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Print74th Romanian Mathematical Olympiad
Romania counting and probability
Problem
Is it possible to arrange the first 49 positive integers in a array, so that each cell contains exactly one number and no two primes are neighbours?
Two neighbours are numbers from cells with a common side or a common vertex.
Two neighbours are numbers from cells with a common side or a common vertex.
Solution
It is possible. The positive primes less than are , , , , , , , , , , , , , , , that is numbers. Number the lines and the columns from to , set the primes in of the squares with both odd indexes and the other numbers in the remaining cells.
Final answer
Yes, it is possible.
Techniques
Coloring schemes, extremal argumentsPrime numbers