Suppose you have twelve eggs and a balance scale. All of the eggs are identical except for one whose only difference is its weight. Using the scale only three times, determine which egg is the odd egg out and whether it is heavier or lighter than the other eggs.
Weigh four against four. If they’re equal, weigh three of them against three you haven’t weighed. If they balance too, weigh the last remaining egg against any of the others to see if it is lighter or heavier. If the three suspects are heavier, weigh one of them against another and the one that goes down is it. If they balance the remaining suspect is heavy. Use the same process if they’re lighter. If the initial four vs four don’t balance, weigh two heavy eggs and a light egg against one heavy egg, one light one and a known normal egg. If they balance weigh the remaining two light eggs against each other. If they balance the unweighed heavy egg is the odd one out. If the side with two heavy eggs goes down weigh them against each other. If they balance it is the light egg on the other side. If the other side goes down it is either because of one heavy egg on that side or because the one light egg on the other side is lighter than the rest. Weigh one of them against a known normal egg to determine which is true.
This seems unduly complicated. Weigh 4 against 4. Whichever group of 4 is heavier, weigh 2 against 2 within that group. Then weigh 1 against 1 within the heavier group of two. You can start with 2 groups of 5 eggs also.
This problem is overhard
(Thanks, Gabby) It scrambles your mind as your blood boils
It releases your inner devil as the weight of it all falls on your shoulders
Orla. Awesome answer if we knew whether the odd egg is heavier or lighter but since we don’t know 4 on 4 then 2 on 2 and then 1 on 1 doesn’t really work since you will not know which 4 to chose if they are not of equal weight. Is it lighter or heavier?
Weigh four against four. If they’re equal, weigh three of them against three you haven’t weighed. If they balance too, weigh the last remaining egg against any of the others to see if it is lighter or heavier. If the three suspects are heavier, weigh one of them against another and the one that goes down is it. If they balance the remaining suspect is heavy. Use the same process if they’re lighter. If the initial four vs four don’t balance, weigh two heavy eggs and a light egg against one heavy egg, one light one and a known normal egg. If they balance weigh the remaining two light eggs against each other. If they balance the unweighed heavy egg is the odd one out. If the side with two heavy eggs goes down weigh them against each other. If they balance it is the light egg on the other side. If the other side goes down it is either because of one heavy egg on that side or because the one light egg on the other side is lighter than the rest. Weigh one of them against a known normal egg to determine which is true right
There is no one definite answer, it is a tree of decisions for the different outcomes and depending on the outcomes of each weighing you should not forget to take that into consideration when you do your next weighing. I will not write my solution but I will suggest a hint:- start with a group of 3 with 4 eggs in each group.
I wish I could attach a picture that would explain better than words, but I’ll try.
Begin by weighing eggs (1-4) vs. eggs (5-8).
Let’s begin with the first option in which this scale is equal. Its obviously eggs 9-12 then.
1a) You remove egg 12 and weigh eggs 9 & 10 vs. 11 & 1(or any constant) and the scale is equal. Its egg 12 and compare to any egg to determine if it’s heavier or lighter.
1b) You remove egg 12 and weigh eggs 9 & 10 vs. 11 and a constant and the scale tips left telling you 9 & 10 are heavier. Remove egg 10 and weigh eggs 9 & 11 vs. any two constants. If they’re equal, it’s egg 10 and it’s heavier (known from 1b). If the scale tips left again, it has to be egg 9 which is heavier. If it tips right, egg 11 is the lighter one.
Ok back to the beginning now and when we weigh eggs (1-4) vs. eggs (5-8) the scale tips left. Same scenario goes if it tips right, just opposite rules apply.
2a) Eggs 9-12 are ruled out and can act as constants. Remove eggs 3,4 & 8. Weigh 1,5,constant vs. 2,6,7 and the scale is even meaning you’re down to eggs that were removed. Set aside egg 3 and weigh 4,8 vs. any two constants. If it’s even it’s egg 3 and it’s heavier (known from first weigh-in). It it tips left again, it has to be egg 4 which is heavier. If it tips right, egg 8 is the lighter one out.
2b) Eggs 9-12 are ruled out and can act as constants. Remove eggs 3,4 & 8. Weigh 1,5,constant vs. 2,6,7 and the scale tilts left (ruling out 5,2 and the removed eggs) meaning 1 could be the heavier one or maybe 6 or 7 are lighter. Set aside 6 and weigh 1,7 vs two constants. If it’s even, it’s egg 6 and it’s lighter. If the scale tips left still, it’s egg 1 which is heavier. If it tilts right, it’s egg 7 which is lighter.
2c) Eggs 9-12 are ruled out and can act as constants. Remove eggs 3,4 & 8. Weigh 1,5,constant vs. 2,6,7 and the scale tilts right now. It’s either 5 or 2 as they were the only eggs that switched sides. Weigh egg 5 vs. any constant, if it’s even then it’s egg 2 which is heavier. If it’s not even then it’s quite certainly egg 5 which is the lighter one.
This is a mensa problem that they used to test people with from many years ago (about 30 years ago?)
Most of the solutions here are by people who never read the question: The key is that you don’t know if the problem egg is heavier or lighter.
When one side of the beam balance goes up, you will never know if it’s because the problem egg is heavier, or lighter. If you didn’t catch that, you’re far from mensa level.
I rem I had found 2 different solutions to it. Here is one of them.
Solution 1.
Split the eggs into 3 groups A1A2A3A4, B1B2B3B4, C1C2C3C4
Step 1
weigh A1A2A3A4 V B1B2B3B4
possibility (i): balance is even
-> the problem egg is in Group C, and all eggs in A and B group are of a constant weight
step (i).2 Weigh C1C2C3 V BBB
possibility (i).2.(i): balance is even
–> the problem egg is C4
—– step (i).2.(i).3 Weigh C4 Vs B
=======> C4 will show if it’s heavier or lighter
possibility (i).2.(ii) if C1C2C3 is heavier
–> C1, C2, or C3 is heavier
—– step (i).2.b.3 Weigh C1 vs C2.
=======> C1 or C2 will either be heavier and be the problem egg, otherwise C3 is heavier
possibility (i).2.(iii) if C1C2C3 is lighter
–> C1, C2, or C3 is lighter
—– step (i).2.b.3 Weigh C1 vs C2.
=======> C1 or C2 will either be lighter and be the problem egg, otherwise C3 is lighter
(This will settle the problem if the odd egg is in C group.)
possibility (ii): A1A2A3A4 is heavier than B1B2B3B4
-> C eggs are all constant weight
step (ii).2 Weigh A1A2B1 vs CA3B2 eggs
possibility (ii).2.(i) A1A2B1 is lighter
–> B1 (lighter) or A3 (heavier).
—– weigh B1A3 Vs CC to confirm.
=======> B1A3 will go up if B1 is lighter, go down if A3 is heavier
possibility (ii).2.(ii) A1A2B1 is heavier
–> A1, A2 heavier, or B2 lighter.
—– weigh A1 vs A2 to confirm.
=======> A1 or A2 will go down if heavier, if not B2 is lighter.
possibility (ii).2.(iii) balance is even
–> A4 heavier or B3B4 lighter.
—– weigh B3 vs B4 to confirm.
=======> B3 or B4 will go up if it’s lighter, else A4 is heavier.
(the above will be swapped for if B group is heavier than A group)
16 Comments on "Balance Twelve Eggs"
Orla says
December 28, 2015 @ 11:30
This seems unduly complicated. Weigh 4 against 4. Whichever group of 4 is heavier, weigh 2 against 2 within that group. Then weigh 1 against 1 within the heavier group of two. You can start with 2 groups of 5 eggs also.
Gabby says
June 4, 2016 @ 23:08
Tried it and got scrambled eggs.
kay says
August 21, 2016 @ 18:03
Lol @Gabby… that’s the best answer
Dan says
August 22, 2016 @ 20:11
@Orla You didn’t explain what to do if the first two groups are the same weight.
Me says
December 18, 2016 @ 13:36
This problem is overhard
(Thanks, Gabby) It scrambles your mind as your blood boils
It releases your inner devil as the weight of it all falls on your shoulders
Jason says
January 15, 2017 @ 17:35
Orla. Awesome answer if we knew whether the odd egg is heavier or lighter but since we don’t know 4 on 4 then 2 on 2 and then 1 on 1 doesn’t really work since you will not know which 4 to chose if they are not of equal weight. Is it lighter or heavier?
Lisa says
March 29, 2017 @ 11:13
This is hard I tried it for my class and no one got it
Here says
February 7, 2018 @ 17:35
Weigh four against four. If they’re equal, weigh three of them against three you haven’t weighed. If they balance too, weigh the last remaining egg against any of the others to see if it is lighter or heavier. If the three suspects are heavier, weigh one of them against another and the one that goes down is it. If they balance the remaining suspect is heavy. Use the same process if they’re lighter. If the initial four vs four don’t balance, weigh two heavy eggs and a light egg against one heavy egg, one light one and a known normal egg. If they balance weigh the remaining two light eggs against each other. If they balance the unweighed heavy egg is the odd one out. If the side with two heavy eggs goes down weigh them against each other. If they balance it is the light egg on the other side. If the other side goes down it is either because of one heavy egg on that side or because the one light egg on the other side is lighter than the rest. Weigh one of them against a known normal egg to determine which is true right
Xmen says
May 11, 2018 @ 06:29
There is no one definite answer, it is a tree of decisions for the different outcomes and depending on the outcomes of each weighing you should not forget to take that into consideration when you do your next weighing. I will not write my solution but I will suggest a hint:- start with a group of 3 with 4 eggs in each group.
Good Luck
Deez nuts says
May 21, 2018 @ 14:23
Just put 6 on each side, that simple.
Dan says
May 22, 2018 @ 09:45
If the left side is heavier, which egg is the odd one out and is it heavier or lighter than the rest?
Laura says
August 12, 2018 @ 23:19
I wish I could attach a picture that would explain better than words, but I’ll try.
Begin by weighing eggs (1-4) vs. eggs (5-8).
Let’s begin with the first option in which this scale is equal. Its obviously eggs 9-12 then.
1a) You remove egg 12 and weigh eggs 9 & 10 vs. 11 & 1(or any constant) and the scale is equal. Its egg 12 and compare to any egg to determine if it’s heavier or lighter.
1b) You remove egg 12 and weigh eggs 9 & 10 vs. 11 and a constant and the scale tips left telling you 9 & 10 are heavier. Remove egg 10 and weigh eggs 9 & 11 vs. any two constants. If they’re equal, it’s egg 10 and it’s heavier (known from 1b). If the scale tips left again, it has to be egg 9 which is heavier. If it tips right, egg 11 is the lighter one.
Ok back to the beginning now and when we weigh eggs (1-4) vs. eggs (5-8) the scale tips left. Same scenario goes if it tips right, just opposite rules apply.
2a) Eggs 9-12 are ruled out and can act as constants. Remove eggs 3,4 & 8. Weigh 1,5,constant vs. 2,6,7 and the scale is even meaning you’re down to eggs that were removed. Set aside egg 3 and weigh 4,8 vs. any two constants. If it’s even it’s egg 3 and it’s heavier (known from first weigh-in). It it tips left again, it has to be egg 4 which is heavier. If it tips right, egg 8 is the lighter one out.
2b) Eggs 9-12 are ruled out and can act as constants. Remove eggs 3,4 & 8. Weigh 1,5,constant vs. 2,6,7 and the scale tilts left (ruling out 5,2 and the removed eggs) meaning 1 could be the heavier one or maybe 6 or 7 are lighter. Set aside 6 and weigh 1,7 vs two constants. If it’s even, it’s egg 6 and it’s lighter. If the scale tips left still, it’s egg 1 which is heavier. If it tilts right, it’s egg 7 which is lighter.
2c) Eggs 9-12 are ruled out and can act as constants. Remove eggs 3,4 & 8. Weigh 1,5,constant vs. 2,6,7 and the scale tilts right now. It’s either 5 or 2 as they were the only eggs that switched sides. Weigh egg 5 vs. any constant, if it’s even then it’s egg 2 which is heavier. If it’s not even then it’s quite certainly egg 5 which is the lighter one.
My brain hurts but how did I do??
Frank says
October 28, 2018 @ 10:00
Weigh 6 against 6
From the heaviest 6 weigh 3 against 3
From the heaviest 3 weigh 1 against 2
If 1 is heavier you have the answer
If the 2 are balanced the same then the remaining egg is the heaviest.
Simple in 3 weighs on the scale
Tony says
December 25, 2019 @ 14:04
I made that riddle 22 years ago, a lot of good answers but not the right one yet…
Keep trying, some of you are very close to the step…
Raheem says
May 10, 2020 @ 19:27
Devide into 3+3+3+3
And weigh 3 with 3 if both equal
Other two sets has difference
So first take one set of weighted 3 eggs with one set of not weighted 3
If both equal the odd is in the unweighted 3
So from the unweighted three one egg can weight with another one may be one side is higher or liggter so that one is odd
If both both side equal the renainin one is odd
Jason says
December 1, 2022 @ 23:32
This is a mensa problem that they used to test people with from many years ago (about 30 years ago?)
Most of the solutions here are by people who never read the question: The key is that you don’t know if the problem egg is heavier or lighter.
When one side of the beam balance goes up, you will never know if it’s because the problem egg is heavier, or lighter. If you didn’t catch that, you’re far from mensa level.
I rem I had found 2 different solutions to it. Here is one of them.
Solution 1.
Split the eggs into 3 groups A1A2A3A4, B1B2B3B4, C1C2C3C4
Step 1
weigh A1A2A3A4 V B1B2B3B4
possibility (i): balance is even
-> the problem egg is in Group C, and all eggs in A and B group are of a constant weight
step (i).2 Weigh C1C2C3 V BBB
possibility (i).2.(i): balance is even
–> the problem egg is C4
—– step (i).2.(i).3 Weigh C4 Vs B
=======> C4 will show if it’s heavier or lighter
possibility (i).2.(ii) if C1C2C3 is heavier
–> C1, C2, or C3 is heavier
—– step (i).2.b.3 Weigh C1 vs C2.
=======> C1 or C2 will either be heavier and be the problem egg, otherwise C3 is heavier
possibility (i).2.(iii) if C1C2C3 is lighter
–> C1, C2, or C3 is lighter
—– step (i).2.b.3 Weigh C1 vs C2.
=======> C1 or C2 will either be lighter and be the problem egg, otherwise C3 is lighter
(This will settle the problem if the odd egg is in C group.)
possibility (ii): A1A2A3A4 is heavier than B1B2B3B4
-> C eggs are all constant weight
step (ii).2 Weigh A1A2B1 vs CA3B2 eggs
possibility (ii).2.(i) A1A2B1 is lighter
–> B1 (lighter) or A3 (heavier).
—– weigh B1A3 Vs CC to confirm.
=======> B1A3 will go up if B1 is lighter, go down if A3 is heavier
possibility (ii).2.(ii) A1A2B1 is heavier
–> A1, A2 heavier, or B2 lighter.
—– weigh A1 vs A2 to confirm.
=======> A1 or A2 will go down if heavier, if not B2 is lighter.
possibility (ii).2.(iii) balance is even
–> A4 heavier or B3B4 lighter.
—– weigh B3 vs B4 to confirm.
=======> B3 or B4 will go up if it’s lighter, else A4 is heavier.
(the above will be swapped for if B group is heavier than A group)
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