Certain numbers of identical circles (pennies, for example) can be packed together closely in a triangular shape.
Those numbers -- 1, 3, 6, 10, 15, 21, and so on -- are called triangular numbers.
If you count the greentriangles in each of these designs, the sequence of numbers you see is 1, 3, 6, 10, 15, 21, ..., again the triangular numbers.
If you count the white triangles that are in the "spaces" between the green ones, the sequence of numbers starts with 0 (because the first design has no gaps) and then continues: 1, 3, 6, 10, 15, ..., again triangular numbers!
If you count all the tiny triangles in each design -- both the green and the white -- the numbers are square numbers!
This connection between square numbers and triangular numbers can be seen another way, too.
Build a stair-step arrangement of Cuisenaire rods, say W, R, G. Then build the very next stair-step: W, R, G, P.
Each stair-step arrangement is "triangular" (if we ignore the stepwise edge). It's "area" -- the number of little white rods that would be needed to cover it exactly -- is a triangular number. Put the two consecutive triangles together, and they make a square: .
Here's another example: . When placed together, these make a square whose area is 64, again the square of the length (in white rods) of the longest rod. (The brown rod is 8 white rods long, and 64 is 8 times 8, or "8 squared.")
Triangular numbers appear in many contexts.
- the number of distinct towers you can make of varying heights using exactly two blue blocks and all the rest white
- the number of distinct "shortest paths" you can take to a destination that is n blocks east and two blocks north of your starting position
- the numbers along the second slant line in Pascal's triangle
- half the product of adjacent (natural) numbers on the number line (see Asher's theorem)