanced - in which case, it must be that 11 was the odd one out, and it was too heavy;
   ii) The balls are unbalanced, with 9 rising - in which case, it must be that 9 is odd, and it's too light (as 10 has already been in a rising set, so can't be too heavy);
   iii)The balls are unbalanced, with 10 rising - in which case, it must be that 10 is odd, and it's too light (for similar reasons as in the previous case).

I hope that makes some kind of sense. If not, or if you'd like any parts of my flow chart explaining in greater detail, feel free to get in touch (leave a comment, or send me a message on Twitter (@TeaKayB), or something).

* This is the first weigh.

** This is the second weigh.

*** This is the third weigh.

12 balls: A clue

Carrying on from the 12 pool balls problem posted the other day, here's a clue for anyone who's not too sure where to get started.

The clue comes in the form of a photograph of my whiteboard after a colleague and I had spent part of a free period muddling the problem through. It's not complete and it's not explicit, but then it wouldn't be a clue if it was, would it?

If you want to have a go without seeing the clue first, then CLICK HERE and close your eyes for a few seconds!



The clue
Warning! This is a clue and contains spoilers!

Click to view full-size image!

I'll post the full solution in another few days... subscribe to make sure you catch it!

I've you've got a maths-related question of your own, you could ask it in the comments or use the form at my main blog.

The 12 pool balls problem

Not my pic... click for originator

O.k, here's one for ya...

You have twelve pool balls and visually they're all exactly identical. One of them, however, has a different mass/weight to the other eleven, but you don't know whether it's heavier or lighter.

You have at your disposal a set of weighing scales so you can weigh the balls against each other, but no weights to go with it. You can only use these scales three times*.

Your job is to find out two things:

1. Which ball is the odd one out?
2. Is it heavier or lighter?

The rules

1. You're only allowed to use the instrumentation and apparatus as described above. No magicking weights or digital scales out of thin air, or inventing science prep rooms nearby.
2. No cheating. There are solutions available online, but if you look them up you're only cheating yourself. If you look it up and post it here, then you degrade yourself to a point a little beneath most traffic wardens.
3. There is a solution.
4. Feel free to comment and ask for clues. My HOD and I have worked out what we think is a pretty nifty solution, so I might post a clue-post in a few days or so if anybody asks for one.
5. I will post my solution at a future date as yet undecided, then edit this post to link to it. If you're reading this after that's happened, please give it a shot on your own (or with friends/colleagues) first. It's more fun that way!
6. Feel free to post your own solutions on your own blogs or websites and then let me know so I can link to them.
7. Subscribe to MathsQS for updates and new posts!

* No, I don't know why. Maybe they're set to explode, or something. Think Die Hard II with the jugs problem.