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 all-time favorite---no trickery or phony logic involved. Be warned your head may explode if you think about it too much. You ...

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Thread: the Godzilla of logic puzzles

  1. #1
    Distinguished Member Array Anubis's Avatar
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    the Godzilla of logic puzzles

    This one is my all-time favorite---no 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 2-pan 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. Anti-hint: for example, you could put 6 balls in each pan. Assume the left pan sinks. Of course that wouldn't help---it 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!

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  3. #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 if-then 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

  4. #3
    Distinguished Member Array Anubis's Avatar
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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 1-4 in left pan, 5-8 in right pan
    result: balls 1-4 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.

  5. #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

  6. #5
    Member Array cpmiv's Avatar
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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)

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    Distinguished Member Array Anubis's Avatar
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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 1-4 in left pan, 5-8 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 pan---I 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.

  8. #7
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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
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    "To own a gun and assume that you are armed
    is like owning a piano and assuming that you are a musician!."


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    Distinguished Member Array Anubis's Avatar
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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 heavy--ball 1 is heavy
    right pan heavy, right pan heavy, right pan heavy--ball 1 is light

    left pan heavy, left pan heavy, right pan heavy--ball 2 is heavy
    right pan heavy, right pan heavy, left pan heavy--ball 2 is light

    left pan heavy, balance, balance--ball 3 is heavy
    right pan heavy, balance, balance--ball 3 is light

    left pan heavy, right pan heavy, right pan heavy--ball 4 is heavy
    right pan heavy, left pan heavy, left pan heavy--ball 4 is light

    right pan heavy, right pan heavy, balance--ball 5 is heavy
    left pan heavy, left pan heavy, balance--ball 5 is light

    right pan heavy, balance, right pan heavy--ball 6 is heavy
    left pan heavy, balance, left pan heavy--ball 6 is light

    right pan heavy, balance, left pan heavy--ball 7 is heavy
    left pan heavy, balance, right pan heavy--ball 7 is light

    right pan heavy, left pan heavy, balance--ball 8 is heavy
    left pan heavy, right pan heavy, balance--ball 8 is light

    balance, right pan heavy, right pan heavy--ball 9 is heavy
    balance, left pan heavy, left pan heavy--ball 9 is light

    balance, left pan heavy, balance--ball 10 is heavy
    balance, right pan heavy, balance--ball 10 is light

    balance, right pan heavy, left pan heavy--ball 11 is heavy
    balance, left pan heavy, right pan heavy--ball 11 is light

    balance, balance, left pan heavy--ball 12 is heavy
    balance, balance, right pan heavy--ball 12 is light

  10. #9
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    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
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    "To own a gun and assume that you are armed
    is like owning a piano and assuming that you are a musician!."


    http://www.rkba-2a.com/ - a portal for 2A links, articles and some videos.

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    Senior Member Array tanksoldier's Avatar
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    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.

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    Distinguished Member Array 4my sons's Avatar
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    Uuuum, what he said.



    Quote Originally Posted by tanksoldier View Post
    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.
    "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

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    Distinguished Member Array Anubis's Avatar
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    Quote Originally Posted by tanksoldier View Post
    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 hammer-and-tongs 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!

  14. #13
    Distinguished Member Array Anubis's Avatar
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    Quote Originally Posted by P95Carry View Post
    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 half-expected 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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