(Non-trivial) Maths problems that don't require any formal maths education 
Does anyone know good brain teasers that show the nature of
  mathematics to people with little or no mathematical knowledge? Or do
  you know a book with such problems?

Criteria are that the problem should be interesting, require no maths knowledge and be solvable by a mathematical core principle, e.g. a real proof. Ideally, the problem does not require calculations or at least not lengthy ones. 
Example Remove two diagonal corner squares of a chess board. Is it possible to tile the board with a given number of domino stones? See this question.
 A: Being very visual, I like synthetic figures.
One of my favorite figure is known as Monge circles' theorem:

Being given three non-intersecting circles in the general case (different radii), consider the external tangent lines of these circles taken two by two. They intersect in 3 points ; they look aligned. How can we prove it ?
If you have an audience, let them search for a short time. Then give the following hint "Think in 3D". Most often, one of the people in the audience will have the idea :
Imagine this "scene" as an "aerial view" of 3 spheres with the same radii as the circles before, placed on a plane floor, to which all the spheres are tangent. But there is a second plane tangent to the 3 spheres (this is our intuition, but it can also be established rigorously (*)). The intersection of this plane with the floor is "the" line we are looking for. If somebody has doubts, one can invoke a supplementary convincing argument dealing with the 3 "icecream cones" containing 2 balls.
(*) Remark: the centers of the spheres determine a plane which is the medial plane of the two tangent planes.
Edit :
Let us present now the Desargues configuration which has a striking similarity with the Monge problem. I will explain it using the notation of the following figure :

Consider two triangles $ABC$ and $A'B'C'$ which are perspective from a certain point P, meaning that lines $AA', \ BB', \ CC'$ meet in this point. Then intersection points
$$Q:= AB \cap A'B', \ R:= AC \cap A'C', \ S:= BC \cap B'C'$$
are aligned (the reciprocal is true).
How can this property can be established ? (question to the reader)
Answer : By the same way as before, this time by interpreting this figure as a 3D scene in which triangular pyramid with basis $ABC$ and apex $P$ is cut by a transversal plane. This plane intersects the base plane along a line on which necessarily $Q, \ R, \ S$ are situated.
Reference : this page
of an excellent geometry site. See as well fig. 2.9 page 18 of this well-written article by Bobenko here.
A: Martin Gardner in the introduction of his book,  Entertaining Mathematical Puzzles, writes that: 

[....] I have done
  my best to find puzzles that are unusual and entertaining, that call for only the most elementary knowledge of
  mathematics, but at the same time provide stimulating glimpses into
  higher levels of mathematical thinking. [......]

Hence I think this is the book you are searching.
You can download PDF version of book from  here  , but if you like the book please do buy the book. 
A: Imagine that all points af the plane are coloured either red or blue.  Prove that there is one color such that for any positive real $a$ there exist two points of that color with distance $a$.
Now prove the same provided all points of $\mathbb R^3$ are coloured in three colours.
