There’s an easy way!

 

“There  is  a famous story  about John von  Neumann, the  mathematician  and  physicist who turned Alan Turing’s  idea  (what we  now  call  a Turing machine) into an actual electronic computer (what we now call a Von Neumann machine, such as your laptop or smart phone). Von Neumann was a virtuoso thinker,  legendary for  his  lightning  capacity for  doing  prodigious  calculations  in  his  head. According to the  story—and  like  most famous stories, this one has many versions—a colleague approached him one day with a puzzle that had two paths to a solution, a laborious, complicated calculation and an  elegant, Aha!-type solution. This colleague  had a theory:  in such a case,  mathematicians work out the  laborious solution while the  (lazier,  but  smarter)  physicists  pause  and  find  the  quick-and-easy  solution.  Which  solution  would  von  Neumann  find?  You  know  the  sort  of puzzle: Two trains, 100 miles apart, are approaching each other on the same track, one going 30 miles per hour, the other going 20 miles per hour. A bird flying 120 miles per hour starts at train A (when they are 100 miles apart), flies to train B, turns around and flies back to the approaching train A ,  and  so  forth,  until  the  two  trains  collide.  How  far  has  the  bird  flown  when  the  collision  occurs?  “Two  hundred  and  forty  miles,”  von  Neumann answered almost instantly. “Darn,” replied his colleague, “I predicted you’d do it the hard way, summing the infinite series.” “Ay!” von Neumann cried
in embarrassment, smiting his forehead. “There’s an easy way!” (Hint: How long until the trains collide?)”