Tempted by Triangles

How many different triangles are in a $10\times 10\times 10$ triangular grid?
Source: NCTM Problem to Ponder 2/16/2011

SOLUTION
We study the problem in a smaller $6\times 6\times 6$ triangular grid as shown below

For ease in discussion, we label the rows of triangles as rows 1, 2, 3, 4, 5, 6.
The top triangle sits at row 1. All other triangles below it are either a translation or a reflection of the top triangle. Triangles formed by translation point upward and those formed by reflection point downward.

Size 1
At row 1 there is 1 translated triangle. At row 2 there are 2 triangles. At row 3 there are 3 (not colored) and so on until at row 6 there are 6 triangles. Thus, the number of translated triangles equals
$1+2+3+4+5+6=21$

At row 2 there is 1 reflected triangle. At row 3 there are 2 triangles (not colored) and so on until at row 6 there are 5 triangles. Thus, the number of reflected triangles equals
$1+2+3+4+5=15$
The total number of triangles of size 1 equals $21+15=36$.

Size 2

Because the size of each triangle now equals 2, the top triangle takes up rows 1 and 2. For ease in notation, we will mention only the beginning row. So we say that at row 1 there is 1 translated triangle. At row 2, there are 2 triangles (not colored) and they overlap each other. At row 3 there are 3 triangles and so on until at row 5 there are 5 triangles (not all are colored). Thus, the number of translated triangles equals
$1+2+3+4+5=15$

The first reflected triangle doesn’t begin until at row 3. At row 4 there are 2 triangles (not colored) and at row 5 there are 3 triangles. Thus, the number of reflected triangles equals
$1+2+3=6$
The total number of size 2 triangles equals $15+6=21$.

Size 3

At row 1 there is one translated triangle. At row 2 there are 2; at row 3 there are 3 and at row 4 there are 4. Thus, the number of translated triangles equals
$1+2+3+4=10$

The figure above shows that there is a single reflected triangle at row 4. In fact there will be no reflected triangle of size greater than 3.
The total number of size 3 triangles equals $10+1=11$.

Size 4

At row 1 there is 1 translated triangle; at row 2 there are 2; at row 3 there are 3. The number of translated triangles equals
$1+2+3=6$
Reflected triangle = 0.
The total number of size 4 triangles equals 6.

Size 5

At row 1 there is one translated triangle; at row 2 there are 2. The total number of translated triangles equals $1+2=3$.
Reflected triangle = 0. The total number of size 5 triangles equals 3.

Size 6
There is only one triangle of size 6.

Summary

 Size Translated Reflected Total 1 21 15 36 2 15 6 21 3 10 1 11 4 6 0 6 5 3 0 3 6 1 0 1 Grand Total 78

Let’s apply what we have learned to the $10\times 10\times 10$ triangular grid.

Size 1
Translated triangles = $1+2+3+\cdots+10=\frac{\left (10+1\right )10}{2}=55$
The rows of translated triangles begin at row 1 and end at row 10, that’s why we add ten consecutive counting numbers starting from 1.
Reflected triangles = $1+2+3+\cdots+9=\frac{\left (9+1\right )9}{2}=45$
Since the top triangle takes up one row, the rows of reflected triangles begin at row 2 and end at row 10. Thus, we add nine consecutive numbers.
Total = $55+45=100$.

Size 2
Translated triangles = $1+2+3+\cdots+9=\frac{\left (9+1\right )9}{2}=45$
The rows of translated triangles begin at row 1 and end at row 9 because had we ended at row 10 the size 2 of the triangle will cause the bottom of the triangles to stick out of the grid at row 11.
Reflected triangles = $1+2+3+\cdots+7=\frac{\left (7+1\right )7}{2}=28$
Since the top triangle takes up two rows, the rows of reflected triangles begin at row 3 and end at row 9.
Total = $45+28=73$.

Size 3

Translated triangles = $1+2+3+\cdots+8=\frac{\left (8+1\right )8}{2}=36$
The rows of translated triangles begin at row 1 and end at row 8.
Reflected triangles = $1+2+3+4+5=15$
Since the top triangle takes up three rows, the rows of reflected triangles begin at row 4 and end at row 8.
Total = $36+15=51$.

Size 4

Translated triangles = $1+2+3+\cdots+7=\frac{\left (7+1\right )7}{2}=28$
The rows of translated triangles begin at row 1 and end at row 7.
Reflected triangles = $1+2+3=6$
Since the top triangle takes up four rows, the rows of reflected triangles begin at row 5 and end at row 7.
Total = $28+6=34$.

Size 5

Translated triangles = $1+2+3+4+5+6=21$
The rows of translated triangles begin at row 1 and end at row 6.
Reflected triangles = $1$
Since the top triangle takes up five row, there is only 1 reflected triangle beginning at row 6.
Total = $21+1=22$.

Size 6

Translated triangles = $1+2+3+4+5=15$
The rows of translated triangles begin at row 1 and end at row 5.
Reflected triangles = $0$. From now on there will be no more reflected triangle.
Total = $15$.

Size 7

Translated triangles = $1+2+3+4=10$
The rows of translated triangles begin at row 1 and end at row 4.
Total = 10.

Size 8

Translated triangles = $1+2+3=6$
The rows of translated triangles begin at row 1 and end at row 3.
Total = 8.

Size 9

Translated triangles = $1+2=3$
The rows of translated triangles begin at row 1 and end at row 2.
Total = 3.

Size 10
There is only one triangle of this size.

Summary

 Size Translated Reflected Total 1 55 45 100 2 45 28 73 3 36 15 51 4 28 6 34 5 21 1 22 6 15 0 15 7 10 0 10 8 6 0 6 9 3 0 3 10 1 0 1 Grand Total 315

By definition triangular numbers are numbers that are sum of consecutive counting numbers. For example, 15 is a triangular number because $15=1+2+3+4+5$. Thus, the numbers of translated triangles and the numbers of reflected triangles are triangular numbers.