Question 127
Camilla has a box containing 2015 paperclips. She takes a positive number $n$ of them and puts them on Federica's desk, challenging her to the following game. Federica has two kinds of moves available: she may remove 3 paperclips from the pile on her desk (if the pile contains at least 3 paperclips), or remove half of the paperclips present (if the pile contains an even number of them). Federica wins if, with a sequence of moves of the types described above, she manages to remove all the paperclips from her desk.
a) For how many of the 2015 possible values of $n$ can Federica win?
b) The girls change the rules of the game and decide to award the victory to Federica if she manages to leave a single paperclip on the desk. For how many of the 2015 values of $n$ can Federica win with the new rules?
a) For how many of the 2015 possible values of $n$ can Federica win?
b) The girls change the rules of the game and decide to award the victory to Federica if she manages to leave a single paperclip on the desk. For how many of the 2015 values of $n$ can Federica win with the new rules?