What are some productive things that can be done (other than directly studying Mathematics) during leisure time that has a side effect to improve oneself at Mathematics? For example, reading Mathematics Stack Exchange.
Inspired by What to do while compiling?
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Learn a little coding (for example python) and practise with old problems from the google coding championships.
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Carry a book about an area of maths, and read bits of it? Do smallish mathematical puzzles? Practice mental arithmetic?
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Project Euler: is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems.
Solving Project Euler problems is an interesting and addicting pastime.
This Mathoverflow question has a good collection of games that are both popular and mathematical. Some examples are set, hex and lights out.
Math puzzles for dinner is another Mathoverflow post which has lot of interesting answers.
How about inventing games with dice or coins, and then analyzing them?