, 2016

Consider the diagonal lines that begin on the left edge of the triangle and move downward to the right.

The first number in the $n^{th}$ diagonal line is $n$, and it lies in the $n^{th}$ horizontal row. For example, the first number in the $3^{rd}$ diagonal line $({\bf3},6,9,12,\dots)$ is 3 and it lies in the $3^{rd}$ horizontal row $({\bf3},4,3)$. The second number in the $n^{th}$ diagonal line is $n+n$ or $2n$ and it lies in the horizontal row numbered $n+1$. The third number in the $n^{th}$ diagonal line is $n+n+n$ or $3n$ and it lies in the horizontal row numbered $n+2$ (each number lies one row below the previous number in the diagonal line). Following this pattern, the $m^{th}$ number in the $n^{th}$ diagonal line is equal to $m\times n$ and it lies in the horizontal row numbered $n+(m-1)$. The table below demonstrates this for $n=3$, the $3^{rd}$ diagonal line. $m$ $m^{th}$ Diagonal Number Horizontal Row Number 1 3 3 2 $2(3)=6$ $3+1=4$ 3 $3(3)=9$ $3+2=5$ 4 $4(3)=12$ $3+3=6$ 5 $5(3)=15$ $3+4=7$ ⋮ ⋮ ⋮ $m$ $m\times n$ $3+(m-1)$ The number 2016 lies in some diagonal line(s). To determine which diagonal lines 2016 lies in, we express 2016 as a product $m\times n$ for positive integers $m$ and $n$. Further, if $2016=m\times n$, then 2016 appears in the triangle in position $m$ in diagonal line $n$, and lies in the horizontal row numbered $n+m-1$. We want the horizontal row in which 2016 first appears, and so we must find positive integers $m$ and $n$ so that $m\times n=2016$ and $n+m$ (and therefore $n+m-1$) is as small as possible. In the table below, we summarize the factor pairs $(m,n)$ of 2016 and the horizontal row number $n+m-1$ in which each occurrence of 2016 appears. Factor Pair $(m,n)$ Horizontal Row Number $n+m-1$ $(1,2016)$ 2016 $(2,1008)$ 1009 $(3,672)$ 674 $(4,504)$ 507 $(6,336)$ 341 $(7,288)$ 294 $(8,252)$ 259 $(9,224)$ 232 $(12,168)$ 179 $(14,144)$ 157 $(16,126)$ 141 $(18,112)$ 129 $(21,96)$ 116 $(24,84)$ 107 $(28,72)$ 99 $(32,63)$ 94 $(36,56)$ 91 $(42,48)$ 89 (Note: By recognizing that when $m\times n=2016$, the sum $n+m$ is minimized when the positive difference between $m$ and $n$ is minimized, we may shorten the work shown above.) We have included all possible pairs $(m,n)$ so that $m\times n=2016$ in the table above.

We see that 2016 will appear in 18 different locations in the triangle.

However, the first appearance of 2016 occurs in the horizontal row numbered 89.