## Exercises and problems in Informatics |

## Please read The Conditions of the Problem Solving Competition.

**I. 4.** A prime number is called *very prime*, if all of
its initial slices are also prime numbers. 239 for example, is a very
prime number, because 2, 23 and 239 are all primes. The prime number
241 on the other hand, is not a very prime number, because--although 2
is prime--24 is not. Write a program that displays all very prime
numbers having *N* digits (1\(\displaystyle le\)*N*\(\displaystyle le\)8) and all of their initial slices. Correct
solutions are ranked according to their speed. (10 points) Example
(*N*=3): 2, 23, 233 2, 23,
239 2, 29, 293 3, 31, 311...

**I. 5.** Write a program which displays a unit cube, firstly as
a wire frame object and secondly with invisible hidden parts. The
centre of the cube should be the origin and it is to be viewed from
the direction of the *z*-axis from a given distance. The program
should be able to rotate the cube around an arbitrary co-ordinate
axis. (10 points)

**I. 6.** We want to simulate the oscillations of a spring using
the following model. By carefully hanging a body with mass *M* on
the ``ideal'' spring (i.e. its mass is neglected), the spring expands
until it keeps its balance. Now we stretch the spring with the body to
have length *L*, and let it oscillate. This motion is to be
simulated by choosing an appropriately short time interval (*t*) and
computing the instant elongation, the resulting force (which comes
from the weight of the body with mass *M* and the elongation of
the spring), the acceleration and the velocity. Write an Excel
spreadsheet (RUGO.XLS) to simulate this phenomenon.

*a)* Give the quantities *F* (resulting force), *a*
(acceleration), *v* (velocity) and *l* (elongation) in the
first 200 timesteps.

*b)* Plot the change of the elongation.

*c)* The parameters of the model (the spring constant
*D*, the constant of friction *KE*, the initial elongation
*L*, the mass *M* of the body and the time interval *t*)
should be indicated in the upper left corner of your sheet (and they
are possibly modifiable).

*d)* How does the simulation change if friction is also taken
into account? (10 points)