Problem 5

In a circus,

n

clowns dress and paint themselves using a selection of 12 distinct colours. Each clown must use at least five different colours. The ringmaster requires that no two clowns have exactly the same set of colours and that no more than 20 clowns use any one particular colour. Find the largest possible

n

Step 5 of 6: Assign five consecutive colours

Previous step Next step

\text{clown }j:\quad\{c_j,c_{j+1},c_{j+2},c_{j+3},c_{j+4}\}\quad(1\le j\le48)

Detailed analysis

Give clown $j$ the five colours $c_j,c_{j+1},c_{j+2},c_{j+3},c_{j+4}$ for $1\le j\le48$ . Every clown uses five colours; the explicit 52-term sequence makes these 48 five-element sets distinct and each colour occurs exactly 20 times, so all conditions hold with $n=48$ .