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 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.