Solution
Centre or Edge
Show the problem again
A point is chosen uniformly at random inside the unit square. What is the probability that it is closer to the centre of the square than to the nearest side of the square? Give an exact closed form, or a decimal rounded to 4 places.
Worked solution
The answer is (4√2 − 5)/3 ≈ 0.2190. The diagonals split the square into four congruent triangles, and in each triangle the nearest side is the one forming its base, so compute in one triangle and multiply by 4. Put the centre at the origin with the base on y = −1/2; the triangle is −1/2 ≤ y ≤ −|x|. Closer to the centre than to the base means x² + y² < (y + 1/2)², i.e. y > x² − 1/4: the region above a parabola. The parabola meets the diagonal y = −|x| where |x|² + |x| − 1/4 = 0, so |x| = a = (√2 − 1)/2. The area in one triangle is 2∫₀^a (−x − x² + 1/4) dx = 2(a/4 − a²/2 − a³/3). Multiplying by 4 and substituting a gives (4√2 − 5)/3. Even though the centre feels "central", most of the square is closer to an edge.
Source: Classic geometric-probability exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.