The image shows two regular octagons, each has 4 red and 4 blue beads, one placed on each vertex.
We say that there is a 'match' if a vertex has the same colour in both octagons. In the diagram, we can see that the upper-right vertices are both blue and the lower-right vertices are both red - all the other vertex pairs have different colours. In this case, we have a matching value of 2.
However, if we rotate the inner octagon by 45 degrees clockwise then all the vertices match, giving us a maximum matching score of 8 for these two arrangements.
Now, if we randomly allocate the beads to the two octagons (each with 4 of each colour) and we are allowed to rotate one of the octagons, what is the smallest guaranteed maximum matching score?
Prove that in each set of ten consecutive integers there is one which is coprime with each of the other integers.
For example, taking 114, 115, 116, 117, 118, 119, 120, 121, 122, 123 the numbers 119 and 121 are each coprime with all the others. [Two integers a, b are coprime if their greatest common divisor is one.]