Putnam: Does the Simplex Contain the Center?

n random points on S^(d−1) form a simplex containing the origin with probability 1/2^(n−1). 2D (circle, n=3): P = 1/4.   3D (sphere, n=4): P = 1/8.

Speed
5/sec
0
Total trials
0
Contain center
0 / 0 = —
Empirical probability
Theory
P = 1/2² = 0.250
2D: no arc ≥ π → triangle wraps center
Coin flip tree (4 outcomes, 1 favorable)
P1: 2D → 0.25 ± 0.01 after 1000 trials.
P2: 3D → 0.125 ± 0.01 after 1000 trials.