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.