Lateral Thinking XXI

[Holmes]

Oh My God! Such a Long time after the last Lateral Thinking Problem, I had to find a new one, maybe the old Masters of Reasoning will come out!

Here it is:

  You have twelve marbles. Eleven of the marbles are of equal weight, but one is heavier or lighter. You have a balancing scale you can use to find this marble and figure out if it weighs more or less than the others. What is the minimum number of weighings required to do this?

[S.H.]

Oh My God! Such a Long time after the last Lateral Thinking Problem, I had to find a new one, maybe the old Masters of Reasoning will come out!

Are you talking about ayw and Sayumi? 

Question: What do you mean by "What is the minimum number of weighings required to do this?"

Is it with the minimum number of weighings with luck or not? 

[bash7353]

[...]
  You have twelve marbles. Eleven of the marbles are of equal weight, but one is heavier or lighter. You have a balancing scale you can use to find this marble and figure out if it weighs more or less than the others. What is the minimum number of weighings required to do this?

Okay, so assuming you mean, what is the minimum number of weighings required to do this, for a method that works always - because if you don't the answer is obviously 1, since you can put 1 marble on each side of the balancing scale and get lucky - my answer is

Spoiler:

4,

because

Spoiler:

that is the way I would solve this: I would create 3 groups of marbles each consiting of 4 marbles. Now I would put 1 of those grups on both sides of the balancing scale and either find out that both groups weigh the same - which means the marble lighter or heavier then the others has to the in the third group, and the second weighing would have the purpose of finding out whether it is heavier or lighter - or the 2 sides don't weigh the same - which means the second weighing is to determine in which group the special marble is, and by doing that you automatically know whether it is heavier or lighter.
So after two weighings you have 4 marbles in which the special one is present and you know if it is lighter or heavier.
In a third weighing you put two marbles on each side of the scale, and you see in which 2 marbles the special one is present and the last fourth weighing puts each of this two remainig marbles on the scale and now we completely know which the special marble is.

[S.H.]

Okay, so assuming you mean, what is the minimum number of weighings required to do this, for a method that works always - because if you don't the answer is obviously 1, since you can put 1 marble on each side of the balancing scale and get lucky

Actually, it is not 1... 

[bash7353]

Actually, it is not 1... 

I realize that, you gotta look inside my spoilers...

[S.H.]

What I mean is that even with the help of luck, the answer is not 1 
Anyway, good job!

[c-square]

Oh My God! Such a Long time after the last Lateral Thinking Problem, I had to find a new one, maybe the old Masters of Reasoning will come out!

Here it is:

  You have twelve marbles. Eleven of the marbles are of equal weight, but one is heavier or lighter. You have a balancing scale you can use to find this marble and figure out if it weighs more or less than the others. What is the minimum number of weighings required to do this?

My answer is:

Spoiler:

An average of 3.083 weighings

because:

Spoiler:

Okay, this is the best I could do to describe the procedure:

Step 1)  Take marbles [1,2,3,4] and weigh them against marbles [5,6,7,8].  Leave marbles [9,10,11,12] off the balance
        If it balances, then you know the odd marble is in the set [9,10,11,12].  Go to Step 2a.
        If it doesn't balance, record whether [1,2,3,4] is heavier or lighter than [5,6,7,8]. Go to Step 2b.
       
Step 2a) Weigh marbles [9,10] against marbles [11,1].  Leave marble [12] off the balance
        If it balances, you know the odd marble is [12].  Got to Step 3a.
        If it doesn't balance, record whether [9,10] is heavier or lighter than [11,1], Go to step 3b.
       
Step 3a) Weigh marble [12] against marble[1].  If [12] is heavier, than the odd marble is heavier.  Otherwise it's lighter.
        Done! (3 weighings 8.3% occurance)
       
Step 3b) Weigh marble [9] against marble [10].  Leave marble [11] off the balance.
        If it balances, you know that marble [11] is the odd marble, and that it's heavier/lighter as observed in step 2a.
        Done! (3 weighings 8.3% occurance)
       
        If it doesn't balance, then if [9,10] in step 2a was heavier, the heavier marble is the odd marble. 
        Otherwise the lighter marble is the odd marble.
        Done! (3 weighings 16.7% occurance)
       
Step 2b) Weigh marbles [1,2,3] against marbles [4,11,12]. 
        If they balance, then the odd marble is in the set [5,6,7,8].  Go to step 3c.
        If they don't balance and [4,11,12] matches being heavier/lighter as [1,2,3,4] was to [5,6,7,8] in step 1
              then marble [4] is the odd marble and is heavier/lighter as observed.
        Done! (2 weighings 8.3% occurance)
       
        If they don't balance and [1,2,3] matches being heavier/lighter as [1,2,3,4] was to [5,6,7,8] in step 1
              then Go to Step 3d.

Step 3c) Weigh marbles [5,6] against marbles [7,1].  Leave marble [8] off the blanace
        If they balance, then the odd marble is [8] and it is heavier/lighter as observed in step 1.
        Done! (3 weighings 8.3% occurance)
       
        If they don't balance and [7,1] matches being heavier/lighter as [5,6,7,8] was to [1,2,3,4] in step 1,
              then [7] is the odd marble, and is heavier/lighter as observed in.
        Done! (3 weighings 8.3% occurance)
       
        If they don't balance and [9,10] matches being heavier/lighter as [5,6,7,8] was to [1,2,3,4] in step 1,
              then go to step 4.
             
Step 4)  Weigh marble [9] against marble [10]. 
        The marble that matches being heavier/lighter as [5,6,7,8] was to [1,2,3,4] in step 1 is the odd marble,
              and is heavier/lighter as observed.
        Done! (4 weighings 16.7% occurance)
       
Step 3d) Weigh marble [1] against marble [2].  Leave marble [3] off the balance.
        If they balance, then marble [3] is the odd marble and is heavier/lighter as observed in step 1.
        Done! (3 weighings 8.3%)
       
        If they don't balance and if [1,2,3,4] was heavier in step 1, then the heavier of [1] and [2] is the odd marble.
              Otherwise the lighter of [1] and [2] is the odd marble.
        Done! (3 weighings 16.7%)


Average weighings: 3.083

Unfortunately, I can't find a way to get it down to an average of 3, or to avoid a fourth weighing somewhere.

[Holmes]

Oh My God! Such a Long time after the last Lateral Thinking Problem, I had to find a new one, maybe the old Masters of Reasoning will come out!

Are you talking about ayw and Sayumi? 

And both two of you and googleearth! I was hoping you to come come out, this problem was the bait.

The last thing: googleearth and c-square, almost! But it is not the answer, keep thinking 

[bash7353]

Question: Do you want us to come up with a method to find the "special marble" that gets us to the solution in always the same number of weighings, where is doesn't make any difference whether I get lucky or not.
Because if I were to be confronted with this particular situation I would certainly use a method that, if I get lucky, reveals the marble in, say 3 weighings but if I don't I would need, say 4 weighings.
I hope you undertsand what I'm saying 'cause I'm Germany...

googleearth

[S.H.]

I thought 4 was already the correct answer and was too lazy to think..

Anyway..This is my answer

Spoiler:

3 steps


I will use letters to indicate the marbles..A,B,C,D,E,F,G,H,I,J,K,L...Don't want to use 2 digits for the 11th and 12th marble 
Make 2-4 marbles sets, then put the 2 sets(Let's say ABCD and EFGH) to the balancing scale(1st step)

Possible cases from 1st step:
-They are equal:
     If they are equal, pick one of the 8 marbles used in 1st step, Lets say marble A(any would do..).
     Then make 2 sets again, (Let's say AI and JK)
     Possible cases from 2nd step
     #They are equal:
          Put A and L to the balancing scale.
          If L is heavier, it is the heavier marble.
          If L is lighter, it is the lighter marble.
     #AI is heavier:
          Put J and K to the balancing scale.
          If equal, I is the heavier marble.
          If J is lighter, J is the lighter marble.
          If K is lighter, K is the lighter marble.
     #JK is heavier:
          Put J and K to the balancing scale.(just the same...)
          If equal, I is the lighter marble.             
          If J is heavier, J is the heavier marble.
          If LK is heavier, LK is the heavier marble.

-Not equal, ABCD is heavier:
     Select 2 marbles from ABCD and 1 marble from EFGH making the 1st set(Let's say ABE)
     Select another 1 marble from ABCD and another 1 marble from EFGH and 1 from the IJKL(which 4 marbles are surely not the odd
     one)(Let's say CFI)
     #They are equal:
          Put G and H to the balancing scale.
          If equal, D is the heavier marble.
          If G is lighter, G is the lighter marble.
          If H is lighter, H is the lighter marble.
     #ABE is heavier:
          Put A and B to the balancing scale.
          If equal, F is the lighter marble.
          If A is heavier, A is the heavier marble.
          If B is heavier, B is the heavier marble.
     #CFI is heavier:
          Put C and I to the balancing scale.
          If equal, E is the lighter marble.
          If C is heavier, C is the heavier marble.

-Not equal, EFGH is heavier:
     Select 2 marbles from ABCD and 1 marble from EFGH making the 1st set(Let's say ABE)
     Select another 1 marble from ABCD and another 1 marble from EFGH and 1 from the IJKL(which 4 marbles are surely not the odd
     one)(Let's say CFI)   (just the same again)
     #They are equal:
          Put G and H to the balancing scale.
          If equal, D is the heavier lighter marble.
          If G is heavier, G is the heavier marble.
          If H is heavier, H is the heavier marble.
     #ABE is heavier:
          Put C and I to the balancing scale.
          If equal, E is the heavier marble.
          If C is lighter, C is the lighter marble.
     #CFI is heavier:
          Put A and B to the balancing scale.
          If equal, F is the heavier marble.
          If A is lighter, then A is the lighter marble.
          If B is lighter, then B is the lighter marble.   

Man...typing all of these and double-checking it takes a lot more time than the "thinking" part..Still, I am not sure if everything is correct...


Edit: Just some spelling.. and some corrections posted by c-square, thanks

[c-square]

I think you mean:

Spoiler:

[...]
     #JK is heavier:
          Put J and K to the balancing scale.(just the same...)
          If equal, I is the lighter marble.             
          If J is heavier, J is the heavier marble.
          If L K is heavier, L K is the heavier marble.

[...]

-Not equal, EFGH is heavier:
     Select 2 marbles from ABCD and 1 marble from EFGH making the 1st set(Let's say ABE)
     Select another 1 marble from ABCD and another 1 marble from EFGH and 1 from the IJKL(which 4 marbles are surely not the odd
     one)(Let's say CFI)   (just the same again)
     #They are equal:
          Put G and H to the balancing scale.
          If equal, D is the heavier lighter marble.
          If G is heavier, G is the heavier marble.
          If H is heavier, H is the heavier marble.
[...]

Congrats!  You found a way to get it down to three weighings.  I sure hope you have the right answer, as my brain is definitely exhausted after trying to work this one out.

[Holmes]

Perfect! This one was REALLY HARD, but I trusted you (everyone).

The explanation if anyone wants another:

Spoiler:

The problem can be solved in three weighings.

â—¦Weigh four marbles against four others, leaving four on the table.
â—¦If both sides are equal, all eight marbles on the scale can be eliminated. Put three of the four from the table onto one side and three from the eliminated batch on the other.
â—¦If both sides are equal, the odd marble is the last one; weigh it with any other marble to see if it's heavier or lighter.
â—¦If the side with the marbles still under consideration moves up or down, weigh one of those three marbles against one of the others, and the third marble is set aside.
â—¦If both sides are equal, the third marble is the odd one, and it is heavier or lighter depending on whether or not the scales moved down or up in the previous weighing.
â—¦If the scales move, the odd marble is the one that moves in the same direction that the three marbles under consideration moved in the previous weighing. If it moves up, it's lighter; if it moves down, it's heavier.
â—¦If the scales move, take one marble from each side and switch them. One one side only, remove the other three and set them aside for later. Replace them with three marbles from the four left on the table (now known not to be the odd one).
â—¦If the two sides are equal, the odd marble is among the three set aside. Weigh one against another, and set the third aside.
â—¦If the sides are equal, the odd marble is the third one, and it is heavier or lighter depending on which way the scales moved in the first weighing.
â—¦If the scales move, the odd marble is the one that moved in the same direction as it did in the first weighing, and it is heavier or lighter depending on whether it went down or up.
â—¦If the two sides move in different directions as in the first weighing, the odd marble is one of the two that switched places. Weigh one of the two against any of the other ten.
â—¦If both sides are equal, the odd marble is the one left out. It's heavier or lighter depending on which way the scales moved in the second weighing.
â—¦If the scales move, the marble on the scales that's under consideration is the odd one, and it is heavier or lighter depending on whether it went down or up.
â—¦If the two sides move in the same direction as in the first weighing, the odd marble is one of the three that hadn't moved from its side. Weigh one of the three against another, and set the third aside.
â—¦If the sides are equal, the odd marble is the third one, and it is heavier or lighter depending on which way the scales moved in the previous weighings.
â—¦If the scales move, the odd marble is the one that moved in the same direction as it did in the previous weighings, and it is heavier or lighter depending on whether it went down or up.

[Eve]

3 and because *noticed your post*.... Nevermind ^_^