Problem 5

In a circus,

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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

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Step 4 of 6: Build an explicit colour sequence

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(c_1,\ldots,c_{52})=(1,2,3,4,5,6,7,8,9,10,11,12,4,1,2,3,8,5,6,7,12,9,10,11,3,4,1,2,7,8,5,6,11,12,9,10,2,3,4,1,6,7,8,5,10,11,12,9,1,2,3,4)

Detailed analysis

Use the explicit 52-term sequence displayed above. For each $j=1,\ldots,48$ , the $j$ -th clown receives $c_j,c_{j+1},c_{j+2},c_{j+3},c_{j+4}$ . Every clown uses five colours; direct inspection of the displayed sequence shows that these 48 five-element sets are distinct and each colour occurs exactly 20 times.