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Fibonacci generation
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Find the 44th prime number over 100,000. Bonus: print the next 4 primes (after the initial prime).
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Create a tic-tac-toe (noughts and crosses) game: user versus machine. Print the 3x3 grid in a terminal & accept input from the command-line; or create a web-based version; or any other UI.
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Implement a Sudoku solver
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Some good suggestions here; perhaps we could get the topic in advance, so we can set up the framework and only have to write core code on the night
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Reinforcement learning; make an agent that learns as it proceeds by the means of getting reward for a right move and punishment for a wrong move