Arnaldo and Bernaldo are playing within the classroom as follows: they write initially under a positive integer n. then alternately, beginning with Arnold, erase the number that is on the table and write a new number that can be: • what has just been erased least the largest power of 2 (with exponent non-negative integer) less than or equal to the number off; • what has just been cleared divided by 2, if the number deleted is even. Whoever wins the game gets first zero. a) Determine which of the players has a winning strategy for n = 40 and describe it. b) Determine which of the players has a winning strategy for n = 2012 and describe it.
For instance, if n=3 then 3-2 = 1 if n = 4 then 4-4 = 0 or 4/2 = 2 Wladimir Araujo Tavares *Federal University of Ceará <http://lia.ufc.br/%7Ewladimir/> Homepage <http://lia.ufc.br/%7Ewladimir/> | Maratona<https://sites.google.com/site/quixadamaratona/>| * --
