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 4x4 Grid Difference (Posted on 2009-10-27)
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.

 No Solution Yet Submitted by K Sengupta No Rating

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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

Edited on October 27, 2009, 4:43 pm
 Posted by Jer on 2009-10-27 16:29:25

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