All of us have probably been exposed to questions such as: *"What are the applications of group theory..."*.
This is *not* the subject of this MO question.  



---


[Here](http://www.irishtimes.com/newspaper/letters/2010/1110/1224283015227.html) is a little newspaper article that I found inspiring:



> 
> Madam, – In response to Marc Morgan’s question, “Does mathematics have any practical value?” (November 8th), I wish to respond as follows.
> 
> 
> Apart from its direct applications to electrical circuits and machinery, electronics (including circuit design and computer hardware), computer software (including cryptography for internet transaction security, business software, anti-virus software and games), telephones, mobile phones, fax machines, radio and television broadcasting systems, antenna design, computer game consoles, hand-held devices such as iPods, architecture and construction, automobile design and fabrication, space travel, GPS systems, radar, X-ray machines, medical scanners, particle research, meteorology, satellites, all of physics and much of chemistry, the answer is probably “No”. – Yours, etc,
> 
> 
> PAUL DUNNE,  
> 
> *The Irish Times - Wednesday, November 10, 2010*
> 
> 
> 


  

The above article article seems to provide an ideal source of solutions to a perennial problem:  
 How to tell something interesting about math to non-mathematicians, without losing your audience?  

However, I am embarrassed to admit that I have *no idea* what kind of math gets used for antenna designs, computer game consoles, GPS systems, etc.  



I would like to have a list applications of math,
*from the point of view of the applications*.  



To make sure that each answer is sufficiently structured and developed, I shall impose some restrictions on their format. Each answer should contain the following three parts, roughly of the same size:


• Start by the description of a *practical problem* that any layman can understand.  

• Then I would appreciate to have an explanation of *why* it is difficult to solve without mathematical tools.  

• Finally, there should be a little explanation of the *kind* of math that gets used in the solution.  
  



♦♦  *My ultimate goal is to have a nice collection of examples of applications of mathematics, for the purpose of casual discussions with non-mathematicians.* ♦♦  
  



As usual with community-wiki questions: one answer per post.



Applications of mathematics
Sending a man to the Moon (and back).


Hilbert once remarked half-jokingly that catching a fly on the Moon would be the most important technological achievement. "Why? "Because the auxiliary technical problems which would have to be solved for such a result to be achieved imply the solution of almost all the material difficulties of mankind." (Quoted from [*Hilbert-Courant*](http://books.google.com.ua/books?id=pPNirfJL86MC&printsec=frontcover&dq=reid+hilbert+courant&source=bl&ots=fY4R_ClWcq&sig=ZbYvvAb7G9jwM9tjv2-yXy3uEgw&hl=en&ei=Ra9mTdutJYvrOc7jkZwL&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBIQ6AEwAA#v=onepage&q&f=false) by Constance Reid, Springer, 1986, p. 92). 


The task obviously required solving plenty of scientific and technological problems. But the key breakthrough that made it all possible was [Richard Arenstorf's](http://en.wikipedia.org/wiki/Richard_Arenstorf) discovery of a stable 8-shaped orbit between the Earth and the Moon. This involved the development of a numerical algorithm for solving the restricted three-body problem which is just a special non-linear second order ODE (see also my answer to the previous [MO question](https://mathoverflow.net/questions/52489/on-the-non-rigorous-calculations-of-the-trajectories-in-the-moon-landings/52490#52490)). 


Another orbit, also mapped by Arenstorf, was later used in the dramatic rescue of the Apollo 13 crew. 


