Some Puzzles to think about

  The game needs two partners and an ordinary deck of cards, the first partner (A) has to turn over the cards one at a time, while the second (B) watches and chooses a time to say STOP at which point the card that is on the top of the deck is turned over, if it is red- the player wins, if not- he loses. Player B must say red some time before all the cards are revealed.

This game was invented by R. Connelly, (Maths Cornell), its first exposure to the public was in Martin Gardner's Scientific American column.
1. Which strategy is better for winning ``Say Red?''

Several strategies are often put forth, here are some suggestions:

1.

Say stop straight away. (The chance of winning is then seen to be 50-50).

2.

Wait for a lead, when more blacks have come out, then say stop, but if there are none, then the last card has to be red to win.

3.

Say stop at a random time, then the chance of red is again seen to be 50-50

Probably the best way of solving this is a thought experiment, the cards are exchangeable, it doesn't amtter whether you guess the first the last or the ith card, the chances are always 50-50 of getting that card right.

Banach's MatchBox Problem A pipe-smoking mathematician chooses out of his two boxes of matches, one in the left pocket, one in the right at random. Both boxes start at N matches, what are the chances that when he discovers the right is empty, there are 5 matches left in the other box?

Family Planning : People who have three daughters try for more, And then its fifty-fifty they'll have four, Those with a son or sons will let things be, Hence all these surplus women , QED.

Another version is :suppose the monarch says, as soon as you have a boy you must stop having children, will this change the sex-ratio?