Sixteen distinct positive integers from 1 to 16 are placed in the cells of a 4x4 grid in a random order, with each number occurring in a cell exactly once.

Determine the probability that the absolute difference between an integer and any of its neighbors (including diagonally) is more than 2 but less than 13.

As a bonus, determine the probability that the absolute difference between an integer and any of its neighbors (including diagonally) is more than 2, but less than or equal to 16.

As I interpret the problem we are randomly picking a number and we want the probability that none of its neighbors differs by 1.

A number can be positioned in a corner (p=1/4), along an edge (p=2/4), or in the center (p=1/4)

If the number is a 1 [or 16] and if
- it is in a corner it has 3 neighbors and a 12/15 chance it doesnt border a 2 [or 15]
- it is on an edge is has 5 neighbor so a 10/15 chance
- in the center is has 8 so a 7/15 chance

If the number is 2 through 15 there are two numbers that differ by 1 and if
-it is in a corner there is a 13/15*12/14*11/13 = 132/210 chance of bordering neither
- it is on and edge there is a 90/210 chance
- it is in the center there is a 42/210 chance

For a grand total of
2/16*(1/4*12/15 + 2/4*10/15 + 1/4*7/15) + 14/16*(1/4*132/210 + 2/4*90/210 + 1/4*42/210) = 189/320

Assuming I made no mistakes.

Edited on October 27, 2009, 4:43 pm

Posted by Jer on 2009-10-27 16:29:25