Tuesday, June 06, 2006

The Evil Shylock

There lived a farmer, his daughter, and a broker. The farmer for some reason borrowed a lot of money from the broker by pawning the only farm, the farmer had. The time given to the farmer to repay the money and get his farm released was over and the broker, the evil shylock, proposed the following: he would place a black stone and a white stone in a bag, and the farmer's daughter would need to pick out one of the stones from the bag in front of the entire town. If the daughter drew the white stone, the broker would return the farmer's farm and forgive him of the money. If she drew the black stone, the broker would marry the farmer's daughter and take the farm as well. The farmer had no choice but to agree. The farmer's daughter does not trust the broker and she knows that the broker might as well place 2 black stones in the bag. How can she get out of marrying the broker and save the farm for her father?

16 Comments:

Anonymous Anonymous said...

Bobach writes:
I don't think this is a probability problem, since (if the likelihood that the broker puts 2 black stones in the bag is equal to the probability that the broker puts in 1 black and 1 white stone), the chances of randomly choosing a white stone is no better than 0.25.
The daughter should put a chemical on her hands that turns a black stone white as she selects it.

6:52 PM, June 06, 2006  
Blogger Karthik Nagaraj said...

Bobach,
How can you say that the probability of picking a white stone is better than 0.25? Take the scenario when the guy puts in 2 black stones - p of white = 0, 1B 1W - p of white = 1/2 and in the case which might not be possible - 2W - p of white = 1. So maybe it is not a probability problem but still the p of picking white depends entirely on how the broker has placed the stones inside the bag. Given the condition that the girl does not know what the broker has put in the bag, what should the girl do to ensure that she wins the deal. I dont believe there is any black magic chemical involved in this. Its should be a logical answer based on the data given in the problem. :)

9:53 AM, June 07, 2006  
Anonymous Anonymous said...

Bobach writes:
I need to defend myself. I interpreted your statement that "the broker might as well place 2 black stones in the bag" would mean there is an equal chance that the bag contains 1 black and 1 white stone or that it contains 2 black stones and those are the only 2 possibilities. The probability of having one white stone in the bag is 0.5. The probability of then picking the white stone at random from that bag is also 0.5, so the probablility that both can happen is 0.5 x 0.5 = 0.25.
Have I mis-understood the parameters?

6:34 PM, June 07, 2006  
Anonymous Anonymous said...

Bobach writes:
I suppose if she took a sketch pad and a pencil, she could dump out the contents of the bag and if she sees a white stone, then she draws it on the pad. If there is no white stone, she doesn't draw it.

6:45 PM, June 07, 2006  
Blogger Karthik Nagaraj said...

Bobach,
I did not quite understand your solution - Can you elucidate it a bit more in detail?

10:27 AM, June 09, 2006  
Blogger Kala said...

this is very simple.
She draws the stone and while attempting to see it drops it on to the ground. It falls among the other stones..and says Oops, lets find out the color of the stone by looking at the one in the bag. If the one in the bag is black, she drew the white one. It is pretty obvious the evil shylock wouldn't put a white stone in there ! :)

1:21 PM, June 09, 2006  
Anonymous Anonymous said...

Bobach writes:
Karthik, I was using an alternate meaning of the word "draw." The meaning I exploited was "to transcribe a visual representation of a 3 dimensional object on a 2 surface." Like an artist draws a picture. The daughter will only draw a white stone.
Pretty lame, huh?

5:25 PM, June 09, 2006  
Anonymous Anonymous said...

What if the daughter picks both the stones? Then, if the broker had cheated, it would be for the whole town to see that the broker is a cheat.

- Tommy Stardust -

7:07 AM, June 12, 2006  
Blogger Karthik Nagaraj said...

So far Kala's solution seems more reasonable for the girl to choose. But then there are other constraints to that solution and I will still leave this one open for more discussion. As far as picking both the stones are concerned - let us assume that the girl should pick only one stone. She cannot pick 2 stones to prove the shylock to be a cheat.

10:29 AM, June 12, 2006  
Anonymous Anonymous said...

The daughter could ask to add two more stones (1 white and 1 black) to the bag. She could argue that adding the two stones is only to prevent any fraud by the broker (i.e. in case he had 2 black stones in there this will ensure that she at least has given herself some chance of picking a white stone). The broker has to agree to this since this will not alter the probability of picking a white stone(assuming that he had placed 1 white and 1 black stone to begin with). He would of course want everyone to think he has placed 1 white and 1 black stone in there so he agrees to this request by the daughter.

Now all that the daughter has to do is make sure that the white stone she places is in some way distinguishable from the rest of the stones in the bag (weight, shape etc).

11:15 PM, June 14, 2006  
Anonymous Anonymous said...

Hi,
My first post up here, so here goes. Thought of two ways actually, and both of them are the same at a fundamental level.

First way: She says that the stone she picks (as a choice for Shylock to decide whether he keeps the land or not) is the one that she leaves in the bag. So in effect, you actually reverse your act of picking by saying that the stone you picked, is actually the one in the bag.
Since she would pick a black, she can announce, relieved to the crowd, that the one left in the bag is a white!(even though it isn't)

(OR she asks Shylock to pick and says that her choice is opposite of what he picks. Pretty similar to the first solution actually)

Secondly,
What she can also do is announce to the crowd that since it doesn't matter what a stone stands for, she could actually reverse the conditions for the stone. So now, a black would result in victory for her and she would be grateful that Shylock was EVIL :D
*evil laughter*

Ok types?

Abhay Kesheorey
NPS Inr Batch of 2003

7:31 AM, June 15, 2006  
Blogger Karthik Nagaraj said...

Welcome aboard Abhay Kesheorey,
In all practical cases this might work out but looks ta this way - If indeed the shylock is evil and he places 2 balck stones, why would he agree to the reversal roles of the picking? Hw would not agree. Lets say the conditions are met with and that the rules of the picking are already laid forward by the shylock. What else can the girl do to make sure she wins the deal with a higher probability? Take a look at Kala's solution and add another constraint to that - it might prove to be more logical. Good try though!

9:39 AM, June 15, 2006  
Anonymous kavya said...

How about it if she already had a white stone in one of her hands, before drawing one from the bad..

9:47 PM, June 19, 2006  
Anonymous Anonymous said...

Bobach writes:
We've been assuming all along that since the farmer owed money, then the daughter had no money. The statement of the problem does not specify that.
So, the daughter can use her own money to pay off the broker. Or appeal to the townspeople to deal with the broker.
Alternatively, she could assassinate the broker.

6:32 PM, June 20, 2006  
Blogger Karthik Nagaraj said...

Bobach,
Its a nice thought but lets say for now our domain of the problem does specify that the girl does not have money and that she cannot assasinate the broker. What else can she do ?

6:34 PM, June 20, 2006  
Blogger Karthik Nagaraj said...

Kavya,
Thats a good suggestion - But what if the broker tells everyone that the stones he put in the bag are still there. Maybe you would need to refine your solution a bit - She picks the stone and if it is black, then she shows the stone she has been hiding. If it is white she she displays whatever she picked. This is pretty much similar to Kala's - one addendum to Kala's solution - she would drop the stone only if the stone is black, lets not assume that the broker cannot be honest at all. So I guess that brings us an end to this problem. This puzzle is SOLVED! Credits go to Kavya and Kala! Cheers guys! Lets get on with the other stuff.

6:40 PM, June 20, 2006  

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