prestontw / lotterywinners

Explores the 100 lottery winners riddle

LotteryWinners

Explores the 100 lottery winners riddle

Riddle Description

One hundred participants sign up for a lottery event. Each participant is assigned a unique number, from 1 to 100. Corresponding to each participant is a lottery ticket with their number in a box in another room. However, the boxes are randomly ordered. Each participant can open no more than 50 boxes. If each participant finds their number, each of them wins a million dollars. If even one of them does not find their number, none of them gets any money.

Specific Details

Each participant faces the exact same room. For example, one participant cannot leave some boxes open, rearrange the boxes, leave a message, etc. Additionally, once a participant has entered the room, they are not allowed to communicate with any participant who has not tried to find their number.

The Question

What is the best strategy for the participants that maximizes their probability of each finding their number?

A Caveat

To avoid this being an awful probability problem, we allow a 'humanitarian' to go into the room once before all participants go. This humanitarian is allowed to open all of the boxes, switch two boxes, then the participants continue as normal.

What is the strategy for the participants and the humanitarian? I claim that once the humanitarian goes, if they optimally switch two boxes, then each participant will always get their number. The probability without the humanitarian for 100 participants is ~33%.