How do professional mathematicians solve problems?

Depiction of the mental image many professional mathematicians create in their mind to solve the bat-and-ball puzzle.
The mathematical puzzles in the BrainQuake app are based on the kinds of mental images professional mathematicians create to solve problems. The puzzles are designed so that solving the puzzle on the screen requires following the same logical steps the pro does in their mind. Image by BrainQuake.
Singapore Mathematics uses (static) depictions of the mental images professional mathematicians create to solve problems, to provide visual support for the mathematical reasoning the student follows in symbolic form.
The three kinds of solution to the bat-and-ball problem. The celebrated Australian mathematician Terrence Tao has called the three kind of thinking “pre-rigorous”, “rigorous”, and “post-rigorous”.

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