A puzzle
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From: Yorba Linda, CA
Originally posted by Skorpion
Ok heres how to solve this.
Lets number the blocks 1-12
Place 1,2,3,4 on one side 5,6,7,8 on the other.
Either they balance or they don't.
a)If they balance, you weigh 9,10 against 11,12 ( For your second weigh in )
Lets say 9,10 were the light side on the left and 11,12 were the heavy side on the right
Weigh 9,11 against 8,12,( For the third and final weigh in )
thus moving a potential heavy to the left side and replacing a potential heavy with a known normal block.
If the left side is still light, then 9 is the odd/light block
If the right side is light, then 11 that you moved is an odd/heavy block.
If its balanced, then 10 is the odd/light block.
b)Now if the first weigh in was uneven we have this situation
Lets say the left side was light.
We have 1,2,3,4 as potential lights and 5,6,7,8 as potential heavies.
Weigh 1,2,6 against 5,3,9 ( weigh in number 2 )
(This is 2 potential lights and a potential heavy against
1 potential heavy, 1 potential light, and 1 normal block)
If this one is even, then we just weigh 7 against 8
If its those are even then 4 is the light one
Otherwise the heavy one between 7 and 8 is the odd/heavy one.
If 1,2,6 vs 5,3,9 was uneven
1) left side light
Either 1 or 2 is a light one, or 5 is a heavy one.
Weigh 1 against 2. (weigh in number 3)
If its even, 5 is the heavy
If its uneven, the light side is the light block.
2) right side light
Either 6 that we moved was a heavy, or 3 that we moved was a light.
So weigh 6 against a standard block and if its heavy, its the heavy.
If its even, 3 is a light.
So thats it. It works with 3 measurements. I understand it even if I can't write it down all that wonderfully.
If you have a right side light situation you just swap the numbers around and it works the same way.
Thanks for helping me pass the last half an hour up here at work.
Can I be an honorary mathematician?
Ok heres how to solve this.
Lets number the blocks 1-12
Place 1,2,3,4 on one side 5,6,7,8 on the other.
Either they balance or they don't.
a)If they balance, you weigh 9,10 against 11,12 ( For your second weigh in )
Lets say 9,10 were the light side on the left and 11,12 were the heavy side on the right
Weigh 9,11 against 8,12,( For the third and final weigh in )
thus moving a potential heavy to the left side and replacing a potential heavy with a known normal block.
If the left side is still light, then 9 is the odd/light block
If the right side is light, then 11 that you moved is an odd/heavy block.
If its balanced, then 10 is the odd/light block.
b)Now if the first weigh in was uneven we have this situation
Lets say the left side was light.
We have 1,2,3,4 as potential lights and 5,6,7,8 as potential heavies.
Weigh 1,2,6 against 5,3,9 ( weigh in number 2 )
(This is 2 potential lights and a potential heavy against
1 potential heavy, 1 potential light, and 1 normal block)
If this one is even, then we just weigh 7 against 8
If its those are even then 4 is the light one
Otherwise the heavy one between 7 and 8 is the odd/heavy one.
If 1,2,6 vs 5,3,9 was uneven
1) left side light
Either 1 or 2 is a light one, or 5 is a heavy one.
Weigh 1 against 2. (weigh in number 3)
If its even, 5 is the heavy
If its uneven, the light side is the light block.
2) right side light
Either 6 that we moved was a heavy, or 3 that we moved was a light.
So weigh 6 against a standard block and if its heavy, its the heavy.
If its even, 3 is a light.
So thats it. It works with 3 measurements. I understand it even if I can't write it down all that wonderfully.
If you have a right side light situation you just swap the numbers around and it works the same way.
Thanks for helping me pass the last half an hour up here at work.
Can I be an honorary mathematician?
Pretty good! However, under a) you cannot tell whether 9 is light or 12 is heavy. You're really close. I didn't check b) closely (I'll need that six-pack), but you're definitely on the right track.
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From: Berkeley
Weigh 4 blocks on each side. There are two possibilities: the two sides are either balanced or they are not.
If the two sides are balanced, it means that the block sought is among the other 4 blocks off the balance. So, for the second weighting, weigh 1 block on each side. If the two sides are balanced, then add the 2 blocks off the balance to different sides for a third and final weighting. This determines which block has a different weight and if it is lighter or heavier. Otherwise, replace either block with one of the blocks off the balance for the third weighting. Again, this determines which block has a different weight and if it is lighter or heavier. (If the two sides balance, the block sought is the one replaced and its weight is known to be either greater or less from the previous weighting.)
If the two sides are not balanced, then 5 of the 8 blocks must be separated for the second weighting. Take 2 blocks from the heavier side, calling them "highs" to denote the possibility that one of them might be heavier than all the rest, and 3 blocks from the lighter side, calling them "lows". Call the 4 blocks off the balance "normals". Then weigh 2 highs and 2 lows on one side and 1 low and 3 normals on the other. There are two possibilities: the two sides are either balanced or they are not.
If the two sides are balanced, then the block sought is among those 3 blocks off the balance, 2 highs and 1 low. To identify the block sought, simply weigh the 1 low against 1 of the highs. Notice that information regarding the type of the block sought, heavier or lighter, has been provided by the first weighting.
If the two sides are not balanced, then there are two possibilities again. If the lighter side is the side with the 1 low and the 3 normals, then the block sought is either the 1 low on this lighter side or one of the 2 highs on the other. To identify the block, simply proceed as described in the paragraph immediately above. Otherwise, if the lighter side is the side with the 2 highs and the 2 lows, then the block sought must be one of the lows and a final third weighting of the 2 lows determines the block.
If the two sides are balanced, it means that the block sought is among the other 4 blocks off the balance. So, for the second weighting, weigh 1 block on each side. If the two sides are balanced, then add the 2 blocks off the balance to different sides for a third and final weighting. This determines which block has a different weight and if it is lighter or heavier. Otherwise, replace either block with one of the blocks off the balance for the third weighting. Again, this determines which block has a different weight and if it is lighter or heavier. (If the two sides balance, the block sought is the one replaced and its weight is known to be either greater or less from the previous weighting.)
If the two sides are not balanced, then 5 of the 8 blocks must be separated for the second weighting. Take 2 blocks from the heavier side, calling them "highs" to denote the possibility that one of them might be heavier than all the rest, and 3 blocks from the lighter side, calling them "lows". Call the 4 blocks off the balance "normals". Then weigh 2 highs and 2 lows on one side and 1 low and 3 normals on the other. There are two possibilities: the two sides are either balanced or they are not.
If the two sides are balanced, then the block sought is among those 3 blocks off the balance, 2 highs and 1 low. To identify the block sought, simply weigh the 1 low against 1 of the highs. Notice that information regarding the type of the block sought, heavier or lighter, has been provided by the first weighting.
If the two sides are not balanced, then there are two possibilities again. If the lighter side is the side with the 1 low and the 3 normals, then the block sought is either the 1 low on this lighter side or one of the 2 highs on the other. To identify the block, simply proceed as described in the paragraph immediately above. Otherwise, if the lighter side is the side with the 2 highs and the 2 lows, then the block sought must be one of the lows and a final third weighting of the 2 lows determines the block.
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From: Palm Springs
Damn. I had to search the internet for the answer, but I evenutally got it. The difficulty level is rated at 2 out of 4, and can be answered by Grades 5 - 7. 
This is the same question, using golf balls:
Among 12 identical looking golf balls there is one that is defective in weight. It is either heavier or lighter than the standard one. You have a balance. You can only weigh 3 times to find out which one is defective and whether it is heavier or lighter.
This is the solution (it's REALLY long), and you can also use this pic to check what's happening as you go along:
In order to explain better, give each ball a number, said from 1 to 12. We also use ball zero to indicate a standard ball if that ball is identified as a standard ball.
Step 1.
Take 8 balls to weight, ball 1, 2, 3, and 4 on the left, ball 5, 6, 7, and 8 on the right.
If they are balanced, go to Step 2-1.
If they are not balanced, go to Step 2-2.
Step 2-1.
At this time, you know the ball 1 to 8 are OK. Take ball 9, 10 and 11, plus a standard ball - ball zero. Just assume you have ball zero and 9 on the left and ball 10 and ball 11 on the right.
If they are balanced, go to Step 3-1.
If left side is heavier, go to step 3-2.
If right side is heavier, go to Step 3-3.
Step 2-2.
At this time, you know the ball 9 to bal 12 are statndard balls. Assume the left side is heavier. Now you need to take 6 balls to weight the second time. You need to choose a standard ball - ball zero, ball 6 which might be lighter, and ball 3 which might be heavier, put them on the left side. Also you choose ball 2 which might be heavier, ball 5 and 7 which might be lighter. Notice that we change sides for ball 2 and ball 6.
If they are balanced, go to step 3-4.
If the left side is heavier, go to step 3-5.
If the right side is heavier, go to step 3-6.
Step 3-1.
From step 2-1, we know ball 9, 10 and 11 are standard balls. We just need to determine the ball 12 is heavier or lighter. We put a standard ball on the left, put ball 12 on the right.
If right side is heavier, the ball 12 is heavier.
If the right side is leghter, the ball 12 is lighter
Step 3-2.
From step 2-1, you know that either ball 9 is heavier, or one of statements should be true: the ball 10 is lighter or the ball 11 is lighter. We can put the ball 10 on the left, the ball 11 on the right.
If they are balanced, they should be OK, the ball 9 is heavier.
If the ball 10 is heavier, the ball 10 is Ok, ball 11 is lighter.
If the ball 10 is lighter, then the ball 11 is OK.
Step 3-3.
From Step 2-1, you know that either ball 9 is lighter, or one of statements should be true: the ball 10 is heavier or the ball 11 is heavier. We can put the ball 10 on the left, the ball 11 on the right.
If they are balanced, they should be OK, the ball 9 is lighter.
If the ball 10 is lighter, the ball 10 is Ok, ball 11 is Heavier.
If the ball 10 is heavier, then the ball 11 is OK.
Step 3-4.
From Step 2-2, you know ball 2, 3, 5, 6, 7 are standard balls. You can determine the ball 1 or ball 4 is heavier, or ball 8 is lighter. We can put ball 1 on the the left and the ball 4 on the right.
If they are balanced, we are sure the ball 8 is lighter.
If the ball 1 is lighter, the ball 1 is Ok and ball 4 is heavier.
If the ball 4 is lighter, the ball 4 is OK, the ball 1 is heavier.
Step 3-5.
From Step 2-2, you know ball 2 and ball 6 are OK, because the change side does not affect the consequence. You can determine the ball 3 is heavier, or one of the statements is true: ball 5 is lighter or the ball 7 is lighter. We can put ball 5 on the the left and the ball 7 on the right.
If they are balanced, we are sure the ball 3 is heavier.
If the ball 5 is heavier, the ball 5 is Ok and ball 7 is lighter.
If the ball 7 is heavier, the ball 7 is OK, the ball 5 is lighter.
Step 3-6.
From step 2-2, you know that the weighting result is changed because the ball 2 and ball 6 are exchanged. It really means that either ball 2 is heavier or ball 6 is lighter. You still have one more weight, I am sure you know how to do it!
-----------------------------------------------------
WOW! My brain hurts. Please Mr Magician, don't post any more of these. If we can't figure out this one, we are all doomed!
This is the same question, using golf balls:
Among 12 identical looking golf balls there is one that is defective in weight. It is either heavier or lighter than the standard one. You have a balance. You can only weigh 3 times to find out which one is defective and whether it is heavier or lighter.
This is the solution (it's REALLY long), and you can also use this pic to check what's happening as you go along:
In order to explain better, give each ball a number, said from 1 to 12. We also use ball zero to indicate a standard ball if that ball is identified as a standard ball.
Step 1.
Take 8 balls to weight, ball 1, 2, 3, and 4 on the left, ball 5, 6, 7, and 8 on the right.
If they are balanced, go to Step 2-1.
If they are not balanced, go to Step 2-2.
Step 2-1.
At this time, you know the ball 1 to 8 are OK. Take ball 9, 10 and 11, plus a standard ball - ball zero. Just assume you have ball zero and 9 on the left and ball 10 and ball 11 on the right.
If they are balanced, go to Step 3-1.
If left side is heavier, go to step 3-2.
If right side is heavier, go to Step 3-3.
Step 2-2.
At this time, you know the ball 9 to bal 12 are statndard balls. Assume the left side is heavier. Now you need to take 6 balls to weight the second time. You need to choose a standard ball - ball zero, ball 6 which might be lighter, and ball 3 which might be heavier, put them on the left side. Also you choose ball 2 which might be heavier, ball 5 and 7 which might be lighter. Notice that we change sides for ball 2 and ball 6.
If they are balanced, go to step 3-4.
If the left side is heavier, go to step 3-5.
If the right side is heavier, go to step 3-6.
Step 3-1.
From step 2-1, we know ball 9, 10 and 11 are standard balls. We just need to determine the ball 12 is heavier or lighter. We put a standard ball on the left, put ball 12 on the right.
If right side is heavier, the ball 12 is heavier.
If the right side is leghter, the ball 12 is lighter
Step 3-2.
From step 2-1, you know that either ball 9 is heavier, or one of statements should be true: the ball 10 is lighter or the ball 11 is lighter. We can put the ball 10 on the left, the ball 11 on the right.
If they are balanced, they should be OK, the ball 9 is heavier.
If the ball 10 is heavier, the ball 10 is Ok, ball 11 is lighter.
If the ball 10 is lighter, then the ball 11 is OK.
Step 3-3.
From Step 2-1, you know that either ball 9 is lighter, or one of statements should be true: the ball 10 is heavier or the ball 11 is heavier. We can put the ball 10 on the left, the ball 11 on the right.
If they are balanced, they should be OK, the ball 9 is lighter.
If the ball 10 is lighter, the ball 10 is Ok, ball 11 is Heavier.
If the ball 10 is heavier, then the ball 11 is OK.
Step 3-4.
From Step 2-2, you know ball 2, 3, 5, 6, 7 are standard balls. You can determine the ball 1 or ball 4 is heavier, or ball 8 is lighter. We can put ball 1 on the the left and the ball 4 on the right.
If they are balanced, we are sure the ball 8 is lighter.
If the ball 1 is lighter, the ball 1 is Ok and ball 4 is heavier.
If the ball 4 is lighter, the ball 4 is OK, the ball 1 is heavier.
Step 3-5.
From Step 2-2, you know ball 2 and ball 6 are OK, because the change side does not affect the consequence. You can determine the ball 3 is heavier, or one of the statements is true: ball 5 is lighter or the ball 7 is lighter. We can put ball 5 on the the left and the ball 7 on the right.
If they are balanced, we are sure the ball 3 is heavier.
If the ball 5 is heavier, the ball 5 is Ok and ball 7 is lighter.
If the ball 7 is heavier, the ball 7 is OK, the ball 5 is lighter.
Step 3-6.
From step 2-2, you know that the weighting result is changed because the ball 2 and ball 6 are exchanged. It really means that either ball 2 is heavier or ball 6 is lighter. You still have one more weight, I am sure you know how to do it!
-----------------------------------------------------
WOW! My brain hurts. Please Mr Magician, don't post any more of these. If we can't figure out this one, we are all doomed!
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Joined: Jul 2001
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From: Yorba Linda, CA
OK, gents, here's the solution I thought up; sorry for the delay but work, the weekend trip to San Diego to look at a horse, and a last-minute show on Sunday kept me way to busy:
Balance blocks 1, 2, 3 & 4 against 5, 6, 7 & 8.
A. If they balance, the bad block is 9, 10, 11 or 12. Balance 9, 10 & 11 against 3 good blocks (1 - 8).
1. If they balance, 12 is bad: balance it against any other.
a. If 12 is heavier, 12 is heavy.
b. If 12 is lighter, 12 is light.
2. If 9, 10 & 11 are heavy, balance 9 against 10.
a. If they balance, 11 is heavy.
b. If 9 is heavier than 10, 9 is heavy.
c. If 10 is heavier than 9, 10 is heavy.
3. If 9, 10 & 11 are light, balance 9 against 10.
a. If they balance, 11 is light.
b. If 9 is lighter than 10, 9 is light.
c. If 10 is lighter than 9, 10 is light.
B. If 1, 2, 3 & 4 are heavier than 5, 6, 7 & 8, balance 1, 2, 3, 5 & 6 against 4, 9, 10, 11 & 12.
1. If they balance, the bad one is 7 or 8 and it's light. Balance 7 against 8.
a. If 7 is lighter than 8, 7 is light.
b. If 8 is lighter than 7, 8 is light.
2. If 1, 2, 3, 5 & 6 are heavier, the bad one is 1, 2 or 3 and it's heavy. Balance 1 against 2.
a. If they balance, 3 is heavy.
b. If 1 is heavier than 2, 1 is heavy.
c. If 2 is heavier than 1, 2 is heavy.
3. If 4, 9, 10, 11 & 12 are heavier, then either 4 is heavy or 5 or 6 is light. Balance 5 against 6.
a. If they balance, 4 is heavy.
b. If 5 is lighter than 6, 5 is light.
c. If 6 is lighter than 5, 6 is light.
C. If 1, 2, 3 & 4 are lighter than 5, 6, 7, & 8, balance 1, 2, 3, 5 & 6 against 4, 9, 10, 11 & 12 (just like in B.).
1. If they balance, the bad one is 7 or 8 and it's heavy. Balance 7 against 8.
a. If 7 is heavier than 8, 7 is heavy.
b. If 8 is heavier than 7, 8 is heavy.
2. If 1, 2, 3, 5 & 6 are lighter, the bad one is 1, 2, or 3 and it's light. Balance 1 against 2.
a. If they balance, 3 is light.
b. If 1 is lighter than 2, 1 is light.
c. I f 2 is lighter than 1, 2 is light.
3. If 4, 9, 10, 11 & 12 are lighter then either 4 is light or 5 or 6 is heavy. Balance 5 against 6.
a. If they balance, 4 is light.
b. If 5 is heavier than 6, 5 is heavy.
c. If 6 is heavier than 5, 6 is heavy.
Yuck!
Here's a list of the second and third weighings (after the first, 1, 2, 3 & 4 vs 5, 6, 7 & 8) depending on which block is light or heavy:
1 is light: C2
2 is light: C2
3 is light: C2
4 is light: C3
5 is light: B3
6 is light: B3
7 is light: B1
8 is light: B1
9 is light: A3
10 is light: A3
11 is light: A3
12 is light: A1
1 is heavy: B2
2 is heavy: B2
3 is heavy: B2
4 is heavy: B3
5 is heavy: C3
6 is heavy: C3
7 is heavy: C1
8 is heavy: C1
9 is heavy: A2
10 is heavy: A2
11 is heavy: A2
12 is heavy: A1
I think I'll stick to magic: it's easier!
Balance blocks 1, 2, 3 & 4 against 5, 6, 7 & 8.
A. If they balance, the bad block is 9, 10, 11 or 12. Balance 9, 10 & 11 against 3 good blocks (1 - 8).
1. If they balance, 12 is bad: balance it against any other.
a. If 12 is heavier, 12 is heavy.
b. If 12 is lighter, 12 is light.
2. If 9, 10 & 11 are heavy, balance 9 against 10.
a. If they balance, 11 is heavy.
b. If 9 is heavier than 10, 9 is heavy.
c. If 10 is heavier than 9, 10 is heavy.
3. If 9, 10 & 11 are light, balance 9 against 10.
a. If they balance, 11 is light.
b. If 9 is lighter than 10, 9 is light.
c. If 10 is lighter than 9, 10 is light.
B. If 1, 2, 3 & 4 are heavier than 5, 6, 7 & 8, balance 1, 2, 3, 5 & 6 against 4, 9, 10, 11 & 12.
1. If they balance, the bad one is 7 or 8 and it's light. Balance 7 against 8.
a. If 7 is lighter than 8, 7 is light.
b. If 8 is lighter than 7, 8 is light.
2. If 1, 2, 3, 5 & 6 are heavier, the bad one is 1, 2 or 3 and it's heavy. Balance 1 against 2.
a. If they balance, 3 is heavy.
b. If 1 is heavier than 2, 1 is heavy.
c. If 2 is heavier than 1, 2 is heavy.
3. If 4, 9, 10, 11 & 12 are heavier, then either 4 is heavy or 5 or 6 is light. Balance 5 against 6.
a. If they balance, 4 is heavy.
b. If 5 is lighter than 6, 5 is light.
c. If 6 is lighter than 5, 6 is light.
C. If 1, 2, 3 & 4 are lighter than 5, 6, 7, & 8, balance 1, 2, 3, 5 & 6 against 4, 9, 10, 11 & 12 (just like in B.).
1. If they balance, the bad one is 7 or 8 and it's heavy. Balance 7 against 8.
a. If 7 is heavier than 8, 7 is heavy.
b. If 8 is heavier than 7, 8 is heavy.
2. If 1, 2, 3, 5 & 6 are lighter, the bad one is 1, 2, or 3 and it's light. Balance 1 against 2.
a. If they balance, 3 is light.
b. If 1 is lighter than 2, 1 is light.
c. I f 2 is lighter than 1, 2 is light.
3. If 4, 9, 10, 11 & 12 are lighter then either 4 is light or 5 or 6 is heavy. Balance 5 against 6.
a. If they balance, 4 is light.
b. If 5 is heavier than 6, 5 is heavy.
c. If 6 is heavier than 5, 6 is heavy.
Yuck!
Here's a list of the second and third weighings (after the first, 1, 2, 3 & 4 vs 5, 6, 7 & 8) depending on which block is light or heavy:
1 is light: C2
2 is light: C2
3 is light: C2
4 is light: C3
5 is light: B3
6 is light: B3
7 is light: B1
8 is light: B1
9 is light: A3
10 is light: A3
11 is light: A3
12 is light: A1
1 is heavy: B2
2 is heavy: B2
3 is heavy: B2
4 is heavy: B3
5 is heavy: C3
6 is heavy: C3
7 is heavy: C1
8 is heavy: C1
9 is heavy: A2
10 is heavy: A2
11 is heavy: A2
12 is heavy: A1
I think I'll stick to magic: it's easier!
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