Welcome to BrainDen.com - Brain Teasers Forum

 Welcome to BrainDen.com - Brain Teasers Forum. Like most online communities you must register to post in our community, but don't worry this is a simple free process. To be a part of BrainDen Forums you may create a new account or sign in if you already have an account. As a member you could start new topics, reply to others, subscribe to topics/forums to get automatic updates, get your own profile and make new friends. Of course, you can also enjoy our collection of amazing optical illusions and cool math games. If you like our site, you may support us by simply clicking Google "+1" or Facebook "Like" buttons at the top. If you have a website, we would appreciate a little link to BrainDen. Thanks and enjoy the Den :-)
Guest Message by DevFuse

bushindo

Member Since --
Offline Last Active Today, 06:59 AM

Blue eyes and green eyes

29 March 2013 - 07:28 PM

Let's say that in a certain country, there are two types of people- blue-eyed and green-eyed.

Let's assume that eye color are governed by a single gene, which can be in the dominant form (B) or recessive form (g).  This means that people in this country have the genotype BB (blue-eyed), Bg (blue-eyed), or gg (green-eyed). The relative frequency of these genotypes in the population is BB (50%), Bg (25%), and gg( 25%).

Judy (blue-eyed) is married to Jack (blue-eyed). Judy's parents are both blue-eyed; one of Jack's parents is blue-eyed while the other is green-eyed. Judy and Jack's first child has blue eyes.

1) Given the above, what is the probability that Judy's father has the dominant genotype BB?
2) What is the chance that Judy and Jack's second child is green-eyed?

More divisibility

12 February 2013 - 12:11 AM

This is an extension of Prime's excellent puzzle One's Divisibility

What is the smallest 101-plus-digit number consisting of only 4's, 5's, and 7's that is divisible by '7777.....777' (one hundred 7's)?

The solution *must contain* at least one 4, one 5, and one 7.

EDIT: clarified the properties of the required solution

01 February 2013 - 07:34 PM

Here is yet another puzzle based on Colorful Foreheads

Suppose that there is a game as follows

* There is a host with 10 stamps, 5 red and 5 blue. There are 4 players- A, B, C, and D.
* In the beginning, the host affixes two stamps to each of the 4 players' head. The choice of stamps for each player is completely random (i.e. the host puts all stamps into an opaque bag and then draws them one by one). The remaining 2 stamps go into the host's pocket. Each player can see the stamps on the remaining 3 players, but can not see his own stamps nor the two in the host's pocket.
* Starting from A to D (and then looping back to A and so on), the host asks if each player definitively knows his color (RR, BB, or RB). If the player does not know, the host goes on to the next player. First player to know his color wins. No guessing is allowed.

Suppose that the host likes you, so he secretly offers you a side bet before the game. You have to pay the host 1
dollar before the game starts, and then you can choose whether to be A, B, C, or D. The payout by position if you win is as follows

A: 3.5 dollars
B: 2.5 dollars
C: 6   dollars
D: 7   dollars

Which position should you choose for the greatest expected winnings?

31 January 2013 - 06:51 PM

This is based on bonanova's puzzle Colorful Foreheads

Here's a game that goes as in the following

* There is a host with 14 stamps, 7 red and 7 blue. There are five players- A, B, C, D, and E.
* In the beginning, the host affixes two stamps to each of the 5 players' head. The remaining 4 stamps go into the host's pocket. Each player can see the stamps on the remaining 4 players, but can not see his own stamps nor the four in the host's pocket.
* Starting from A to E (and then looping back to A and so on), the host asks if each player definitively knows his stamps distribution (RR, BB, or RB). If the player does not know, the host goes on to the next player. No guessing is allowed.
* The game goes as follows

1st turn- player A: I do not know
2nd turn- player B: I do not know
3rd turn- player C: I do not know
4th turn- player D: I do not know
5th turn- player E: I do not know
Host- Alright, to help you, I'll now reveal two stamps from my pocket. *At this point, the host pulls out two of the four stamps in his pocket and shows everyone. The game then continues as before*
6th turn- Player A: I do not know.

Question- what is the longest possible number of turns required before a player definitely knows his color?

Closed form expression on steroids

27 January 2013 - 04:53 PM

This is based on a previous puzzle.

Find a closed form expression for function f(N) such that

f( N ) = 0 for  1  <= N <= 11
f( N ) = 1 for  12 <= N <= 22
f( N ) = 2 for  23 <= N <= 33
f( N ) = 3 for  34 <= N <= 44

f( N ) = 4 for  45 <= N <= 55
...

No modulus or rounding operators are allowed.