the Godzilla of logic puzzles
This is a discussion on the Godzilla of logic puzzles within the Off Topic & Humor Discussion forums, part of the The Back Porch category; This one is my alltime favoriteno trickery or phony logic involved. Be warned your head may explode if you think about it too much.
You ...

August 15th, 2006 06:35 PM
#1
Distinguished Member
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the Godzilla of logic puzzles
This one is my alltime favoriteno trickery or phony logic involved. Be warned your head may explode if you think about it too much.
You have 12 billiard balls, all the same size. You know that 11 are the same weight but the twelfth is either heavier or lighter than the rest.
Your goal is to use a 2pan balance only 3 times to determine which ball is different and whether it is in fact heavier or lighter.
The balance will weigh 2 groups of balls with the same number in each group. Antihint: for example, you could put 6 balls in each pan. Assume the left pan sinks. Of course that wouldn't helpit would only indicate that the odd ball could be heavy and in the left pan or be light in the right pan.
I can post the solution later if there is interest. Enjoy!

August 15th, 2006 08:13 PM
#2
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1) Put 4 in each balance. If one side is lighter remove 2 from each side. 2) If one side is lighter 3) take the two balls on that side and put one in each pan. 2) If after removing two from each side one side is not lighter 3)take the 2 balls you removed from the original lighter side and put one in each balance.
1)If when you put the four in each balance one side is not lighter than the other. 2) Put 2 of the remaining 4 balls in each balance. 3) Take the two from the lighter balance and put one in each balance.
This is a simple ifthen puzzle or if you've ever worked on computers or other equipment a troubleshooting flowchart.
George
Two things are infinite: the universe and human stupidity; and I'm not sure about the universe. Albert Einstein

August 15th, 2006 09:43 PM
#3
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I appreciate your interest; however, your proposed solution is incorrect. The solution must work for all possible cases. Here's one case of your method failing:
Number the balls, and let's stipulate that ball #5 is heavier than the rest.
1st weighing: balls 14 in left pan, 58 in right pan
result: balls 14 are lighter
Given this result, your next step is to arbitrarily pick 2 balls from the light group to compare against 2 balls from the heavy group. Let's assume you compare 1,2 against 5,6.
2nd weighing: balls 1,2 in left pan, 5,6 in right
result: balls 1,2 are lighter
Given this result, your next step is to compare ball 1 against ball 2.
3rd weighing: ball 1 against ball 2
result: they are equal, so all you know is 5 or 6 is the heavy, but not which one. Busted.
(I too have several years' experience of programming logic and pure math logic. I tried your solution and a few others en route to the valid solution. Trust me, this one is not easy.)
Last edited by Anubis; August 15th, 2006 at 09:52 PM.

August 15th, 2006 09:50 PM
#4
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I came back a while ago and reread the puzzle. I discovered that in my original reading I had skipped the word "heavier". Being that I had already posted my incorrect answer I refused to go back and change it.
Just didn't want to edit it and appear to have solved it at the time shown when the original post was incorrect.
Thanks for posting it. Now that everyone knows my answer was wrong, I'll look at the puzzle again about midnight and see if I can solve it, not one that wasn't posted.
George
Two things are infinite: the universe and human stupidity; and I'm not sure about the universe. Albert Einstein

August 16th, 2006 09:09 AM
#5
Member
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I'll give it a go
1st balance: 4 balls on each side. This will narrow it down to 4 balls as either one of the two groups on the balance are heavier or the last 4 not on the balance.
2nd balance: 2 balls on each side. This will narrow it down to 2 balls.
3rd balance: 1 ball on each side. This will identify the heavy ball.
I hope I haven't missed anything.
There are over 550 million firearms in worldwide circulation. That's one firearm for every twelve people on the planet. The only question is: How do we arm the other 11? (Yuri Orlov [Nicolas Cage] Lord of War)

August 16th, 2006 07:30 PM
#6
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All right! More interest in this gem. Unfortunately your proposed solution doesn't work because the odd ball could be either heavier or lighter, we don't know which in advance. In your first weighing, you have 4 balls on each pan and 4 balls not weighed. Suppose the left pan sinks, so at this point you know only that one of those balls is heavy or one of the balls on the right pan is light.
Number the balls like the example in post #3, and this time #6 is the odd ball, lighter than the others.
1st weighing: balls 14 in left pan, 58 in the right.
result: left pan heavy
Arbitrarily picking 2 balls from each group for the next weighing:
2nd weighing: 1 & 2 in left pan, 5 & 6 in right
result: left pan heavy
3rd weighing: one ball in each panI am not sure what you meant for this one, so here are all possibilities:
case 1: 1 in left, 2 in right, result: they balance
case 2: 1 in left, 5 in right, result: they balance
case 3: 1 in left, 6 in right, result: either 1 is heave or 6 is light
case 4: 2 in left, 5 in right, result: they balance
case 5: 2 in left, 6 in right, result: either 2 is heavy or 6 is light
case 6: 5 in left, 6 in right, result: either 5 is heavy or 6 is light
In all cases, this scenario fails to identify 6 as the light ball.
Maybe I misunderstood your choices for the second weighing; please correct me if that's the case. Even if I did misunderstand, I think other choices for the 2nd weighing run into the same problems in the 3rd weighing.
A serious hint: since the odd ball could be any of the 12, the solution must generate unique information about each one to solve all cases.

August 16th, 2006 08:40 PM
#7
Assistant Administrator
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Steve  you are a cruel man
I love these teasers but these days they make my head hurt, and for sure my math dexterity is leagues below yours.
I used to try solutions by converting the whole puzzle to algebraic terms and then writing an aquation or two  now seem too lazy any more!!
Chris  P95
NRA Certified Instructor & NRA Life Member.
"To own a gun and assume that you are armed
is like owning a piano and assuming that you are a musician!."
http://www.rkba2a.com/  a portal for 2A links, articles and some videos.

August 16th, 2006 11:15 PM
#8
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the solution
I am trying to cut back on infliction of cruelty, so here's the solution.
For a weighing, a particular ball can be in the left pan, the right pan, or neither pan. The table below lists the possibilities for each ball in 3 weighings (using L for left, R for right and O for out). LLL (1) means this weighing pattern is selected for ball #1, etc.
LLL (1)
LLO
LLR (2)
LOL
LOO (3)
LOR
LRL
RRR
RRO (5)
RRL
ROR (6)
ROO
ROL (7)
RLR
LRO
LRR (4)
OLL
OLO (10)
OLR
OOL (12)
RLO (8)
RLL
ORR (9)
ORO
ORL (11)
ROO
OOO
Further table entries would be duplicates of previous entries. Each ball is tagged with a number, assigned to the sequences indicated above, and is included in at least 1 weighing  this precludes using sequence OOO. We must select a unique sequence for each ball; one, furthermore, that is not the reverse of another ball’s sequence. For example, LLL and RRR are opposites. If ball 1’s sequence is LLL and ball 2’s is RRR and the left pan is heavier in 3 weighings of equal numbers of balls, it could be that either ball 1 is heavy or ball 2 is light. Ball 5’s sequence of RRO means that it is in the right pan for the 1st and 2nd weighings, and is omitted in the 3rd weighing; so if the right pan is heavy in the 1st and 2nd weighing and the pans balance in the 3rd, ball 1 is heavy. If the left pan is heavy in the 1st and 2nd weighing and the pans balance in the 3rd, ball 1 is light. The list below shows the arrangement of balls for weighing.
1st weigh: left(1 2 3 4), right(5 6 7 8), out(9 10 11 12)
2nd weigh: left(1 2 8 10), right(4 5 9 11), out(3 6 7 12)
3rd weigh: left(1 7 11 12), right(2 4 6 9), out(3 5 8 10)
From this, we can construct the following list, relating each possible weighing pattern to the ball producing the pattern. The result of the first, second, third weighing, and the cause.
1:left pan heavy, 2:left pan heavy, 3:left pan heavyball 1 is heavy
right pan heavy, right pan heavy, right pan heavyball 1 is light
left pan heavy, left pan heavy, right pan heavyball 2 is heavy
right pan heavy, right pan heavy, left pan heavyball 2 is light
left pan heavy, balance, balanceball 3 is heavy
right pan heavy, balance, balanceball 3 is light
left pan heavy, right pan heavy, right pan heavyball 4 is heavy
right pan heavy, left pan heavy, left pan heavyball 4 is light
right pan heavy, right pan heavy, balanceball 5 is heavy
left pan heavy, left pan heavy, balanceball 5 is light
right pan heavy, balance, right pan heavyball 6 is heavy
left pan heavy, balance, left pan heavyball 6 is light
right pan heavy, balance, left pan heavyball 7 is heavy
left pan heavy, balance, right pan heavyball 7 is light
right pan heavy, left pan heavy, balanceball 8 is heavy
left pan heavy, right pan heavy, balanceball 8 is light
balance, right pan heavy, right pan heavyball 9 is heavy
balance, left pan heavy, left pan heavyball 9 is light
balance, left pan heavy, balanceball 10 is heavy
balance, right pan heavy, balanceball 10 is light
balance, right pan heavy, left pan heavyball 11 is heavy
balance, left pan heavy, right pan heavyball 11 is light
balance, balance, left pan heavyball 12 is heavy
balance, balance, right pan heavyball 12 is light

August 16th, 2006 11:23 PM
#9
Assistant Administrator
Array
Wow  just wow
Indeed, somewhat per my old approach  sort of algebraic  come even a hint of Boolean.
Thx Steve  appreciate the details!
Chris  P95
NRA Certified Instructor & NRA Life Member.
"To own a gun and assume that you are armed
is like owning a piano and assuming that you are a musician!."
http://www.rkba2a.com/  a portal for 2A links, articles and some videos.

August 16th, 2006 11:49 PM
#10
Senior Member
Array
Solution: Take all balls and scale to range, shoot the crap out of everything. Buy new set of billiard balls manufactured to spec.
No worries.
"I am a Soldier. I fight where I am told, and I win where I fight." GEN George S. Patton, Jr.

August 17th, 2006 09:48 AM
#11
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"fundamental principle of American law that a government and its agents are under no general duty to provide public services, such as police protection, to any individual citizen." [Warren v. District of Columbia,(D.C. Ct. of Ap., 1981)]
If I have to explain it, you wouldn't understand

August 17th, 2006 07:48 PM
#12
Distinguished Member
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Originally Posted by
tanksoldier
Solution: Take all balls and scale to range, shoot the crap out of everything. Buy new set of billiard balls manufactured to spec.
No worries.
Good plan; this is similar to my hammerandtongs solution of Rubik's cube. Never figured out a general solution, but I learned how to disassemble it and reassemble with the colors properly arranged!

August 17th, 2006 07:57 PM
#13
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Originally Posted by
P95Carry
Wow  just wow
Indeed, somewhat per my old approach  sort of algebraic  come even a hint of Boolean.
Thx Steve  appreciate the details!
You're welcome! It is an elegant solution. I can't take credit for either the puzzle or the solution. Although I did eventually dope it out, I am sure many others did it first.
It has been around for decades. I halfexpected at least one reply like "I solved that dinosaur years ago". I Googled the words "12 ball puzzle" and got 12,600,000 hits (alleged by Google). I only looked at a few and found that most solutions listed were actually invalid.
Last edited by Anubis; August 17th, 2006 at 08:06 PM.
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