## Task 5 - 030925

a) How must 1023 balls be distributed into 10 bags so that any number of balls from 1 to 1023 can be assembled without opening a bag.

b) How many bags are needed at a minimum to be able to assemble any number of balls from 1 to 3000?
a) When distributing the balls into the bags, one can orient oneself to the binary system, as for each bag there are only 2 possibilities when putting together a number: either you take it, or you don't.

The bags must therefore correspond to the powers of two.

This results in the following distribution of balls:

$$
1-2-4-8-16-32-64-128-256-512
$$

b) To be able to represent numbers up to 3000 using the same method, 2 more bags must be added, filled with 1024 and 2048 balls.

Solutions of the II. Round 1963 adopted from [5]

### 6.5.3 III. Round 1963, Class 9