2006 iTest Problems/Problem 10

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Solution

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

$1.$ The black triangles with $3$ red dots in them. There are $27$ of these. $2.$ The three small red triangles with a dot in the middle separated by black in between. There are $9$ of these. $3.$ The three red dots with a red triangle in the middle separated by black in between. There are $6$ of these. $4.$ The medium red triangles. There are $10$ of these. $5.$ The large red triangles. There are $3$ of these.

For the first figure, there are $3$ multiples of $4$ represented by the three red dots. For the second figure, notice the first one of those is on the $8$th row, meaning there are $9$ total numbers in that row. Then subtract the $3$ black numbers to get $6$ multiples, but that's for both of those lines, so each one is $3$ 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 $2$ numbers long and the last row has $1$ number. Each one of those triangles therefore has $6$ numbers.

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