Question 34
Two people play a game with black chips, white chips and two boxes. The first player puts several of his chips into one of the boxes, and several others into the other box. It is allowed to put no chips in one of the boxes and, moreover, it is not compulsory to put all the available chips into the boxes. Then the second player chooses a box and takes all the chips of that box. From the other box, he doubles the number of chips of each colour and gives them to the first player, leaving both boxes empty. They continue playing in this way in turns.
The goal of the first player is to achieve that his chips of one of the colours are exactly twice his chips of the other colour (in particular, if he is left without chips, he wins). If the first player starts with $a$ black chips and $b$ white chips, for which values of $a$ and $b$ can the first player achieve his goal, regardless of the strategy of the second player?
The goal of the first player is to achieve that his chips of one of the colours are exactly twice his chips of the other colour (in particular, if he is left without chips, he wins). If the first player starts with $a$ black chips and $b$ white chips, for which values of $a$ and $b$ can the first player achieve his goal, regardless of the strategy of the second player?