Math Forum :: View topic – colouring of triangle board

Let’s consider for each natural similar “triangle board” as in the picture (there’s the situation for =3, for arbitrary it only suffices to divide the side of the original (“greatest” equilateral triangle in the picture into same parts). By this we obtain cells in the board, now let’s imagine that we’ll inscribe all numbers 1, 2, …, in the cells so that each cell contains exactly one number and no two cells include the same number. Find out the minimal number of colours (dependent on ) that are necessary for the colouring of the cells of the triangle board that satisfies these two conditions: 1) Each two neighbouring cells (only by edge, not by vertice) are coloured by different colours.

2) Each two cells that have inscribed any two consecutive natural numbers are coloured by different colours (so the cells with numbers and for each are coloured by different colours).

No quiero meter las cabras en el corral a tu.