The Monty Hall

If you have ever struggled with the Monty Hall, check this out:

http://www.blog.assignmenthelpsite.com/probability-the-monty-hall-paradox/

Prepare to square off

Consider an 8x8 chess board as shown in the adjacent figure. How many squares can you count without repetition?





Now consider the infinite chess board shown in the next figure. What is the inverse of the sum of the squares of all the distinct areas of squares that can formed from this board? What is the inverse of the sum of squares of all the square areas that can be formed from this board?

Playing with dominoes - II

This one is often attributed to Martin Gardner.

How many dominoes would you require to fill up a 8x8 chess board?

Now, let us give vent to the suppressed child in us, and mangle the chess board by removing the opposite corner squares (see picture). How many dominoes would you require now?

Playing with dominoes

A domino is a 2x1 piece of rectangle (see picture). Now consider a rectangular area of size nx2. The question is simple: given a large number of identical dominoes, how many different ways of arrangements of the 2x1 dominoes into the nx2 area are possible? Assume all the dominoes are identical. Note that as always, key ideas are more important in this blog than complete solutions.

DARPA mathematical challenges

Alright dudes. Its time to show you some real $$$. Check out the DARPA mathematical challenge:

https://www.fbo.gov/utils/view?id=4a43eacdb1c247bc9809e581987b8f31

As the one who told you about the problems, I get a 15% cut. Thanks.

People on a plane!

(Motivated by snakes on a plane) Here are two of the standard people on plane problem just to get things rolling again:
1) There are (say) 100 seats on a plane and since people with disabilities board first, the blind person boards first. Let us say this isnt Southwest and that each person is assigned a fixed seat (even the blind person is assigned a seat) and if his seat is occupied, he will sit in any seat that is free. Let us further say that the blind person picks his seat randomly of all the ones on the plane. The question is: What is the probability that the last (100th) person gets to sit in his assigned seat?
2) (Easier) Say there are n seats and n people. What are the number of ways in which each person sits in a different seat?

Queens!

Apologies for the long absence. Lets try to solve the problem of the queens. This one is derived from the question asked by Vichu.

We know that the problem of placing 8 queens on a 8x8 chessboard without any queen cutting any other is quite simple. The question to be solved now is: How many such unique arrangements are possible?

Bit by Bit...

What is the smartest way of counting the number of bits in a byte? (psuedocode please)