What are “triangular numbers” and “square numbers”? How can you change one into the other? The basics of triangular and square numbers are not as complex as you’d think.

## How to Create Triangular Numbers Using Pennies

The easiest way to think of triangular numbers is to start placing objects into the shape of a triangle. Check the first image: 10 pennies in the shape of a triangle. Each side has 4 pennies, and the center is filled with a penny.

Let’s call this a “4-10″ triangle, because there are 4 pennies on a side and 10 pennies in total. “10” is the fourth triangular number.

In the rather strange and arbitrary way that mathematicians start things, one could say that a triangle with 1 penny on each side forms a triangle with 1 penny in total. This would be a “1-1″ triangle. In any case, the first triangular number is “1”. Again, check that first picture and just look at the bottom penny.

Put 2 pennies on each side to get a total of 3 pennies. “3” is the third triangular number.

## How to Create Triangular Numbers Using Mathematics

The rule to create the [n]th triangular number, T, is: T(n) = sum (1, 2, 3…n). So T[1] = 1, T[2] = 1+2 = 3, T[3] = 1+2+3 = 6, T[4] = 1+2+3+4 = 10, and so on.

One might notice that this is exactly the same as saying that **T[n] = T + n**. For example, T[4] = (1+2+3)+4 = T[3]+4 = 10.

As a special bonus, let’s say that the zero-th triangular number is T[0], and that T[0] = zero. Rather than provide a picture here, just close your eyes and count to three. Welcome back.

## How the Pennies Add Up Mathematically

Simply add the next row to the “top” of the triangle. The new row has one more penny than the previous row. That is a physical representation of the rule that T[n] = T + n.

## Square Numbers

A square number is the product of multiplying a number by itself. Without having fancy fonts, we can say S[n] = n*n.

The list of square numbers based on positive integers starts as {1, 4, 9, 16, 25…}.

## How to Square a Triangular Number

The old joke answers “How do I get to Carnegie Hall?” with “Practice, practice, practice”. Squaring triangular numbers is surprisingly easy by comparison.

It turns out that S[n] = T + T[n].

So the real answer is: add two consecutive triangular numbers to create a square number. In fact, these numbers are joined at the hip: there are no squares of integers between the sums of consecutive triangular numbers.

## Prove It!

Mathematical induction is the method of proving that S[n] = T + T[n].

First, this works for the first values of “n”:

S[1] = 1*1 = 1. Meanwhile, T + T[1] = 0+1 = 1. (That is why I mentioned T a few paragraphs earlier!)

S[2] = 2*2 = 4. Meanwhile, T[1] + T[2] = 1+(1+2) = 1+3 = 4.

S[3] = 3*3 = 9. Meanwhile, T[2] + T[3] = 3+(3+3) = 3+6 = 9.

Second, assume that the equation is true for n=N. In other words, assume that S(N) = T(N-1) + T(N).

With that assumption, and the rules for S(n) and T(n), we must try to prove that it is true for n=N+1. In other words, is it true that S(N+1) = T(N) + T(N+1)?

Start with the left side of the equation to be proven:

S(N+1) = (N+1)*(N+1) because that is the rule.

= N*N + 2*N + 1 by multiplying and combining the “N”s.

= S(N) + 2*N + 1 because S(n) = n*n by its rule.

Switch to the right side of the equation to be proven:

T(N) + T(N+1) = ( T(N-1) + N ) + ( T(N) + (N+1) ) by the rule for T(n): just so we can get back to the assumed-to-be-true “T(N-1) + T(N)”.

= ( T(N-1) + T(N) ) + ( N + (N+1) ) by rearranging.

= ( T(N-1) + T(N) ) + ( 2*N + 1) ) by combining the “N”s.

= T(N-1) + T(N) + 2*N + 1 by simply dropping the extra brackets.

Let’s put the left and right sides together again:

S(N+1) = S(N) + 2*N + 1 ?=? T(N-1) + T(N) + 2*N + 1 = T(N) + T(N+1) where the “?=? means we are not quite sure this is truly equal. But subtract “2*N + 1″ from both sides of the “?+?” and a miracle happens.

S(N) + 2*N + 1 ?=? T(N-1) + T(N) + 2*N + 1

= S(N) ?=? T(N-1) + T(N) which was true by the assumption, and was calculated for N = 1, 2 and 3

## Visual Confirmation

The images above and below demonstrate how the sum of consecutive triangular penny formations actually create square formations of pennies.

Adding the second and third triangular numbers gives the third square number. T[2] + T[3] = 3+6 = 9 = 3*3 = S[3].

## One Practical Example of a Triangular Number

Each frame of ten-pin bowling starts with a “4-10″ triangle of bowling pins.

**References**:

Weisstein, Eric W., MathWorld-A Wolfram Web Resource, “Triangular Number.”, 1999-2011, referenced June 3, 2011.

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