Counterfeit coin

[Toadman]

Suppose you have 12 coins identical in every way except one is either slightly more or slightly less than the other 11 coins. You can not tell this different by hand. You have a balance beam scale to weigh coins against each other but you may only use it 3 times. Using only 3 tries on the scale, how can you figure out which is the counterfeit coin AND tell whether it is lighter and heaver than all the other coins?

[phaze]

Spoiler

Weigh 1

4-4-4

a) if it is unbalanced you have 4 coins suspected of being heavy and 4 of being light

b) if it is balanced you have 4 balls left that you dont know if they are heavy or not 

Weigh 2a

Place two suspected heavy coin and one suspected light coin on both sides of the weight (2 light coins will remain off the scales)

a) if it is unbalanced the two heavy coin on the down scale and the one light coin on the up scale are still suspect the rest are balanced (proceed with 3a)

 if it is balanced then only the two light coins are still suspect (I'll leave it to the reader to figure out how to proceed)

Weigh 2b

Place three of the unknown coins onto the scale along with one balanced one (one unknown coin will remain off the scale)

a) if the two unknown coins are on the down scale you have one suspected light coin and two heavy coins (proceed with 3a)

b) if the two unknown coins are on the up scale you have one suspected heavy coin and two light coins (proceed with  3a)

if it is balanced you still have one unknown coin and 1 weighing (I'll leave it to the reader to figure out how to proceed)

Weigh 3a

You have two coins that you suspect to be heavy or two coins that you suspect to be light and one that is opposite

place the opposite one aside and compare the two equally suspect ones

if they are unbalanced the way they are unbalanced will be instructive which coin is counterfeit

if they are not unbalanced  your suspicion is confirmed about the one put aside

If you want just a hint

Spoiler

Try doing it backward first (It may help you figure it out)

 

Edited March 29, 2016 by phaze

[BASKARAN R]

Dr.BASKARAN R

THOOTHUKUDI

INDIA

aarbaskaran@gmail.com

 

 

 keep 4 separately. put 4 and 4 in each pan.  weigh (i).

1) If they are not equal, the 4 coins kept aloof are originals.

     take 3 coins which appear to be with less weight and keep them marked.

     have the remaining 1 supposed with less weight and 2 supposed with more weight on one pan.

     have 1 original and the remaining 2 supposed with more weight on the other pan. weigh (ii).

     a) EQUAL: one of the 3 coins which appear to be with less weight and kept marked is fake.

                        put one in each pan and  weigh (iii).  if equal, third one is the fake,

                        otherwise, one with less wight is the fake one.

                        Let O,L and M denote original, less weight and more weight coins.

     b) if L1 + M2 weighs less then L1 or one of M2 in the other pan is the fake coin. 

                         put 1 M in each pan and weigh (iii). if equal, L1 is the fake coin; 

                          otherwise one with more weight is the fake coin.

     c) if L1 + M2 weighs more then one of M2 with L1 is the fake coin.

                           put 1 M in each pan and weigh (iii). 

                            one with more weight is the fake coin.

 

 

2) if they are equal, one of the 4 coins kept aloof is fake; the other 8 are originals.

                            keep 1 aloof from these 4. put 2 in one pan.

                            put the remaining 1 with 1 O in the other pan. weigh (ii). 

         a) if equal, one kept aloof is fake, weigh (iii) with that in one pan and 1 O in the other

                            to decide whether it is of less weight or more weight.

          b) if not equal, suppose the 2 unknown coins weigh more; (similar argument will hold for weighing less)

                             1 of these M2 is fake or 1 unknown in the other pan is fake with less weight.

                              put 1 M in each pan and weigh (iii). if equal, the other L1 is fake. otherwise 1 with more weight is fake.

Edited April 2, 2016 by BASKARAN R
spelling mistake

[Mr.Dog]

A balance beam scale compares mass not weight.

1. 6 - 6

2. 3 - 3

3. 1 - 1

Edited March 24, 2016 by Mr.Dog

[bonanova]

On 3/24/2016 at 2:35 PM, Mr.Dog said:

A balance beam scale compares mass not weight.

1. 6 - 6

2. 3 - 3

3. 1 - 1

@ Mr. Dog, Toadman's puzzle is well stated. It needs no correction.

Not that this point has anything to do with the puzzle, but a balance beam does in fact compare weight and not mass.

Spoiler

At least not directly. It compares the forces exerted on the two arms when masses are placed there, and those forces are exactly the weights of those masses. It differs from a scale in that it compares weights, rather than telling us what the weights are. Since both instruments respond to weight, and not directly to mass, they both require a gravitational field: neither would properly function in space, for example, where things are weightless.

How is 6-6 / 3-3 / 1-1 a solution?

Spoiler

After comparing two groups of six coins, let's say the mass placed in the left arm has the greater weight. Then the counterfeit coin is among the coins on the left side, if its weight is greater than the others, or among the coins on the right, if its weight is less. Which groups of three coins are compared next?

 

[Mr.Dog]

OK, i confused. But i put the blame on this article  :

http://www.livestrong.com/article/343183-the-differences-between-balance-beam-scales-electronic-scales/

...and not on me... at all.

and previous solution was just a typo:

Here's the new one:

Spoiler

x=mrdogrocks=any value greater than 0

1. 6-6

so you take the devil group and:

2. 4-2

Proportions would either be 3.x : 2,  4.x : 2,  4 : 1.x, or  4 : 2.x,

 

if the cf coin is the group of 2 coins:

3. 1-1

Proportions would be: 1.x : 1 = heavier coin or 0.x : 1 = lighter coin

 

if the cf coin is in the group of 4 coins

3. 2-2

Proportions would be either 2.x : 2 = Rock solid Malphite heavier coin or 1.x : 2 = 420 blaze lighter coin

 Dont mess with a dog, or you'll get a bark! bark!

[bonanova]

Regarding the linked article

Spoiler

It contains the following paragraph

Mass Vs. Weight

The University of California Department of Physics and Astronomy explains that weight is the force of the gravity puts on the mass of an object. Weight measurements calculate how much gravity it takes to hold an object to the earth. Mass, on the other hand, is about solidity. Mass is a quantifiable measurement of the amount of substance it takes to make an object solid. Balance scales measure mass. An electronic scale measures weight.

The part in red is pure rubbish. The author clearly listened to an expert and then printed her own ideas.

The difference between (electronic, or other) scales and a balance is that the latter compares (same, heavier, lighter,) while the former measures (235.9 pounds.) That's all.  Both instruments require the objects involved to exert a force. That force (in a gravitational field) is their weight. Weight goes away outside of gravity, but mass does not. An astronaut could place her feet on an electronic scale while she is in en route to Mars, and the scale will register a reading of zero. That indeed is her weight outside a gravitational field. The electronic scale did its job. It measured her weight.

She could also get onto one side of a teeter-totter (with no one sitting on the other side) and it would balance! If you believe that a balance measures mass, you must conclude that she has zero mass. But that is incorrect. Since a balance compares forces, you can conclude, correctly, that masses are weightless in space. On earth, of course, where she is not weightless, she would fall to the floor.

The author's definition of mass (red underlined) is shameful. Gasses have mass, as do liquids, not just solids. Mass is simply an object's resistance to acceleration. That's what F=ma means. Mass (m) is the amount of force (F) required to provide one unit of acceleration (a) to an object. Consider the relative masses of a toy, a car and a freight train: You can flick a toy car with your finger and give it speed of perhaps 30 mph almost instantaneously. In a few seconds you could get a car to a speed of a few mph, by putting your shoulder to it and pushing with your legs. Push on a freight train, however, and you'd have to bring a lunch, perhaps several of them, and eat them before you could get it to observably budge, even if the wheels were frictionless. Applying a force and measuring acceleration is how you measure mass directly. You'd get the same results in space as you do on earth. You can also infer mass from weight in a gravitational field by measuring the force it takes to keep if from falling (its weight) and divide the weight by g, the acceleration due to gravity.

 

Regarding the new solution,

Spoiler

1. What is the Devil group? It's one of the groups of 6, but which?
2. How does 6-6 provide useful information?
3. Why is x positive?

 

 

[Mr.Dog]

On 3/26/2016 at 1:48 PM, bonanova said:

Regarding the linked article

  Reveal hidden contents

It contains the following paragraph

Mass Vs. Weight

The University of California Department of Physics and Astronomy explains that weight is the force of the gravity puts on the mass of an object. Weight measurements calculate how much gravity it takes to hold an object to the earth. Mass, on the other hand, is about solidity. Mass is a quantifiable measurement of the amount of substance it takes to make an object solid. Balance scales measure mass. An electronic scale measures weight.

The part in red is pure rubbish. The author clearly listened to an expert and then printed her own ideas.

The difference between (electronic, or other) scales and a balance is that the latter compares (same, heavier, lighter,) while the former measures (235.9 pounds.) That's all.  Both instruments require the objects involved to exert a force. That force (in a gravitational field) is their weight. Weight goes away outside of gravity, but mass does not. An astronaut could place her feet on an electronic scale while she is in en route to Mars, and the scale will register a reading of zero. That indeed is her weight outside a gravitational field. The electronic scale did its job. It measured her weight.

She could also get onto one side of a teeter-totter (with no one sitting on the other side) and it would balance! If you believe that a balance measures mass, you must conclude that she has zero mass. But that is incorrect. Since a balance compares forces, you can conclude, correctly, that masses are weightless in space. On earth, of course, where she is not weightless, she would fall to the floor.

The author's definition of mass (red underlined) is shameful. Gasses have mass, as do liquids, not just solids. Mass is simply an object's resistance to acceleration. That's what F=ma means. Mass (m) is the amount of force (F) required to provide one unit of acceleration (a) to an object. Consider the relative masses of a toy, a car and a freight train: You can flick a toy car with your finger and give it speed of perhaps 30 mph almost instantaneously. In a few seconds you could get a car to a speed of a few mph, by putting your shoulder to it and pushing with your legs. Push on a freight train, however, and you'd have to bring a lunch, perhaps several of them, and eat them before you could get it to observably budge, even if the wheels were frictionless. Applying a force and measuring acceleration is how you measure mass directly. You'd get the same results in space as you do on earth. You can also infer mass from weight in a gravitational field by measuring the force it takes to keep if from falling (its weight) and divide the weight by g, the acceleration due to gravity.

 

Regarding the new solution,

  Hide contents

1. What is the Devil group? It's one of the groups of 6, but which?
2. How does 6-6 provide useful information?
3. Why is x positive?

Spoiler

1.- 2. I figured out that doesn't provide any progress, since the coin could be either lighter or heavier and the scale could go either way (up or down)

3. cause x is not a downer - but he is now terminated for the uprising foundation.

Answer attempt 3 - the foundation of hope:

Spoiler

Ok Ok

lets split the 12 coins into groups of 3 coins making 4 groups in total, so:

1. 3-3

if they balance the cf coin would be in the 2 other groups of 3 coins each. 

if they don't balance the cf coin is in within these 6 coins, making the other 6 coins normal.

so with 1 weigh we get rid of 6 suspicious coins.

 

2. 2-2

Now, as we know that the cf coin will be within 6 coins. We split the them into groups of 2 coins each making 3 groups.

now we weigh 2 groups against each other - If they don't balance then the coin within those 4 coins - and you're screwed, well for now.

but lets look away from that - yes far away - and think optimistic and lets say they if they balance, the coin will be in the 2 coins not weighted. Then we proceed victoriously to the next legendary step.

 

3. 1-1

pick 1 normal coin from the other normal 6 coins and compare it with 1 of the 2 coins - if they balance the coin is the other 1 of the 2 coins, if they don't balance - well then its obvious.

 

This is the foundation - i am gonna rework this when i get into physics and probability next year.

 

 

On 3/26/2016 at 1:48 PM, bonanova said:

 

 

 

 

[wolfgang]

We should devide the 12 coins into three groups, 5,5,and 2

1- for the first  five, we weigh 2 against 2, leaving one  coin aside.

   a- if they are blanced,we put them  aside.

   b.- if they are´nt  balanced.

        we should weigh the heavier two with any other two coins :- now... if they balanced so the counterfeit coin is one of the  two light coins,which will be easy to be find with the               third weigh.....and if they are heavier,then we should weigh them one against one,,,and they heavier is the counterfeit coin.

2- for the 2nd five .we will do the same 2 against 2,leaving one coin aside:-

    a- if  they are balanced,  so one of the two coins left aside is the counterfeit coin, which will be easy to find with the 3rd weigh.

    b- if they are not balanced....see above.

[Mr.Dog]

Bro, it doesn't work like this. Let me teach and educate you why:

Ok, so you say that we split the coins into 3 groups: 5, 5 and 2

Then you take the one of the group of 5 coins and:

1. 2 - 2...  and you put 1 coin aside. 

if they are balanced the cf coin will be in other 8 coins

2. You do the same with the second group of 5 coins - if they are balanced the cf coin will be in the remaining 4 coins( 2 you put aside + third group of 2 coins)

3. now how do you find the cf coin in these 4 coins?

Tell me, homeboy.

 

[bonanova]

Baskaran has it.

[BASKARAN R]

On 4/2/2016 at 3:30 PM, bonanova said:

Baskaran has it.

 Thank you, sir.