Robots Guessing the Numbers on Their Foreheads
Source: 2024 Taiwan TST Round 3 Independent Study 1-C
April 24, 2024
combinatorics
Problem Statement
Dexter's Laboratory has robots, each with a program setup by Dexter. One day, his naughty sister Dee Dee intrudes and writes an integer in on each of the robot's forehead. Each robot detects the numbers on all other robots' foreheads, and guess its own number base on its program, individually and simultaneously.
Find the largest positive integer such that Dexter can setup the programs so that, no matter how the numbers distribute, there are always at least robots who guess their numbers right.Proposed by sn6dh