Three problems are discussed and solved using DERIVE. Problem 1: We want to put the pairs of all rational numbers up to a chosen denominator on the set $\lbrack0,1\rbrack\times\lbrack0,1\rbrack$ of the plane. We want to investigate the possible patterns appearing. Problem 2: Find a function, which puts the digits of an integer into reverse order, eg f(123)=321, f(420)=024=24, f(1000)=0001=1. Problem 3: Start with any number, say 128 and form the following sequence: $1^2+2^2+8^2$=69, $6^2+9^2$=117, $1^2+1^2+7^2$=51. Then you will end either with 1, 1, 1, ... or with a loop starting with 16:16, 37, 58, 89, 145, 42,20, 4, 16, ... How can you do this with DERIVE?