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On this page we describe a few examples of what we see as an actual or potential awakening of a mathematical brain, and what it is that example exemplifies.

Example	Exemplifying
Seeing a square array of dots as composed of a series of diagonals.	Insight into a proof that $1+2+...+n-1 = n(n-1)/2$
Determining how many 2/11 there are in 3.	Insight into the connection between a remainder and a unit.
“This might be crazy, but I think there might be a connection between towers, grid walks and binomial expansions.”	Insight into connections between seemingly different combinatorial problems.
Determining from which squares on a $5\times 5$ grid there is a walk, through adjacent squares, that visits each square exactly once.	The persistent urge to understand why.

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