**Booth Id:**

MATH025T

**Category:**

Mathematics

**Year:**

2019

**Finalist Names:**

Kwag, Seo Yeong (School: Blair Academy)

Park, Taeyang (School: Peddie School)

**Abstract:**

Given S, a set of n points contained in the unit square Q = [0, 1]^2, let f(S) denote
the area of the largest axes-parallel rectangle that does not contain any of the points of S in
its interior. Further, let f(n) be the minimum value of f(S) over all sets S of n points in Q.
In 2009, Dumitrescu and Jiang proved that f(2) = (3 −√5)/2, f(4) = 1/4, and the following general bounds for f(n):
(1.25 − o(1)) ·1/n ≤ f(n) ≤ 4 ·1/n.
We show that f(3) = 0.3079 . . . , 0.2192 < f(5) < 0.2215, and we
improve the bounds in the general case:
(1.31 − o(1)) ·1/n ≤ f(n) ≤ 1.91 ·1/n.

**Awards Won:**

Third Award of $1,000

American Mathematical Society: Third Award of $500