Finally Home: A Logical Conundrum by the Doctor

Good morning.

Today is prolific mathematician Paul Erdős’ 112th birthday. Erdős published 1500 math papers with over 500 different co-authors during his lifetime. He solved and created numerous problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory.

This week’s problem is inspired by Erdős’ journey. Erdős had no permanent home: he would travel from one math conference to another, staying at colleagues’ homes along the way.

Backstory: Dr. Rational, after journeying through the mean streets, trekking the deep woods, and sailing across the river, finally reached his lab. He decides to make a puzzle for you, his venerable audience. As a professor of physics at the University of Demonstrandum, he lands on an interesting conundrum.

Problem:

A colleague of Dr. Rational has built a strange machine with a single button and a digital display. After pressing the button a few times, Dr. Rational observes the following:

1. The display only shows three digit numbers.

2. The display initially showed the number 432

3. When you press the button, the number changes like so:

   1. 432→2160

   2. 2160→105840

   3. 9621.8181…→274808749.091

   4. 274808749.091→193.547

Dr. Rational posits that the machine follows a set of secret rules that determine how the number changes when the button is pressed. If Dr. Rational presses the button again, what number will appear on the display? Can you determine what rule the machine follows?

Hint: No hints this time. Whoever solves this post first will automatically gain a score equivalent to double the score of the top scorer on our leaderboard. Note that the numbers in the series are rounded.

When you solve the above problem, please email your solution as a real number to [email protected] to be mentioned on our GitHub leaderboard. Submissions close 4/2 at 9:59 am PST.

That's all, folks!

Angad Singh