Author Topic: Car Talk Puzzler.  (Read 3558 times)

0 Members and 1 Guest are viewing this topic.

Offline Thrifty

  • Grendel's Mom
  • ***
  • Posts: 9275
  • Liked: 1272
  • Now available in non-prescription strength.
Car Talk Puzzler.
« on: February 18, 2010, 11:47:51 AM »
Well, the puzzler almost always stumps me (I actually managed to figure out last week's puzzler).  Usually I just give up and wait until next week.  The current puzzler, though, I find particularly intriguing,

Quote
Imagine this: It's the eve of the annual Car Talk 'We Haven't Been Canceled Yet' Banquet. John 'Bugsy' Lawlor has procured the food and the wine for the following evening's gala, and he's extremely pleased with himself because he knows a guy, who knows a guy, who knows a guy, and he's obtained for $50, thirteen one-half gallon jugs of the finest red wine. He's very excited about the following evening's festivities until Doug 'Punkin Lips' Berman informs him that he received a note suggesting that one of those 13 bottles of wine contains a deadly poison. In fact, it's one that kills within 24 hours. 'Is someone out to get us? Who could be out to get us? Why? Why not? But why the whole staff?'

Anyway, because they're hopeless cheapskates and unwilling to discard 12 bottles of perfectly good cheap wine, they head off to MIT to consult with a scientist friend of theirs. 'So let's see,' he says, 'you have these thirteen bottles of wine and one of them contains a poison that will prove fatal within 24 hours after it's been consumed. And the Car Talk Extravaganza is when?'

'Tomorrow night.'

The scientist goes on to suggest that from his knowledge of poisons, even the smallest sample is usually enough to cause certain death, even if mixed and diluted with the wine from the untainted bottles. 'Hmm,' he says, 'I'll be right back.'

In a flash, he returns with four small cages, each one containing your standard lab rat. What a moral dilemma, they think, we have to sacrifice lab rats to save Car Talk? 'But wait, wait,' they say. 'We have 13 bottles of wine. How are we supposed to save the entire Car Talk staff and empire, with just 4 rats?'

'You can do it,' he says and then disappears into the inky shadows.

So, you have 4 rats in little cages, 13 bottles of wine and one of them has got poison in it. Now you can obviously take samples from any of the bottles and you can give as much or as little as you want to any of the rats. And don't forget, you're not going to know if the poison is fatal until the night of the banquet because it takes 24 hours to kill human or rat. So, how do you do it?


Yeah, I'm stumped.  4 rats, 12 clean bottles, 1 poison bottle.


Offline Steve-O

  • Mayor of Nilbog
  • *****
  • Posts: 3569
  • Liked: 56
    • The Incomparable
Re: Car Talk Puzzler.
« Reply #1 on: February 18, 2010, 02:57:50 PM »
I think I've worked it out by thinking in terms of binary numbers.  4 binary digits can represent any decimal number from 0 to 15.  The numbers from 1 to 13 break down as follows:

4321
====
0001 = 1
0010 = 2
0011 = 3
0100 = 4
0101 = 5
0110 = 6
0111 = 7
1000 = 8
1001 = 9
1010 = 10
1011 = 11
1100 = 12
1101 = 13

Now think of each 1 in the table above as a rat that tries that wine.  The key is to give each rat a mixture of wines such that the combination of dead rats uniquely identifies the poisoned bottle.

Number each bottle from 1 to 13, and each rat from 1 to 4.  Then give each rat a mixture from the following bottles:

Rat 1 tries 1, 3, 5, 7, 9, 11, 13
Rat 2 tries 2, 3, 6, 7, 10, 11
Rat 3 tries 4, 5, 6, 7, 12, 13
Rat 4 tries 8, 9, 10, 11, 12, 13

Let's say that rats 1, 2, and 3 die.  That gives you a binary number of 0111 = 7, so bottle number 7 is your poison.  Rats 4, 2, and 1 keel over?  That's 1011, or bottle 11.
« Last Edit: February 18, 2010, 03:12:28 PM by Steve-O »


Offline Steve-O

  • Mayor of Nilbog
  • *****
  • Posts: 3569
  • Liked: 56
    • The Incomparable
Re: Car Talk Puzzler.
« Reply #2 on: February 18, 2010, 03:18:57 PM »
Also, I'm pretty sure you could potentially save one ratty little life by leaving either wine 1, 2, 4, or 8 out of the mixtures.  If no rats die, the wine you left out is the only one nobody tried, and therefore, the poison.  (It's basically the same thing as numbering the bottles from 0 to 12 instead of from 1 to 13.)


Offline Plastic Self-Cleaning Duck

  • Mr Bungle
  • *
  • Posts: 13883
  • Liked: 2
  • Duck!!!
Re: Car Talk Puzzler.
« Reply #3 on: February 18, 2010, 03:47:40 PM »
That solution sounds perfect......

But it's hard to imagine them explaining that on air in the space of an hour long program.  But who knows....


Offline Steve-O

  • Mayor of Nilbog
  • *****
  • Posts: 3569
  • Liked: 56
    • The Incomparable
Re: Car Talk Puzzler.
« Reply #4 on: February 18, 2010, 04:15:02 PM »
That solution sounds perfect......

Yeah, I have to imagine they've worked out an easier way to explain the solution than I did.  It basically boils down to: divvy the wines up so that each bottle kills a unique combination of rats.


Offline Thrifty

  • Grendel's Mom
  • ***
  • Posts: 9275
  • Liked: 1272
  • Now available in non-prescription strength.
Re: Car Talk Puzzler.
« Reply #5 on: February 18, 2010, 04:37:59 PM »
Binary numbers!  Genius!  Why didn't I think of that?


Offline Thrifty

  • Grendel's Mom
  • ***
  • Posts: 9275
  • Liked: 1272
  • Now available in non-prescription strength.
Re: Car Talk Puzzler.
« Reply #6 on: February 19, 2010, 02:15:38 PM »
I still can't follow it, but I think it would be easier if I could remember my Boolean algebra.


Offline Steve-O

  • Mayor of Nilbog
  • *****
  • Posts: 3569
  • Liked: 56
    • The Incomparable
Re: Car Talk Puzzler.
« Reply #7 on: February 19, 2010, 03:06:31 PM »
Let me see if I can come up with a better explanation, since even I have trouble following my funky logic in the previous one.

Whichever wine is the poison, the rats that have tried it will die, and the ones that haven't, won't.

So all we have to do is make sure that each wine is given to a unique set of rats.  That way, the combination of rats that die and rats that don't will tell us exactly which wine was the deadly one.

To simplify, let's say we only have 4 wines and 2 rats.  There are four different ways that we can either give or not give two rats a taste of something, and we'll assign each combination to a different wine:

          Rat
          1 0
        +-----
Wine 0  | 0 0   = neither rat gets a taste
Wine 1  | 0 1   = rat 0 gets a taste, rat 1 doesn't
Wine 2  | 1 0   = rat 1 gets a taste, rat 0 doesn't
Wine 3  | 1 1   = both rats get a taste


Now in 24 hours, if:

 -- Neither rat dies: We know the poison was wine 0.  It's the only one neither rat tried.
 -- Rat 0 dies, rat 1 lives: The poison was wine 1.  Rat 0 tried it, rat 1 didn't.  Rat 0 tried wine 3 as well, but since rat 1 also tried wine 3 and lived to squeak another day, it can't be poison.
 -- Rat 1 dies, rat 0 lives: The poison was wine 2.  Rat 1 tried it, rat 0 didn't, etc., etc.
 -- Both rats die: The poison was wine 3.  It's the only wine they both tried.

With 13 wines and 4 rats, it's the exact same deal.  You just have more possible combinations of rats trying/not trying each wine.


Offline FLOCK of RABID SHEEP?!?!

  • Mayor of Nilbog
  • *****
  • Posts: 3055
  • Liked: 1
Re: Car Talk Puzzler.
« Reply #8 on: February 20, 2010, 11:33:37 AM »
yeah it was base 2 :P just heard the answer today


Offline Steve-O

  • Mayor of Nilbog
  • *****
  • Posts: 3569
  • Liked: 56
    • The Incomparable
Re: Car Talk Puzzler.
« Reply #9 on: February 21, 2010, 01:16:33 PM »
yeah it was base 2 :P just heard the answer today

Boo!  The answer they picked always kills a rat.  That 26 dollar gift certificate is rightfully mine!


Offline Plastic Self-Cleaning Duck

  • Mr Bungle
  • *
  • Posts: 13883
  • Liked: 2
  • Duck!!!
Re: Car Talk Puzzler.
« Reply #10 on: February 21, 2010, 01:33:23 PM »
yeah it was base 2 :P just heard the answer today

Boo!  The answer they picked always kills a rat.  That 26 dollar gift certificate is rightfully mine!

E-mail and tell them that


Offline Thrifty

  • Grendel's Mom
  • ***
  • Posts: 9275
  • Liked: 1272
  • Now available in non-prescription strength.
Re: Car Talk Puzzler.
« Reply #11 on: February 21, 2010, 01:56:19 PM »
Here's the answer (streaming MP3 file).


Offline Thrifty

  • Grendel's Mom
  • ***
  • Posts: 9275
  • Liked: 1272
  • Now available in non-prescription strength.
Re: Car Talk Puzzler.
« Reply #12 on: February 21, 2010, 02:01:23 PM »
Also, answer in written format.  With this diagram.