Puzzles modulo math

I am not impressed with the math puzzle in the Times today.  (Background: the “Wordplay” blog that the Times hosts to discuss crossword puzzles occasionally does “Numberplay” instead; that is, math puzzles.)  Here’s the statement:

Player 1 writes a sequence of eight positive integers. Player 2 then writes a + or – sign in each of the seven spaces between the integers. If the final numeric result is odd, player 1 wins. If even, player 2 wins. Who should win this game?

What if Player 2 can use a × sign?

I thought about this for all of thirty seconds and realized that it has no depth.  If you know what modular arithmetic is you can answer it; if you don’t, you will be stuck with a boring case-by-case analysis and presumably doomed to argue with math skeptics in the comments section for the rest of the week.

The thirty-second solution: mod 2, player 1 is just choosing a sequence of 0’s and 1’s, and player 2 is doing nothing at all (without multiplication) since + and – are the same.  So player 1 has only to choose an odd number of 1’s (that is, odd numbers), and therefore wins.  If multiplication is allowed, then player 2 can multiply everything if there is at least one even number, and add everything if all the numbers are odd, and therefore wins.

On the other hand, the linked puzzle (the “Five Pirates Puzzle”) from earlier used to be a favorite of mine.  Of course, I heard it as the “Hundred Pirates Puzzle”, and the answer gets very strange once you have more than 100 pirates.

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