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Grind

Indian Penny Puzzle

If you've not already seen this in one of its many incarnations, it's a bit of a bastard.

You are given twelve seemingly identical pennies, but one of them is a cunning forgery.

Not quite cunning enough though: the forged penny has a discernibly different weight (or, more pedantically, mass) to the other eleven "normal" coins.

The only means you have at your disposal to determine this rogue penny is a tipping balance scale (i.e. the see-saw type).

You are only allowed to use this scale three times.

Soooo. Under all possible circumstances.......

Could you uniquely determine which is the rogue penny?

In addition, can you tell if it is lighter or heavier than the genuine articles?

* Have I said it's a bit of a bastard? *
Pond Life

6 on one side 6 on the other. Take the lighter bunch then split them 3 on one side 3 on the other.

You are left with three coins put one one one side and one on the other if they balance the fake is not on the scales otherwise the lighter one shows up on the scales.

Then plug the pc in switch it on, log on the plug in the ereader....

sorry....wrong forum.
Grind

Pond Life wrote:
6 on one side 6 on the other. Take the lighter bunch then split them 3 on one side 3 on the other.

You are left with three coins put one one one side and one on the other if they balance the fake is not on the scales otherwise the lighter one shows up on the scales.

Then plug the pc in switch it on, log on the plug in the ereader....

sorry....wrong forum.


Hmmm.

* Might avoid buying Sony for a while...... *  
Pond Life

I'm right though.
Grind

Pond Life wrote:
I'm right though.


In what sense?  
Pond Life

In the sense that i am right.
Grind

Pond Life wrote:
In the sense that i am right.


So if the problem coin happens to be heavier and not lighter than the rest, your plan would definitely work?

Methinks thou spouteth bollocks.  

If that was your intention then, yes, you are right.  
Grind

I would even go as far as saying your first weighing basically reveals diddly squat....

And Masterchef is well gay.
Pond Life

What the fuck you doing trying to pass off fake Indian pennies on me anyways? I thought we had a relationship?
Grind

Pond Life wrote:
What the fuck you doing trying to pass off fake Indian pennies on me anyways? I thought we had a relationship?


Good point.  

We'll leave this one to Bearing then?  
Grind

Notices that Bearing's been gone a while.........

* I said it was a bastard. *
bearing

My brain hurts, I think I'm on it's tail but it'll take me a while.
Grind

bearing wrote:
My brain hurts, I think I'm on it's tail but it'll take me a while.


I thought someone was messing with my head too only to have an "Aha!" moment thinking about it whilst driving.

If you're logical, you'll definitely get it. Which means you'll get it.  

* It's one of those puzzles for one of those books I've not got around to writing yet. *
Fin

Split the pennies into three fours,put two lots of four on the scales then take one set of four off and put on the last set of four on  make a note of which set of four pennies is a different weight to the other two ie heavier or lighter,THEN I'M STUCK  
Pond Life

Just ask an Indian.
Grind

Fin wrote:
Split the pennies into three fours,put two lots of four on the scales then take one set of four off and put on the last set of four on  make a note of which set of four pennies is a different weight to the other two ie heavier or lighter,THEN I'M STUCK  


Not a bad start.  
fartcatcher

Warning! Warning!  Danger Will Robinson! Does not compute!

bearing

Fin wrote:
Split the pennies into three fours,put two lots of four on the scales then take one set of four off and put on the last set of four on  make a note of which set of four pennies is a different weight to the other two ie heavier or lighter,THEN I'M STUCK  


Yeah, I'm a bit further in to it mathematically.

There's 27 permetations.  

Each of the three weighings has three outcomes, those three outcomes each have three outcomes.

Just need to work it out into English.
Fin

bearing wrote:
Fin wrote:
Split the pennies into three fours,put two lots of four on the scales then take one set of four off and put on the last set of four on  make a note of which set of four pennies is a different weight to the other two ie heavier or lighter,THEN I'M STUCK  


Yeah, I'm a bit further in to it mathematically.

There's 27 permetations.  

Each of the three weighings has three outcomes, those three outcomes each have three outcomes.

Just need to work it out into English.


Bearing i think we should just weigh two sets of four,if they are the same the rogue penny is in the other four then it would be a case of weighing three of them against three "proper"pennies to find out which is the rogue,if they weigh the same it is the unweighed penny which is the rogue if the three are heaveir/lighter simply weigh one against one to find the heaviest/lightest if they are the same it is the one of the three that you have taken off the scales which is the heaviest/lightest penny

If at the start the two sets of four are different weights you would have to move half from one side to the other and vice versa THEN MY HEAD EXPLODED
Pond Life

Take them to a bank and let them sort it out.
Grind

Fin wrote:
bearing wrote:
Fin wrote:
Split the pennies into three fours,put two lots of four on the scales then take one set of four off and put on the last set of four on  make a note of which set of four pennies is a different weight to the other two ie heavier or lighter,THEN I'M STUCK  


Yeah, I'm a bit further in to it mathematically.

There's 27 permetations.  

Each of the three weighings has three outcomes, those three outcomes each have three outcomes.

Just need to work it out into English.


Bearing i think we should just weigh two sets of four,if they are the same the rogue penny is in the other four then it would be a case of weighing three of them against three "proper"pennies to find out which is the rogue,if they weigh the same it is the unweighed penny which is the rogue if the three are heaveir/lighter simply weigh one against one to find the heaviest/lightest if they are the same it is the one of the three that you have taken off the scales which is the heaviest/lightest penny

If at the start the two sets of four are different weights you would have to move half from one side to the other and vice versa THEN MY HEAD EXPLODED


You've essentially got one third of the solution.  
Pond Life

I'm not going to understand the answer to this.
grunt

Fack! All that for one third.
Grind

Which, in itself, is pretty good. Fine skills.
Grind

grunt wrote:
Fack! All that for one third.


Unfortunately, the second third includes the "clever" bit.

Too many clues already.  
Grind

Pond Life wrote:
I'm not going to understand the answer to this.


It doesn't really matter, you could just pass the couterfeit off at the local Newsagent's anyway.  
Pond Life

Grind wrote:
Unfortunately, the second third includes the "clever" bit.


grunt

You make the rats eat them?
Grind

grunt wrote:
You make the rats eat them?


It's either that or drink Blue Nun.

They lose either way.......... tryyryo.gif
Grind

Here's the solution:-
Spoiler:

made you look!

bearing

Okay...

1st weigh (equal balance)

Split the coins into three groups of four, weigh two of the groups, if they're equal then you know that the fake coin is in the other group.

2nd weigh

Using one of the coins from the first two groups and add it two three of the other four, weigh them and if they balance then you know that the remaining coin is fake. (use the remaining coin with one of the other coins to find whether it's lighter or heavier)

If on the other hand the balances don't balance then you know that the remaining coin is a good one.

I need to think about the next step and it's late, I'll come back to it.
bearing

Get rid of the spoiler, I'm trying to work it out here!
Grind

bearing wrote:
Get rid of the spoiler, I'm trying to work it out here!


Chuckles.  
Pond Life

I'm going to answer this tonight, so you'd better hurry up if you want to look cleverer than me.
Grind

Okay I'm back - we've just had a five day power cut, which is no fun when you also get about a foot of snow and you need electricity to run your heating, water, sewerage, cooking, sump pumps etc.

And you have no generator.

So, do you give up?

Do you? Do you?
Grind

The answer....
Spoiler:

A solution to The Twelve coins puzzle:

This has an "Aha" element at the second weigh where, depending on the first weigh, you may need to swap some of the already weighed coins between pans.

Other solutions exist, but they follow the same basic logic as that presented - any differences lie in the number of coins used for the second and third weighings and are purely cosmetic in nature.

The twelve coins are arbitrarily labelled from A to L. The following scheme can then be used to tell which is the odd coin out and whether it is heavier or lighter.

It might help to set up a "pretend" set and follow the directions rather than just read.

1) FIRST WEIGH: ABCD v EFGH
There are three possible outcomes and corresponding logical conclusions:-

Outcome 1a) The scales balance. Conclusions: The target coin is one of I, J, K or L.

Outcome 1b) The scales tip to down the left. Conclusions: The target coin is EITHER (A, B, C or D and heavier) OR (E, F, G, or H and lighter).

Outcome 1c) The scales tip down to the right.Conclusions: The target coin is EITHER (A, B, C or D and lighter) OR (E, F, G, or H and heavier).  


2) SECOND WEIGH
The choice of second weigh depends on the outcome of the first round.

2a) Second weigh in the event of outcome 1a): IJ v AK

Outcome 2a1) The scales balance. Conclusions: The target coin is L.

Outcome 2a2) The scales tip down to the left. Conclusions: The target coin is EITHER (I or J and heavier) OR (K and lighter).

Outcome 2a3) The scales tip down to the right. Conclusions: The target coin is EITHER (I or J and lighter) OR (K and heavier).

2b) Second weigh in the event of outcome 1b): ABCE v DIJK

Outcome 2b1) The scales balance. Conclusions: The target coin is F, G or H and lighter.

Outcome 2b2) The scales tip down to the left. Conclusions: The target coin is A, B or C and heavier.

Outcome 2b3) The scales tip down to the right. Conclusions: The target coin is EITHER (E and lighter) OR (D and heavier).

2c) Second weigh in the event of outcome 1c): ABCE v DIJK
(i.e. the same approach as 2b), but with logically different conclusions).

Outcome 2c1) The scales balance. Conclusions: The target coin is F, G or H and heavier.

Outcome 2c2) The scales tip down to the left. Conclusions: The target coin is EITHER (E and heavier) OR (D and lighter).

Outcome 2c3) The scales tip down to the right. Conclusions: The target coin is A, B or C and lighter.  


3) THIRD WEIGH
As you might expect, the choice of third weigh depends on the outcome of the second.  

3a) Third weigh in the event of outcome 2a1): A v L

Outcome 3a1) The scales tip down to the left. Conclusions: The target coin is L and it is lighter.

Outcome 3a2) The scales tip down to the right. Conclusions: The target coin is L and it is heavier.

3b) Third weigh in the event of outcome 2a2): I v J

Outcome 3b1) The scales balance. Conclusions: The target coin is K and it is lighter.

Outcome 3b2) The scales tip down to the left. Conclusions: The target coin is I and it is heavier.

Outcome 3b3) The scales tip down to the right. Conclusions: The target coin is J and it is heavier.

3c) Third weigh in the event of outcome 2a3): I v J

Outcome 3c1) The scales balance. Conclusions: The target coin is K and it is heavier.

Outcome 3c2) The scales tip down to the left. Conclusions: The target coin is J and it is lighter.

Outcome 3c3) The scales tip down to the right. Conclusions: The target coin is I and it is lighter.

3d) Third weigh in the event of outcome 2b1): F v G

Outcome 3d1) The scales balance. Conclusions: The target coin is H and it is lighter.

Outcome 3d2) The scales tip down to the left. Conclusions: The target coin is G and it is lighter.

Outcome 3d3) The scales tip down to the right. Conclusions: The target coin is F and it is lighter.

3e) Third weigh in the event of outcome 2b2): A v B

Outcome 3e1) The scales balance. Conclusions: The target coin is C and it is heavier.

Outcome 3e2) The scales tip down to the left. Conclusions: The target coin is A and it is heavier.

Outcome 3e3) The scales tip down to the right. Conclusions: The target coin is B and it is heavier.

3f) Third weigh in the event of outcome 2b3): A v E

Outcome 3f1) The scales balance. Conclusions: The target coin is D and it is heavier.

Outcome 3f2) The scales tip down to the left. Conclusions: The target coin is E and it is lighter.

3g) Third weigh in the event of outcome 2c1): F v G

Outcome 3g1) The scales balance. Conclusions: The target coin is H and it is heavier.

Outcome 3g2) The scales tip down to the left. Conclusions: The target coin is F and it is heavier.

Outcome 3g3) The scales tip down to the right. Conclusions: The target coin is G and it is heavier.

3h) Third weigh in the event of outcome 2c2): A v D

Outcome 3h1) The scales balance. Conclusions: The target coin is E and it is heavier.

Outcome 3h2) The scales tip down to the left. Conclusions: The target coin is D and it is lighter.

3i) Third weigh in the event of outcome 2c3): A v B

Outcome 3i1) The scales balance. Conclusions: The target coin is C and it is lighter.

Outcome 3i2) The scales tip down to the left. Conclusions: The target coin is B and it is lighter.

Outcome 3i3) The scales tip down to the right. Conclusions: The target coin is A and it is lighter.

Obviously this is a bit verbose just to prove all the possible outcomes - this can be expressed more simply mathematically (which will read as even worse gobbledygook to most) - but it definitely works.

The key is spotting that swapping coins from one pan to the other can reveal more information. Sometimes a "no change" result is good.

sheeps

I was just going to say that.
bearing

sheeps wrote:
I was just going to say that.


Well done sheeps, here have a sheepy snack!

*throws biscuit*
Grind

So, you're all thickies then?

* Hopes they don't remember the wine thing. *
Plastic Man

Grind wrote:
So, you're all thickies then?

* Hopes they don't remember the wine thing. *


I think most, if not all, of us worked out the solution.

But, too be blunt, couldn't be arsed typing the answer out.
grunt

Plastic Man wrote:
Grind wrote:
So, you're all thickies then?

* Hopes they don't remember the wine thing. *


I think most, if not all, of us worked out the solution.

But, too be blunt, couldn't be arsed typing the answer out.


Yes I know I did. There just weren't enough permutations for me to be bothered with it.
Grind

Plastic Man wrote:
Grind wrote:
So, you're all thickies then?

* Hopes they don't remember the wine thing. *


I think most, if not all, of us worked out the solution.

But, too be blunt, couldn't be arsed typing the answer out.


I did the same with Fermat's last theorem. Wasn't worth the virtual ink.

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