James Tanton discusses with Sunil Singh, in the video interview below, the problem he thought about as a young child, that led – years later – to an epiphany and to a deeper understanding of mathematical thinking:
The thought that led James to a solution – a reason why some squares in a 
In the grid colored as shown, there are 13 blue squares and 12 yellow squares. Now a moment’s thought will reveal that starting from a yellow square there is no walk through adjacent squares – which, as you can see, flips alternately between the two colors – that reaches each square exactly once.
BY BY BY BY BY BY BY BY BY BY BY BY B 
(doesn’t rule out such a walk)
YB YB YB YB YB YB YB YB YB YB YB YB Y 
(proves there’s no such walk)