## ODD BALL LOGIC PROBLEM

Philosophical, mathematical and computational logic, linguistics, formal argument, game theory, fallacies, paradoxes, puzzles and other related issues.

### ODD BALL LOGIC PROBLEM

HI,

I am new, the solution to this problem can be found on the internet, but please don't spoil it be going there and posting the solution here!

Work it out, yourselves, please. (It is a really fun puzzle to try!)

The Problem

You have a balance type Scale you often see depicted in law courts, with pans on either side to hold the balls

You can only use this device to work out your solution (Not, for instance, a bathroom type scale)

You can weigh any combination of balls.

BUT YOU MUST SOLVE THE PROBLEM IN JUST THREE COMPARISON EFFORTS

For example, you could use 6 and 6 balls and one side will go up and the other down?

You have 12 identical balls in volume, shape and color.

However , one ball, namely the oddball differs in mass/weight in such a tiny amount that you need a way to find by logic means exactly which one it is

The oddball might be heavier or lighter than the 11 other balls of average weight ?

You must locate the oddball and establish whether it is lighter or heavier than the others ?

Thus, there is a two-paths to a full solution.

Alan
Alan McDougall
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### Re: ODD BALL LOGIC PROBLEM

Nice one. I think I see a way, but will wait a while. What makes it tricky is not knowing which way the differential goes WRT to the average balls.

Braininvat
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### Re: ODDBALL

I like the title.

It was fun, though I usually don't do games.

I don't think matters, solving it alone or looking it up. A posted solution spoils it for others.

Mum's the word.

Limited access (to the scale) is analogous to a limited number of resources. Efficiency is key to maximizing utility.

Faradave
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### Re: ODD BALL LOGIC PROBLEM

Process of elimination is the way to go.
Alan McDougall
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### Re: ODD BALL LOGIC PROBLEM

I seem to need the 4th comparison, or use some guess work.

Watson
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### Re: ODD BALL LOGIC PROBLEM

Think about the logical implications of each comparison, insofar as the ball groups not measured are concerned. SPOILER AHEAD

Your thought on piles of three was right on. Remember after the first comparison, you have a "control" group that consists entirely of normal balls. That's the key later.

Braininvat
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### Re: ODD BALL LOGIC PROBLEM

Yes that seems to get me to the third weigh in, which may disclose the odd ball of the 3 balls. If not, I have 2 balls to check with the 4th try. Then I either find it, or the last ball must be it. Previous weighings would have disclosed if the odd ball was more or less in weight.

Watson
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### Re: ODD BALL LOGIC PROBLEM

If you reached the 3rd weighing with 3 balls left, then you don't need a 4th weighing. Here's why:

SPOILERS AHEAD

You take two balls from the final group of 3, and weigh them. Either they balance, in which case the other ball is the oddball, or they don't, and if they don't the one that goes in the direction of the oddball (already determined) is the oddball.

If steps 1 and 2 were done properly, I think that should work.

Braininvat
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### Re: Solution process

SEMI-SPOILER:

I also started with three groups. Did anyone else end by a swap of two balls between scale pans?
Alan referred to two solutions.

Faradave
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### Re: ODD BALL LOGIC PROBLEM

I think so, but isn't that an extra turn at the scales?

Watson
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### Re: One solution

Yes. Swapping counts as one turn.
.
.
SPOILER (of the simpler case).
.
.
This is the simpler case but it can work in either if you extend the method.

1. Put four balls on each of two scale pans. (1st measure)

2. If they are equal (simple case), all those balls are eliminated as "odd".

3. With two remaining balls on each scale pan, say the left side is lighter. (2nd measure)

4. Swap a ball from each pan, keeping track of them. (3rd measure)

If the left side stays light, you know an odd light ball is on the left, the ball not swapped.
If the right side is now lighter, a light odd ball the one that moved to the right side.

The works similarly, if the odd ball is heavier.

See if you can use this method when the original pans of 4 balls are unequal.

Faradave
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### Re: ODD BALL LOGIC PROBLEM (Solution)

Solution

You are given 12 seemingly identical metal balls, but one of them weighs slightly more or less than the others. With a balance scale and three weighings only, how can it be determined which is the oddball and whether it is heavier or lighter?

This is one those puzzles that'll likely take a fair amount of time to work out, but the solution (at least the only one I know of) is ingenious and very satisfying to discover. No special math or logic skills required, just perseverance and insight.

Solution to Twelve Balls puzzle

This one requires a little work and inspiration to solve. I know of only this one solution here. If anyone knows of another, I'd be happy to see it.

Divide the balls into three groups of four each; label these groups AAAA, BBBB and CCCC.

Weigh AAAA_BBBB. The possible results are:
1. If they balance: One of the C's is heavy or light. Therefore:
weigh CCC_AAA (remember, all A's are now known to be standard):
a. If they balance: The 4th C is the oddball. Therefore, weigh the 4th C against any other ball.
i. If the 4th C falls: The 4th C is heavy.
ii. If the 4th C rises: The 4th C is light.
b. If the CCC side falls: One of the C's is heavy. (Remember, the A's are known to be standard.) Therefore, weigh C_C:
i. If they balance: The other C is heavy.
ii. If one side falls: The fallen C is heavy.
c. If the CCC side rises: One of the C's is light. Therefore, weigh C_C.
i. If they balance: The other C is light.
ii. If one side rises: The risen C is light.
2. If the AAAA side falls: The oddball is either a heavy A or a light B and the C's are all standard. Therefore, arrange the balls into three new groups like so: AAAC BBBA CCCB. (This re-arrangement step, and the one like it below in step 3, are the key to solving this puzzle.) Weigh BBBA_CCCB:
a. If they balance: The oddball is in AAAC. Therefore, weigh A_A.
i. If they balance: The other A in AAAC is heavy.
ii. If one side falls: The fallen side has the heavy A.
b. If the left side (BBBA) falls: The A in BBBA is heavy or the B in CCCB is light. Therefore, weigh A_C (C is known to be standard).
i. If they balance: The B in CCCB is light.
ii. If the A side falls: A is heavy.
iii. If the C falls: Not possible.
c. If the right side (CCCB) falls: The a B in BBBA is light. Therefore, from the BBBA group weigh B_B.
i. If they balance: The other B in BBBA is light.
ii. If the left side falls: The B on the right is light.
iii. If the right side falls: The B on the left is light.
3. If the BBBB side falls: The oddball is either a heavy B or a light A and the C's are all standard. Therefore, arrange the balls into three new groups like so: AAAB BBBC CCCA. Weigh AAAB_CCCA:
a. If they balance: The Oddball is in BBBC. Therefore, weigh B_B.
i. If they balance: The other B in BBBC is heavy.
ii. If one side falls: The fallen side has the heavy B.
b. If the left side (AAAB) falls: The B in AAAB is heavy or the A in CCCA is light. Therefore, weigh B_C (C is known to be standard).
i. If they balance: The A in CCCA is light.
ii. If the B side falls: B is heavy.
iii. If the C side falls: Not possible.
c. If the right side (CCCA) falls: An A in AAAB is light. Therefore, from AAAB weigh A_A.
i. If they balance: The other A in AAAB is light.
ii. If the left side falls: The A on the right is light.
iii. If the right side falls: The A on the left is light.
Alan McDougall
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### Re: ODD BALL LOGIC PROBLEM

Alan McDougall » Sat Mar 26, 2016 7:46 am wrote:ODD BALL LOGIC PROBLEM

Simple, there is no need to even touch the balls on the table, nor to touch the scale or even to weigh them, other than as the means to check the answer!

It's the same as the difference between Laotse and Einstein.

Shall I give it a few moments/days for others to exercise their thoughts/ego (perhaps even intuition) first?
Sure. *__-
nameless
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### Re: ODD BALL LOGIC PROBLEM

If they are all made from the same material, the diameter of the odd ball has to be different.

Line them up on a flat surface and place a steel straight edge along the top of them.

If it's a light ball, it won't touch the straight edge.

If it's a heavy ball, the straight edge will pivot over it.

doogles
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### Re: ODD BALL LOGIC PROBLEM

You have a balance type Scale you often see depicted in law courts, with pans on either side to hold the balls.

You can only use this device to work out your solution.....

- from the OP

Sorry, guys. I am not sure where Einstein or Lao Tse stood on following directions, but it is a requirement in logic puzzles.

It's like chess, where the rook can't take up diagonal travel and the knight can't make its "L" 7 squares tall. That would be an interesting game, though.

Braininvat
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### Re: ODD BALL LOGIC PROBLEM

Oh! Darn! Will David now have to take his 'Thumbs-up" symbol back? Thanks David! You and I know how to get results anyhow.

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### Re: ODD BALL LOGIC PROBLEM

Braininvat » Sun May 28, 2017 3:45 pm wrote:
You have a balance type Scale you often see depicted in law courts, with pans on either side to hold the balls.

You can only use this device to work out your solution.....

- from the OP

Sorry, guys. I am not sure where Einstein or Lao Tse stood on following directions, but it is a requirement in logic puzzles.

Breakthroughs, may I remind you, happen when 'rules' are not followed obsessively.
For instance, what if no 'device' is needed, will you burn us as witches?
Must the scale be used?
Does the OP require the use of the scale?
Let's go check, brb...
...
Nope, it does not, it just requires that IF you use a scale/device, it be that sort, not the bathroom scale (represented by that handy ellipsis).
nameless
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### Re: ODD BALL LOGIC PROBLEM

doogles » May 28th, 2017, 11:16 pm wrote:If they are all made from the same material, the diameter of the odd ball has to be different.

Line them up on a flat surface and place a steel straight edge along the top of them.

If it's a light ball, it won't touch the straight edge.

If it's a heavy ball, the straight edge will pivot over it.

The odd ball puzzle is about about comparing the mass of the 12 balls against each other USING A TWO PAN SCALE and by a process of elimination find our which ball differs in mass/weight from the other 11 balls

You must find the odd ball by using the scale in just three times efforts
Alan McDougall
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### Re: ODD BALL LOGIC PROBLEM

Yes Allan McDougall, David_Oblad and I will just have to go to the naughty corner. I'm ashamed of myself. I'll have an extra glass of wine tonight as penance.

doogles
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### Re: ODD BALL LOGIC PROBLEM

Hi all,

Biv is correct, and Doogles and I are deviants. However, I might be finding myself checking if the Scale has a flat-edge.. lol.

But.. the OP problem also states: "You have 12 identical balls in volume, shape and color". So a straight edge won't be of any use.

So I stand Corrected too. I should have read the problem closer.

Best Regards all,
Dave :^)

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### Re: ODD BALL LOGIC PROBLEM

Dave_Oblad » May 29th, 2017, 1:19 pm wrote:Hi all,

Biv is correct, and Doogles and I are deviants....

Haha. "Deviant" seems like such a harsh and judgmental word. How about "oddball"? ;-)

I'm actually fond of the outside-the-box solution....I like movies where people are trapped or facing some difficult problem that doesn't yield to orthodox solutions, and some eccentric member of the group saves the day coming up with an unorthodox one.

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