Problem:
Two unit squares are selected at random without replacement from an grid of unit squares. Find the least positive integer such that the probability that the two selected squares are horizontally or vertically adjacent is less than .
Solution:
In each row of an grid of squares, there are pairs of adjacent squares. Thus there are pairs of horizontally adjacent squares in the grid. Similarly there are pairs of vertically adjacent squares in the grid. Out of the equally likely ways to select two squares in the grid, there are ways to select the two squares so that they are adjacent. Hence the required condition is , which simplifies to . The least positive integer satisfying this is .
The problems on this page are the property of the MAA's American Mathematics Competitions