MathDB

Turbo the snail sits on a point on a circle with circumference

1

. Given an infinite sequence of positive real numbers

c_1, c_2, c_3, \dots

, Turbo successively crawls distances

c_1, c_2, c_3, \dots

around the circle, each time choosing to crawl either clockwise or counterclockwise. Determine the largest constant

C > 0

with the following property: for every sequence of positive real numbers

c_1, c_2, c_3, \dots

with

c_i < C

for all

i

, Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.

Problem Statement