, 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.