A checker is placed in a random square in the diagonal of an ordinary 8x8 checkerboard (with all squares being equally likely).

Checkers are then placed in the squares that lie below and to the right of the first checker, including those squares that are directly below or directly to the right of the first checker.

What, then, is the expected number of checkers on the board?

Can I have a hint, or the solution?

Max0815 Jul 25, 2018

#1**0 **

Could you start with the amount of options there are for a square to be chosen? Would that matter?

KaBoom Jul 25, 2018

#3**+1 **

Here's my best effort....

Since a checker is placed on a diagonal....we have 8 possibilities for the checker's placement

If we place the checker on Row 1, Column 1 we have 64 total checkers = 8^2

If we pllace the checker on Row2, Column 2, we have 49 total checkers = 7^2

Following this pattern, the expected value is

[8^2 + 7^2 + 6^2 + 5^2 + 4^2 + 3^2 + 2^2 + 1^2 ] / 8 = 204 / 8 = 51/2 = 25.5 checkers

CPhill Jul 26, 2018