2006 iTest Problems/Problem 10

Solution

The pattern for rows of Pascal's Triangle with the multiples of colored red is here: http://www.catsindrag.co.uk/pascal/?r=64&m=4 There are five different figures in this triangle.

The black triangles with red dots in them. There are of these. The three small red triangles with a dot in the middle separated by black in between. There are of these. The three red dots with a red triangle in the middle separated by black in between. There are of these. The medium red triangles. There are of these. The large red triangles. There are of these.

For the first figure, there are multiples of represented by the three red dots. For the second figure, notice the first one of those is on the th row, meaning there are total numbers in that row. Then subtract the black numbers to get multiples, but that's for both of those lines, so each one is numbers long. We know from how these patterns on Pascal's Triangle work that the number of red numbers in the row of the triangle below that one is numbers long and the last row has number. Each one of those triangles therefore has numbers.