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.

Here are the first few steps:

Bank Pocket
1 500 
2 -300 300
3 200 300
4 198 -198
5 398 102
6 -300 300
7 98 402
8 198 -198
9 198 -198
10 494 6

(The subtotals are bolded.) It doesn't take Sofia K.(1850-1891) to recognise that after another 9 lines (one cycle) I will have 12 in my pocket and so on increasing by 6 each time. So we cut a long story short and jump straight over the intermediate cycles to whatever entry is:

xx 2 498

whereafter the cycle repeats with:

398 102. So I am forced to leave at least 2 in the bank.

(Again, I haven't worked this out, but I guess that if the Bank would only increase its deposit limit to 199, a full withdrawal would be possible, after a ridiculous lot of paperwork, since 300 and 199 are relatively prime.)

Posted by broll on 2011-02-02 14:27:39