**Hardest Balance Logic Puzzle - 10 October**

You are given a set of scales and 12 marbles. The scales are of the old balance variety. That is, a small dish hangs from each end of a rod that is balanced in the middle. The device enables you to conclude either that the contents of the dishes weigh the same or that the dish that falls lower has heavier contents than the other.

The 12 marbles appear to be identical. In fact, 11 of them are identical, and one is of a different weight. Your task is to identify the unusual marble and discard it. You are allowed to use the scales three times if you wish, but no more.

Note that the unusual marble may be heavier or lighter than the others. You are asked to both identify it and determine whether it is heavy or light.

**For Solution :**Click Here

Divide the 12 marbles into Set1, Set2, Set3 of 4 marbles each.

ReplyDeleteWeighing 1:

Weigh set1 against set2.

If the dishes don't balance, lets say set2 shows as being lighter than set1 here and it also means that set3 has unbiased marbles. Follow Part1 now.

If the dishes do balance, it means set3 has the defective marble. Follow Part2 for this.

Part1:

Now we know that the defective marble belongs to either set1 or set2. Also, in weighing 1 set2 is lighter(say). We have to do some rearrangements now. Lets call the marbles as A,B,C,D in set1 and E,F,G,H in set2. Before weighing, replace A,B,C with 3 unbiased marbles from set3. Swap D with any marble(say E) from set2 and weigh it - Weighing 2 (3 biased + E vs D,F,G,H).

Now 3 cases arise.

Case1: if the dishes balance now, it means defective marble belongs to A,B,C and is heavier than the rest. Follow part3 now.

Case2: If the set2 is still lighter(same position as earlier), it means defective marble belongs to F,G,H and is lighter than the rest. Follow part3 now for these 3 marbles.

Case3: If the set2 is heavier now, it means D is defective and is heavier.

Part2:

Set3(I,J,K,L) has the defective marble.

Weighing 2: Weigh I,J,K against 3 unbiased marbles from set1 or set2.

If it balances, L is the defective marble. Weighing 3: Weight it against an unbiased marble to find if its heavier or lighter.

If its lighter/heavier, I,J,K has the lighter/heavier defective marble, follow part 3.

Part3:

This is a common case where you know whether the defective marble is lighter or heavier than the rest from the previous experiments. Here you have 1 weighing left.

Say you have A,B,C as the marbles and lets say the defective is heavier. Weighing 3: Weigh A against B, if balances, C is defective and the heavier one. If A is lighter/heavier, means B/A is defective.

What happens when A and B don't weigh the same?

DeleteSplit the marble up into groups 1,2,3 each containing 4 marbles.

Deleteweigh group 1 against 2 to see if there is an imbalance, from here many things can happen, either they can be the same weight, meaning the marble is in group 3 or it can be either in group 1 or 2, whatever the case, take the group containing the unique marble and devide it into 2 groups weighing each group against one another, from here you'll know which group of 2 contains the marble then divide that group to find the marble :)

Part1 Case3:

DeleteIt doesn't just mean D is Heavier.

E could be lighter as well.

For NorthPointingStick

DeleteIf it is in Group 1 or 2, I cannot pick out the group containing the unique marble yet.

Actually this is perfect solution!! But little detail missed in part1 case3.

DeleteIn case 3, if part two is heavier, then either D or E is defective. Take D and weigh with any normal marble. If both are same, then E is lighter. Otherwise, D is heavier :)

Split in half and balance so that there are 6 on each side. One side will be lighter, and therefore, the lighter marble will in that group of 6. Repeat, this time with 3 on each side, and now you have narrowed the light marble to 3. Finally, select two of the three and use your last weighing to compare those two weights. If the scales balance, then the remaining (that was unweighed in weighing #3) is the light marble. Otherwise, if the scales do not balance, the lighter side as indicated by the unbalanced scale is the light marble.

ReplyDeleteIt wasn't said that the marble is lighter than the rest. It may be heavier than the others, so it can be in the heavier group.

DeleteThis comment has been removed by a blog administrator.

ReplyDelete@ Keith Roberts...........As given in question, the unusual marble may be heavier or lighter than the others.....so what about the first case in which you left the heavier one....... ????

ReplyDeleteYour explaination is correct if it is given that unusual one is lighter.........

split into 6-6

ReplyDeletechoose the heavier one

then

split into 3-3

choose the heavier one

then

choose any 2 randomly and weigh

if any 1 lowers its the heavy one

else the third one is heavy

and what if it's lighter and not heavier?? how would we know which group should we take on the first step... lighter or heavier?

Deletedivide 12 marbles into 4 sets (3 each)-A,B,C,D

ReplyDeletecase 1 :if A = B then

compare A or B = c then

split D indiviually -D1,D2,D3

if D1=D2

then

D3 is unidentical

@Faith Departed !

IF A not equal to B or D1 not equal to D2 more than 3 measurements will be required.

Deleteif A is not equal to B, we know that it is in either A or B.

Deletenotice the weight of A and B. say A is heavier and B is lighter.

now , weight A with C.

if both are not of equal weight then we know that it is in A and heavier one. weight any two marbles of A if both equal( thn third one and is heavier) and if not then the heavier one is that marble.

if A and C are of equal weight then we know that it is in B. and we can easily find the marble in B by weighing any two and we already know that the marble is lighter as B is lighter

Split the 12 marbles into 4 sets of 3 each. say s1(1,2,3), s2(4,5,6), s3(7,8,9) and s4(10,11,12).

ReplyDelete1. Compare s1 with s2. Two possible scenarios can occur. They can be either equal or not equal.

If they are not equal, it means either set s1 or set s2 contains a defective item. To check that, compare either s1 or s2 with s3(since s3 cannot be defective for this case). If s1 vs s3 are equal, then s2 contains the defective marble. Select any two marbles from s2 to and see if they are equal to find out the defective marble.

If they are equal, then either s3 or s4 contains the defective marble.

Repeat the above process on s3 and s4 to determine the defective piece.

How its possible give me the clear explanation?

Deleteu can use the scale only for 3 times, but in your comment u r trying to use scale several times.

DeleteThere is a problem witht the last step. If in the first step s1 != s2, in the second s1=s3 and in the third the two marbles from s2 (s2(1) and s2(2)) s2(1) != s2(2) then there is no way of knowing whether it is s2(1) or s2(2) that is defective. Your method will work 1/3 of the time

Delete"Repeat the above process on s3 and s4 to determine the defective piece" is a fourth measurement.

DeleteHere's the solution:

ReplyDeleteI will use > to be mean heavier than, < to mean lighter than, and = to mean equal to. I will also use P and * to mean possibility and possibility of possibility (eg. P1*3*1 would mean possibility 1 of possibility 3 of possibility 1. And finally I will use N to represent the normal marbles. so let's begin

Start by splitting the marbles into 3 groups of 4 so group 1(A,B,C,D), group 2(E,F,G,H) and group 3 (I,J,K,L). Now weigh group 1 VS group 2. There are 2 possible outcomes.

P1. group 1 is > or < than group 2. So now I know the defective marble is in group 1 or 2 and group 3 is good. For this experiment let's say group 1 is > than group 2 (remember this because it is important). Now make 3 new groups of group 1 and 2. so now I have

group I(A,B,E), group II(C,D,F) and group III(G,H). Weigh group I against group II. There are 3 possibilities.

P1*1. group I is > group II. This tells me that F is lighter than all the other marbles or A or B is heavier than all the other marbles. This is because I have moved all other marbles except these 3 but the results are still the same as P1 where the former > then the latter. So now I now that A+B > F+N. Now weigh A vs B. There are 3 possibilities.

P1*1*1. A > B. A is the defective ball and is heavier then the rest. Since I know that A or B has to be heavier or F lighter. I know that the heavier one between A or B is the defective ball

P1*1*2. A < B. B is the defective ball and is heavier than the rest. same reason as above.

P1*1*3. A=B. F is my defective ball and is lighter than the rest. Since if A=B that means A and B are the normal balls meaning C is defective. Since I know from that F+N > A+B. F is defective and lighter

P1*2 group I < group II. Now I know that the defective ball could be E that is lighter than everything else or C or D that will be heavier than everything else. compare C Vs D then. once again 3 possibilities

P1*2*1 C > D. C is the defective ball and is heavier because like P1*1*1. I know that the heavier ball within the C and D is defective and heavier.

P1*2*2 C < D. D is defective and heavier as above.

P1*2*3 C=D. E is defective and lighter same as P1*1*3

P1*3 group I = group II. the defective ball is in group III (G,H) and is lighter than the rest. compare G Vs H. Two possibilities may occur.

P1*3*1 G > H. H is my defective ball and is lighter than the rest. Since I already knew that the defective ball had to be lighter, all I had to do is find the lighter one.

P1*3*2 G < H. G is my defective ball and is lighter than the rest. As a above

You have only shown 8 of 24 possible outcomes.

DeleteG lighter, H lighter, E lighter, D heavier, C heavier, F lighter, B heavier, A heavier.

Where is A lighter, B lighter, C Lighter, C heavier ...

This comment has been removed by the author.

ReplyDeleteP2. group 1 = group 2. Therefore the defective ball is in group 3 (I,J,K,L). Remove L from the group and compare I,J,K to three balls you know are normal so A,B,C. Three possibilities can happen.

ReplyDeleteP2*1 A,B,C > I,J,K. Since I know that A,B,C are all normal balls, I know that the defective ball is I, J or K and is lighter than the others. Now compare I with J. There are 3 possibilities.

P2*1*1. I > J. J is my defective ball and is lighter. I know that the defective ball has to be lighter and since J < I. J is defective and lighter.

P2*1*2. I < J. I is my defective ball and is lighter. As a above

P2*1*3 I = J. K is my defective ball and is lighter. In P2*1 I found out the defective was ball was lighter but didn't know which one. Since I = J, K must be defective because it is the last one left.

P2*2 A,B,C < I,J,K. Now I know that I, J, or K is defective and is heavier than the rest. compare I and J now. There are 3 possibilities

P2*2*1-3: I am sorry but I am kind of pressed for time. These 3 steps are mostly the same as P2*1*1-3 except the answer is heavier opposed to lighter.

P2*3. A,B,C = I,J,K. Now I know that I,J, or K is not the defective ball so therefore L is my defective ball.

Compare L with an N so let's compare L with A. There are only 2 possibilities

P2*3*1. A > L. L is my defective ball and is heavier than the rest. I know that the defective ball has to be L but I didn't know if it weighed more or less. Now I know.

P2*3*2 A < L. L is my defective ball and is lighter than the rest as above.

Problem Solved

what an answer

DeleteBut But But what if Group 1 does not equal Group 2?

DeleteYou've only solved for I,J,K,L.

Solution for finding the odd marble from a set of thirteen (13) seemingly identical marbles

ReplyDelete1ST WEIGHING - 5 ON 5, 2ND WEIGHING 3 ON 3, 3RD WEIGHING 1 ON 1. "S" REPRESENTS A MARBLE KNOWN TO BE OF STANDARD WEIGHT, BUT IS NOT ONE OF THE 13. USE OF THIS ITEM IS CRITICAL TO THE OUTCOME OF THE TEST

"/" => LEFT SIDE IS HEAVIER

"\" => RIGHT SIDE IS HEAVIER

"-" => BOTH SIDES ARE EQUAL

12345/6789S (OUTCOME 1A)

126/347

1/2 => 1H

1\2 => 2H

1-2 => 7L

126\347

3/4 => 3H

3\4 => 4H

3-4 => 6L

126-347

8/9 => 9L

8\9 => 8L

8-9 => 5H

12345\6789S (OUTCOME 1B)

126\347

1/2 => 2L

1\2 => 1L

1-2 =>7H

126/347

3/4 => 4L

3\4 => 3L

3-4 => 6H

126-347

8/9 => 8H

8\9 => 9H

8-9 => 5L

12345-6789S (OUTCOME 1C)

1011/12S (you can use "S" or any of 1 through 9)

10/11 => 10H

10\11 => 11H

10-11 =>12L

1011\12S

10/11 => 11L

10\11 => 10L

10-11 => 12H

1011 - 12S

13/S => 13H

13\S => 13L

13-S => ALL IDENTICAL

S" REPRESENTS A MARBLE KNOWN TO BE OF STANDARD WEIGHT, BUT IS NOT ONE OF THE 13

DeleteSorry but the problem only provides you with 12 (not 13) marbles. There is no marble S around.

As well if outcome 1C which is 12345-6789S is true then

outcome 10 11 - 12 S cannot occur.

ReplyDeleteNumbering the marbels as 1,2,3,4,5,6,7,8,9,A,B,C.

Compare 123 and 456.

If 123 = 456, Odd one is in the remaining ones.

Compare 123 and 789.

If 123 = 789, odd one is in ABC.

If 123 != 789, odd one is in 789.

If 123 != 456, Odd one is in one of these only.

Compare 123 and 789.

If 123 = 789, odd one is in 456.

If 123 != 789, odd one is in 123.

So, we are narrowed down to 3 marbles having that 1 odd marble. Also, since in the 2nd comparison, we knew that the other group had all identicals, we know if the odd one is lighter or heavier.

We know the remaining 3, say AA, BB and CC. Also we know if the odd one is lighter or heavier. Do this:

Compare AA, BB.

If AA = BB, CC is the ODD one.

If AA != BB, based on if the odd one is lighter or heavier, you will know the answer.

u r testing 4 times...while u cn chk only thrice..

Delete"If 123 = 789, odd one is in ABC".

DeleteHowever you do not know if the odd one is light or heavy as ABC has never been on the scales.

Correct

ReplyDeleteFirst marbles to be measured

ReplyDelete000-1

001-2

010-3

011-4

100-5

101-6

110-7

111-8

Second marbles

-00-9

-01-10

-10-11

-11-12

----------------------------------

First scenario

Explanation: Each marble is given a number and the three digit binary number representing where it will be in the balance in each balancing try( 0 is left, 1 is right) exp: marble 5, first try 1, right, second try 0, left, third try 0, left). TAKE NOTE IN EACH BALANCING ACTION(if it goes left write 0, if right write 1). After trying the first time, if the balance go right switch the marbles 1,2,3,4 with 9,10,11,12. After trying the second time if they balance, switch 1,2,3,4 with 5,6,7,8, if not, keep on trying the next balancing.After the third balancing take note. You now should get the result of a 3 digit number exp:101 witch correspond to number 6 which tell it is the heavy one but 010 which is 3 can be the lighter one so you have to see the second try, if the switched marbles balance with the other marbles they cannot be different. exp: if in the second try 5,6,7,8 balanced with 9,10,11,12 they are normal marbles so 010 which is 3 is the lighter one.

Second scenario

If they dont balance in the first place, than if you have understood the first procedure, should be easy

There are three states (Left, Balanced, Right).

Deleteyou only talk of two states (Left, Right)

What if 1,2,3,4 balances with 5,6,7,8 in the first test?

take 8 marbles put on balance 4 each side

ReplyDelete4 l 4

- -

if they are equal then the other 4 have the odd one out in which case take them to the scales

2 l 2

- -

then one set will have an odd one out so divide them and use your last chance at the balance like this

1 l 1

_ _

now you know whether the heavier or lighter is odd thanks to the earlier uses of the scale

now if one of the 8 you selected is odd (most likely)

you have used the scales once and no 4 are out

so divide the set of 4 into 2

2 l 2

- -

and you know the last step but if the groups you divided into 2 are equal that means that the other 4 have the odd

I cannot tell if the lower dish has the heavy odd marble or the higher dish has the light odd marble.

DeleteWhat if 4|4 is not equal?

This comment has been removed by the author.

ReplyDelete1, 2, 3, 4 5, 6, 7, 8 9, 10, 11, 12

ReplyDelete1, 2, 3, 4 vs 5, 6, 7, 8 -------------- 1st weigh

if 1, 2, 3, 4 is heavy then

keep out - 4, 8

5, 2, 3 vs 1, 6, 7 -------------- 2nd weigh

if 5, 2, 3 == 1, 6, 7 then

4 , 5 ------------------- 3rd weigh

if 4 == 5 then 8 is lighter (5 is standard marbel)

if 4 goes up - 4 is lighter (5 is standard marbel)

if 4 goes down - 4 is heavier (5 is standard marbel)

else if 5, 2, 3 is heavy then

remove 4,8,5,1 - all standard marbels

keep out - 7

2, 3 & 6, 9

6, 3 vs 2, 9 ----------------- 3rd weigh

if 6, 3 is heavy then

3 is heavier

if balanced

7 is lighter

if 2,9 is heavier

2 is heavier

else if 1, 6, 7 is heavier

remove 4,8

5 vs 9 ----------------- 3rd weigh

if 5 == 9 then 1 is heavier

else if 5 <= 9 then 5 is lighter

else if 1, 2, 3, 4 == 5, 6, 7, 8 then,

take - 9, 10, 11, 12

keep out 12

(1 is standard marbel)

9,10 vs 11,1 ----------------- 2nd weigh

if 9,10 == 11,1 then,

1 vs 12 -------------------- 3rd weigh

if 12 is heavier then as it is.

else if 12 lighter then as it is.

if 9, 10 is heavier then,

keep out 11, 1

9 vs 10 ----------------- 3rd weigh

if 9 == 10 11 is lighter

else if 9 is heavier then 9 is heavier

else if 9 is lighter then 10 is heavier

else if 11,1 is heavier then

keep out 11,1

9 vs 10 ----------------------- 3rd weigh

if 9 == 10 then 11 is heavier

else if 9 is lighter then as it is.

else if 9 is heavier then 10 is lighter

"if 4 goes up - 4 is lighter (5 is standard marble)"

Deletecannot occur as 4 is in group 1,2,3,4 defined as heavy.

"if 2,9 is heavier

2 is heavier" or 6 is lighter.

You have shown 15 cases. There are 24 in total. I have trouble with knowing 2 is heavier. Where are cases 1 light, 2 light, 3 light, 4 light, 5 heavy, 6 heavy, 6 light, 7 heavy, 8 heavy?

Take the marbles as A B C D E F G H I J K L

ReplyDeleteFirst turn: Weigh A B C D E F against G H I J K L

We will now know which set is faulty, say A B C D E F is lighter.

Second turn: Take out any 2 from this set but do not mix it with second set. Let us take A B out. Now weigh C D against E F. If scales show equality, then one of A B is faulty. If scales show C D or E F as light, then A B are right.

You now know which set is faulty.

Third turn: Now let us assume A B are faulty. Weigh A against B. The marble which is faulty is now revealed.

You don't know if the faulty marble is in A B C D E F or G H I J K L from the first measurement. If the faulty is light it is the high dish. if the faulty is heavy it is in the low dish. which dish can I take out 2 from for the second measurement?

DeleteTry this. Say the faulty marble was heavy and A B C D E F is in the high dish. You weigh C D with E F. It will balance. But the faulty one is not in A B. It's a heavy marble in G H I J K L.

Divide the marbles into 4 Groups of 3. Label the marbles:

ReplyDelete- Red1, Red2, Red3 for the Red Group

- Yellow1, Yellow2, Yellow3 for the Yellow Group

- Blue1, Blue2, Blue3 for the Blue Group

- Black1, Black2, Black3 for the Black Group

The balance scale is shown with:

Left dish <=> Right dish

Marbles “in hand” are important to the test’s conclusion.

There are 3 possible test outcomes:

1. The left dish is heavy, the scale tilts left.

2. The right dish is heavy, the scale tilts right.

3. The dishes weigh the same, the scale balances.

A “Known” marble is not a possible unusual marble as determined by 1st and 2nd test.

######

1st TEST has 4 marbles in each dish.

Red Group, Black1 <=> Yellow Group, Black2

Blue Group, Black3 in hand

######

2nd TEST has 4 marbles in each dish. The Red, Yellow, and Blue groups of marbles rotate and the Black marbles stay in place.

Blue Group, Black1 <=> Red Group, Black2

Yellow Group, Black3 in hand

After the 1st and 2nd test, only Black3 has not been on the scale.

######

3rd TEST has only one marble in each dish. There are 9 setups for the 3rd test depending on the first and second tests.

SETUP 1 - 1st test tilted left - 2nd test tilted left

Unusual marble is either Black1 and heavy or Black2 and light. Scales didn’t change and only the Black balls were not moved.

Known <=> Black1

Black2 in hand

Can’t tilt left as Black1 can’t be light.

Tilts right. Black1 is heavy.

Balanced. Black2 is light.

SETUP 2 - 1st test tilted left - 2nd test tilted right

Unusual Marble is Red and heavy. The Red group was always on the scales on the low side.

Red1 <=> Red2

Red3 in hand

Tilts left. Red1 is heavy.

Tilts right. Red2 is heavy.

Balanced. Red3 is heavy.

SETUP 3 - 1st test tilted left - 2nd test balanced

Unusual marble is Yellow and light. Scales balanced without the Yellow group and high on the Yellow group side when the Yellow group was on the scale.

Yellow1 <=> Yellow2

Yellow3 in hand

Tilts left. Yellow1 is light.

Tilts right. Yellow2 is light.

Balanced. Yellow3 is light.

SETUP 4 - 1st test tilted right - 2nd test tilted left

Unusual marble is Red and light. The Red group was always on the scales on the high side.

Red1 <=> Red2

Red3 in hand

Tilts left. Red1 is light.

Tilts right. Red2 is light.

Balanced. Red3 is light.

SETUP 5 - 1st test tilted right - 2nd test tilted right

Unusual marble is either Black 1 light or Black 2 heavy. Scales didn’t change and only the Black balls were not moved.

Known <=> Black1

Black2 in hand

Tilts left. Black1 is light.

Can’t tilt right as Black1 can’t be heavy.

Balanced. Black2 is heavy.

SETUP 6 - 1st test tilted right - 2nd test balanced

Unusual marble is Yellow and heavy. Scales balanced without the Yellow group and low on the Yellow group side when the Yellow group was on the scale.

Yellow1 <=> Yellow2

Yellow3 in hand

Tilts left. Yellow1 is heavy.

Tilts right. Yellow2 is heavy.

Balanced. Yellow3 is heavy.

SETUP 7 - 1st test balanced - 2nd test tilted left

Unusual marble is Blue and heavy. Scales balanced without the Blue group and low on the Blue group side when the Blue group was on the scale.

Blue1 <=> Blue2

Blue3 in hand

Tilts left. Blue1 is heavy.

Tilts right. Blue2 is heavy.

Balanced. Blue3 is heavy.

SETUP 8 - 1st test balanced - 2nd test tilted right

Unusual marble is Blue and light. Scales balanced without the Blue group and high on the Blue group side when the Blue group was on the scale.

Blue1 <=> Blue2

Blue3 in hand

Tilts left. Blue1 is light.

Tilts right. Blue2 is light.

Balanced. Blue3 is light.

SETUP 9 - 1st test balanced - 2nd test balanced

Unusual marble is Black 3 as it is the only marble that has never been on the scales. It is light or heavy.

Known <=> Black3

Tilts left. Black3 is light.

Tilts right. Black3 is heavy.

Can’t balance as Black3 can only be light or heavy.

Three corrections to my submission:

DeleteCorrection #1 SETUP 3 -

Tilts left. Yellow2 is light.

Tilts right. Yellow1 is light.

Correction #2 SETUP 4 -

Tilts left. Red2 is light.

Tilts right. Red1 is light.

Correction #3 SETUP 8 -

Tilts left. Blue2 is light.

Tilts right. Blue1 is light.

This comment has been removed by the author.

ReplyDeleteHERE IS THE SOLUTION FOLKS !!!

ReplyDeletelet ABCD EFGH KLMN be the names of marblesA

#1---- if ABCD=EFGH , compare ABC & KLM.

if ABC=KLM, compare A&N

if AN , N is light

if ABCL ,K is heavy

if K=L , M is heavy

if ABC>KLM , compare K&L

if KL ,L is light

if K=L , M is light

#2---if ABCD>EFGH ,compare ABE & CFK

if ABE=CFK , compare G&H

if G=H , D is heavy

if GH , H is light

if ABE< CFK , compare E & K

if E=K , C is heavy

if E CFK compare A&B

if A=B , F is light

if A>B , A is heavy

if A<B , B is heavy

#3----if ABCD<EFGH repeat #2 taking heavier set as ABCD

------Best Regards,

Siraj Mangalassery

Ha! I know this one; this was from Professor Layton and the Curious Village. Actually, not exactly; it's a variant, but the way to solve it is the same.

ReplyDelete1.Divide marbles into four sets of 3

ReplyDelete2.Use the balance for any two sets: If not balanced replace anyone with the third one. Do the same for the other case

3.By now you have identified two things, the correct set and weather the incorrect marble is lighter or heavier in just TWO STEPS

4.The incorrect marble could be any out of the three. So,take any two marbles and weigh them against each other: If they dont balance, since we have already deduced whether it is light or heavy, we know the incorrect one. Else if they balance, the third is the incorrect one.

So in THREE steps we can figure out the incorrect marble.

If the first two sets (set1 and set2) balance and we switch a pair of sets, say set2 with set3 and they also balance then we know in TWO STEPS that the incorrect marble is in set4. However set4 has never been on the scales, so we don't know if set4 is heavy or light.

Delete1. Divide marbles into 3 sets of 4.

ReplyDelete2. Let say A,B,C,D,E,F,G,H,I,J,K,L are marbles. Make a group 1. ABCD, 2.EFGH, 3.IJKL.

3. Use balance for set 1 & 2.

4. Case 1, If unbalance, defective marble will be in ABCDEFGH & marble IJKL are clean. Again make three group 4.ABEF, 5.GIJK & 6.CDHL. Use balance for group 4 & 5 (weighing # 2).

4.1. If unbalance, (IJKL are clean as per weighing # 1)

4.1.1. If weighing # 1 between group 1 & 2 and weighing # 2 between group 4 & 5 showing same tendency of group 1 & 4 either heavy or light than defective marble will be in ABG, because these are the only marbles which not replaced from its position in weighing # 1 & 2.

4.1.1.1. Now weighing # 3 will be between A & B.

4.1.1.1.1. If unbalance & tendency of A is same as group 1 & 4 (light or heavy) defective piece will be A and if weighing # 3 reads it heavy/light than defect will be heavy/light.

4.1.1.1.2. If unbalance & tendency of A is opposite to group 1 & 4 (light or heavy) defective piece will be B and if weighing # 3 reads it heavy/light than defect will be heavy/light.

4.1.1.1.3. If balance, defective piece will be G and if weighing # 1 reads group 2 as heavy/light than defect will be heavy/light.

4.1.2. If weighing # 1 between group 1 & 2 and weighing # 2 between group 4 & 5 showing opposite tendency of group 1 & 4 ( 1 light & 4 heavy or 1 heavy & 4 light) than defective marble will be in EF, because these are the only marbles which not replaced from its position in weighing # 1 & 2.

4.1.2.1. Now weighing # 3 will be between E & F.

4.1.2.1.1. If unbalance & tendency of E is same as group 2 & 4 (light or heavy) defective piece will be E and if weighing # 3 reads it heavy/light than defect will be heavy/light.

4.1.2.1.2. If unbalance & tendency of E is opposite to group 2 & 4 (light or heavy) defective piece will be F and if weighing # 3 reads it heavy/light than defect will be heavy/light.

4.2. If balance, defective marble will be in CDH.

4.2.1.1. Now weighing # 3 will be between C & D.

4.2.1.1.1. If unbalance & tendency of C is same as group 1 (light or heavy) defective piece will be C and if weighing # 3 reads it heavy/light than defect will be heavy/light.

4.2.1.1.2. If unbalance & tendency of C is opposite to group 1 (light or heavy) defective piece will be D and if weighing # 3 reads it heavy/light than defect will be heavy/light.

4.2.1.1.3. If balance, defective piece will be H and if weighing # 1 reads group 2 as heavy/light than defect will be heavy/light.

5. If balance in weighing # 1, defective marble will be in IJKL & marble ABCDEFGH are clean. Again make 3 group 7.IA 8. KB & 9.JL. Use balance for group 8 & 9 (weighing #2).

5.1. If unbalance, defective marble will be in JLK. Do the weighing # 3 between J & L.

5.1.1. If unbalance & tendency of J is same as group 9 (light or heavy) defective piece will be J and if weighing # 3 reads it heavy/light than defect will be heavy/light.

5.1.2. If unbalance & tendency of J is opposite to group 9 (light or heavy) defective piece will be L and if weighing # 3 reads it heavy/light than defect will be heavy/light.

5.1.3. If balance, defective piece will be K and if weighing # 2 reads group # 8 as heavy/light than defect will be heavy/light.

5.2. If balance, defective marble will be I. Do the weighing of marble I against marble A. if weighing # 3 reads it heavy/light than defect will be heavy/light.

thank you for 5 through 5.2

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DeleteLet’s divide 12 marbles into three sets of 4. So you will have (4 4 4) marbles.

ReplyDelete~ Colour first set of four marbles to red and number them 1,2,3,4. So you have R1, R2, R3, R4

~ Colour second set to green and number them, so you have G1, G2, G3, G4

~ Colour third set to blue and number them, so you have B1, B2, B3, B4

~ Consider ↑ = Heavier Weight, ↓ = Lighter Weight

Act: Place R1, R2, R3, R4 on one side of the scale and G1, G2, G3, G4 on opposite side. B1, B2, B3, B4 are left out.

Result: Three possible results:

1. ↓ R1 R2 R3 R4 G1 G2 G3 G4 ↑

2. ↑ R1 R2 R3 R4 G1 G2 G3 G4 ↓

3. R1 R2 R3 R4 = G1 G2 G3 G4

Now let’s explore each possible result.

1. ↓ R1 R2 R3 R4 G1 G2 G3 G4 ↑

Fact: Either one of the red is heavier or one of the green is lighter.

Act: Replace 3 Red marbles with 3 green and 3 green with 3 blue

Result: Three possible results:

1.1. ↓ R1 G2 G3 G4 G1 B2 B3 B4 ↑

Fact: Either R1 is heavier or G1 is lighter

Act: Now place R1 and R2 on opposite side of scale

Result: Two possible results:

1.1.1. ↓ R1 R2 ↑ Answer: R1 is heavier than all

1.1.2. R1 = R2 Answer: G1 is lighter than all.

1.2. ↑ R1 G2 G3 G4 G1 B2 B3 B4 ↓

Fact: one of the G2, G3, G4 is lighter.

Act: Now place G2 and G3 on opposite side of scale

Result: Three possible results:

1.2.1. ↑ G2 G3 ↓ Answer: G2 is lighter than all

1.2.2. ↓ G2 G3 ↑ Answer: G3 is lighter than all

1.2.3. G2 = G3 Answer: G4 is lighter than all.

1.3. R1 G2 G3 G4 = G1 B2 B3 B4

Fact: one of the R2, R3, R4 is heavier.

Act: Now place R2 and R3 on opposite side of scale

Result: Three possible results:

1.3.1. ↓ R2 R3 ↑ Answer: R2 is heavier than all

1.3.2. ↑ R2 R3 ↓ Answer: R3 is heavier than all

1.3.3. R2 = R3 Answer: R4 is heavier than all.

2.↑ R1 R2 R3 R4 G1 G2 G3 G4 ↓

Fact: Either one of the green is heavier or one of the red is lighter.

Act: Replace 3 Red marbles with 3 green and 3 green with 3 blue

Result: Three possible results:

2.1. ↑ R1 G2 G3 G4 G1 B2 B3 B4 ↓

Fact: Either G1 is heavier or R1 is lighter

Act: Now place R1 and R2 on opposite side of scale

Result: Two possible results:

2.1.1. ↑ R1 R2 ↓ Answer: R1 is lighter than all

2.1.2. R1 = R2 Answer: G1 is heavier than all.

2.2. ↓ R1 G2 G3 G4 G1 B2 B3 B4 ↑

Fact: one of the G2, G3, G4 is heavier.

Act: Now place G2 and G3 on opposite side of scale

Result: Three possible results:

2.2.1. ↓ G2 G3↑ Answer: G2 is heavier than all

2.2.2. ↑ G2 G3↓ Answer: G3 is heavier than all

2.2.3. G2 = G3 Answer: G4 is heavier than all.

2.3. R1 G2 G3 G4 = G1 B2 B3 B4

Fact: one of the R2, R3, R4 is lighter.

Act: Now place R2 and R3 on opposite side of scale

Result: Three possible results:

2.3.1. ↑ R2 R3 ↓ Answer: R2 is lighter than all

2.3.2. ↓ R2 R3 ↑ Answer: R3 is lighter than all

2.3.3. R2 = R3 Answer: R4 is lighter than all.

3. R1 R2 R3 R4 = G1 G2 G3 G4

Fact: one of the blue is different weight (heavier or lighter)

Act: Replace 3 Red marbles with 3 blue

Result: Three possible results:

3.1. ↑ R1 B2 B3 B4 G1 G2 G3 G4 ↓

Fact: one of the B2, B3, B4 is lighter

Act: Now place B2 and B3 on opposite side of scale

Result: Three possible results:

3.1.1. ↑ B2 B3↓ Answer: B2 is lighter than all

3.1.2. ↓ B2 B3↑ Answer: B3 is lighter than all

3.1.3. B2 = B3 Answer: B4 is lighter than all.

3.2. ↓ R1 B2 B3 B4 G1 G2 G3 G4 ↑

Fact: one of the B2, B3, B4 is heavier.

Act: Now place B2 and B3 on opposite side of scale

Result: Three possible results:

3.2.1. ↓ B2 B3↑ Answer: B2 is heavier than all

3.2.2. ↑ B2 B3↓ Answer: B3 is heavier than all

3.2.3. B2 = B3 Answer: B4 is heavier than all.

3.3. R1 B2 B3 B4 = G1 G2 G3 G4

Fact: B 1 is either heavier or lighter

Act: Now place B1 and B2 on opposite side of scale

Result: Two possible results:

3.3.1. ↓ B1 B2↑ Answer: B1 is heavier than all

3.3.2. ↑ B1 B2↓ Answer: B1 is lighter than all

Correction: ~ Consider ↓ = Heavier Weight, ↑ = Lighter Weight

ReplyDeleteMy solution is divide the 12 into set of 6 and 6.

ReplyDeleteOne set weighs more then divide that set into 3 and 3 and compare and again one set weighs more.

In that set of 3 which weighs more compare any two marbles, if one weighs more then it is odd one and if both are same then the remaining marble is odd one.

I first saw this puzzle in 1987. It took me about 2 hrs to solve and I got it while driving some friends in my car. When they told me I missed the turn I said it didn't matter because I had the solution. The solution presented is correct but it can be seriously simplified. It needs to be made very clear to anyone who attempts to solve that the scale shows that (of 2 marbles or 2 groups of marbles) one side is either heavier (the lower side) or lighter (the higher side) than the rest. For the individual who thinks measuring 6 against 6 is a useful way to start, it only shows that one of the 12 is different. Of course the puzzle parameters already state that, so this strategy results in an unproductive (and one wasted) use of the scale. It is also worthy to note that once a group of three has been isolated and its weight has been determined (i.e. the group is lighter or heavier than a comparable group) then the unique marble can be identified with only one more use of the scale.

ReplyDeleteThanks, man! Thank goodness someone got it right! (maybe someone else did too, but many got it wrong!)

DeleteAs for the answer it's actually pretty simple:

First step: You make 3 groups of 4 marbles each

Second step: On one side you place group 1 (4 marbles) and on the other side, group 2 (4 marbles)

Possibilities:

- If they balance, the odd one out is in the remaining 4 marbles (group 3) Also, we'll call groups 1 and 2 REGULAR MARBLES - So just take 3 marbles of group 3 and weigh them against 3 regular marbles...

- If they balance, the odd one out is the 4th and last marble of group 3 -Just weigh that marble against a regular marble and if this 4th marble's position is higher it means it's THE ODD ONE OUT AND LIGHTER than the rest- If it's lower, it means it is THE ODD ONE OUT AND HEAVIER than the rest...

- If those 3 marbles of group 3 weighed against the 3 regular marbles don't balance, that means the odd one out is one of those 3 marbles of group 3 - Also, that part of the scale will be either higher or lower than the other set of marbles (let's say it's higher, so they are hypothetical lighter marbles- doesn't really matter if you say higher or lower - then they will be hypothetical heavy marbles) - Simply take two of those three and weigh one of them against the other - If one is higher, that's the LIGHTER one - If they balance, the LIGHTER one is the 3rd one you didn't put on the scale (and if it was heavier, same thing - then the one that is lower will be the heavier one; otherwise the remaining marble will be the heavy one)

- If group 1 and group 2 don't balance (that's the first measurement), it means one of the 4 marbles of the side that is lower on the scale is heavier than the rest, or that one of 4 the marbles of the side that is higher on the scale is a lighter marble - So let's call 4 of those hypothetical HEAVY marbles and the other 4 (the ones that are higher on the scale) hypothetical LIGHTER marbles (in the end, only one of these assumptions will be true) - So you place 3 hypothetical HEAVY marbles plus a hypothetical LIGHTER marble on one side, and on the other side, you place 3 REGULAR marbles plus one hypothetical HEAVIER marble... So heavy-heavy-heavy-light VERSUS regular-regular-regular-heavy!

-If they balance, it's none of those marbles - So you have 3 hypothetical LIGHT marbles that you didn't put on the scale and one last try. But it's easy, because you know that since the hypothetical heavy ones balanced with the other set, the odd one out is not heavy but lighter. So you have 3 marbles and know that one of them is lighter than the rest... So easy peasy: you weigh one against another! If they balance, the remaining one is the lighter one - If they don't balance, the one that is higher on the scale will be the lighter one!

- Now what if those heavy-heavy-heavy-light VERSUS regular-regular-regular-heavy DON'T balance? Then two options:

- If the side with regular-regular-regular-heavy is lower, it means that either the hypothetical heavy one is indeed the HEAVY one, or the lighter one on the other side caused the opposite side to be lower - So we've narrowed it to two - Just weigh one of them (say the hypothetical lighter one) against a regular marble - If they balance, the other one is the odd one out, and you already knew it was a hypothetical HEAVY marble, so it is indeed a HEAVY marble- And if they don't balance, then you took a hypothetical light marble, so it will appear higher on the scale because it is indeed the odd one out and LIGHTER than the rest!

- Finally, if the side with regular-regular-regular-heavy is higher (the last possibility of all) it means one of the three hypothetical heavy marbles on the other side is a HEAVY marble- So simply weigh one against another - If they balance, the HEAVY one is the third one - If they don't balance, the one that is lower on the balance is, of course, the HEAVY one!

Have a good day everyone :)