Riddle #2 - Quant + CR (700 Level!)

Question

10 humans are abducted by aliens; each represents 10% of the entire human population. The aliens give each abductee either a purple hat or a green hat. The 10 are lined up in a single file line, each facing forward, such that the last person can see the remaining 9's hats, the second to last person can see the remaining 8's hats and so on. No one can see his or her own hat.

The aliens then proceed, starting from the last person, to ask each of the abductees what the color of their hat is. If they guess correctly, they and the 10% of the human population they represent survives; if not, the opposite happens.

Assuming the abductees are given a chance to develop a strategy before they are lined up and questioned: what is the optimal strategy they can utilize (i.e. the one with the highest expected number of survivals)?

During the questioning, the abductees are not allowed to say anything besides their guess for the color of their hat when it is their turn.

This ones a bit tricky but can be solved using a simple GMAT concept. Good luck!

gmatexam439 · Answer

Minimum 50% of the population can survive as per me.
The abductees can make a strategy that the 10th person will say that color's name which has the greater count and the remaining will speak out that color's name. Therefore minimum 50% of the population will survive.

Please tell the answer.

jkolachi · Answer

Thinks again. It is possible to save more.

gmatexam439 · Answer

I can't really think of any other scenario right now. With the given piece of information I can't think of any other better strategy.

Waiting for other's replies and finally the answer.

yashbabu · Answer

50%??

chetan2u · Answer

Hi,
I can for sure say 60% ATLEAST.
It's more to do with giving a meaning to colour name.

But my mind says there must be some way to save more..

Saha6812 · Answer

Even numbered persons should say hat colour of the odd numbered person standing in front. Worst case scenario , 50% will be saved, best case scenario 100%

gmatexam439 · Answer

It can't be 90% ... If 10 says the color of the 9th's hat ... 9th will say the color of his hat and not the color of 8th's hat. Therefore it can never be 90% sure shot !!

jkolachi · Answer

Lol you guys are going to laugh when you see the solution. This can be solved with a basic 500 level quant trick. Think.

Remember these guys can talk about a strategy to tell the next person what his hat colour is... but ofcourse you can't say the colour.

Saha6812 · Answer

True, that was a wrong strategy.

Sent from my BLN-L22 using  GMAT Club Forum mobile app

gmatexam439 · Answer

But it's not written anywhere that they can signal each other or something like that -_-

jkolachi · Answer

Assuming the abductees are given a chance to develop a strategy before they are lined up and questioned: what is the optimal strategy they can utilize (i.e. the one with the highest expected number of survivals)?

They are given a chance to discuss a strategy.

chetan2u · Answer

Hi,

Moving on with my earlier post where I did feel that the colour meant a meaning and found a way by which 60% can be saved.
But now I think it can be 90% with same concept but going through a solution above which talked of even number and odd numbers.

Say they decide to say Purple if number of purple hats are even otherwise white if purple are odd..
The first guy sees all 9 ahead and mentions purple or white.
If he says purple, the second person counts the purple hats ahead and if they are even still says white. 
The third now knows that his hat included the numbers of purple hats is even and if he sees odd number, he knows he is wearing the purple hats and says purple.
The next similarly knows that the number of purple is odd as the previous has said purple.
This will carry on with each keeping track of the purple hats gone.

But yes it took time and although knowing the colour had to mean some signal, wouldn't have thought in these lines unless the post above would not have mentioned odd and even

gmatexam439 · Answer

Bow to you chetan2u

bb · Answer

I will be honest, I had no idea where to start. If I had 2-3 mins on this, I would just have to guess this question on the GMAT....

chetan2u · Answer

Hi,
I would have answered 60% if I had 2-3 minutes. But if there was 90% in choices, I would have gone for that without actually knowing method..

60% was by making groups of 3 that is 1-3 , 4-6, 7-9 and 10..
So 1 says purple if 2 and 3 are wearing same colour or he says white..
If 1 says purple 2 and 3 know they are wearing same colour and 2 can see 3, so he says what 3 is wearing and 3 repeats it.
If 1 says white, 2 and 3 know they are wearing opposite colour and again 2 can see 3, so he says opposite of what 3 is wearing and 3 says opposite of what 2 said..

Similarly 4 looks at 5-6 and says white or purple.

This way 2,3,5,6,8,9 say correctly and 1,4,7&10 is luck..

But getting to ODD and EVEN was after taking some time and reading a post with these words.
So a good exercise for brain.

broall · Answer

My solution:  The 10th abductee counts the 9 remaining hats with each color, and just says the color with higher count. After that, 9 remaining abductees just repeat what the 10th abductees said. The probability in worse case is 50% (if there are 5 green hats and 5 purple hats) and the best case is 100% (when there are 10 green hats or 10 purple hats).

broall · Answer

If the question means some tricks, I think my solution comes with this way.

This sentence means that abductees could say the color of their hat unlimited times. Hence, if all abductees agree that if the previous abductee says 1 times, his/her hat color will be green and if the previous abductee says 2 times, his/her hat color will be purple, then at least 90% people will be save.

jkolachi · Answer

Chetan is correct. You can determine 9/10 hats by using Odd/Even property. By choosing one of the hat colours to mean Odd & other to mean Even, the last person can let everyone know whether the number of Hats are Odd or even.

Riddle #2 - Quant + CR (700 Level!)

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