Solve one - get one free.

I. Crazy bank

I have not a penny in my pocket and the only money available to me is $500, deposited in a crazy bank with some very strange rules:

a. There is no interest on money deposited.
b. There are no charges for withdrawal, but one can only draw exactly $300,- no overdraft and no other amounts allowed.
c. There is only one specific amount of money one can deposit - $198 to be exact, happily enough no charges for this .

Using only those operations, what is the biggest amount I can rescue from this bank?

II. Shift a token.

On a ruler, marked from 0 to 147 cm, a token is placed on one of the marked places.
You can move this token any time:
either 100cm to the right or 47 cm to the left.
Prove that after 147 permissible moves the token is at its starting point.


If I were you I would start with the token story first.

(In reply to solutions by Charlie)

In fact I have just finished looking at a more manageable ruler of length, L= 0-7 with moves {5,2}, but starting at 0. As in your case,  2 (=47) is reached after L-1 moves, and a move to 0 is possible on the Lth move, completing the cycle. But a move to 7 is also possible, from which the next position is 5 (i.e. the only permissible first move from 0). Strictly speaking, the repeat happens either on the 7th move (=147) or the 8th move (= 148). So it is not necessarily true that 'after 147 permissible moves the token is at its starting point', but only that 'either the token revisits its starting point on the 147th move, or revisits the point it it was at after the first move on the 148th move'.

Posted by broll on 2011-02-02 15:39:51