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Task 1, CJam, 7 bytes

q~,:+_*

I just wanted to get the (presumably) optimal CJam solution for this in. It makes use of the fact that the sum of the first n cubes is the square of the nth triangular number, which is itself the sum of the first n integers.

Test it here.


Task 9, Mathematica, 4 bytes

1##&

This is essentially the same as Sp3000's solution, but using more syntactical trickery (and since I'm losing the tie to him, I don't think there's any harm in that). The ## is a sequence with all parameters of the unnamed function. Now 1## is short for 1*##, which is syntactical sugar for Times[1,##] where ## gets expanded into the number and the nested list. Multiplication threads over lists automatically in Mathematica.