# Triangular Numbers and Combinations

This article explains couple of important properties of triangular numbers, how they can be used in puzzle solving, and how triangular numbers are related to combinations. Concepts are explained in the form of puzzles and graphical illustrations. Triangular numbers are the count of objects that can be arranged in the form of an equilateral triangle. (Just like how square number s are the count of objects that can arranged in the form of a square).

Trianglular Numbers

(image via http://www.mathsmutt.co.uk/files/sroot_files/trian.gif)

So the triangular number series goes like this – 0, 1, 3, 6, 10, 15, 21, 28, 36, 45 and 55 , and so on.

Index |
Triangular Number |

0 |
0 |

1 |
1 |

2 |
3 |

3 |
6 |

4 |
10 |

5 |
15 |

Table of Triangular Numbers This series looks familiar? First, let us first look at some of the properties of Triangular Numbers.

Property #1: Let us find the difference between consecutive triangular numbers.

**0 1 3 6 10 15**

1 - 0 3 – 1 6 – 3 10 – 6 15 - 10

1 2 3 4 5

Figure 3: Difference between consecutive Triangular Numbers

Difference between consecutive Triangular Number If you observe , the difference between consecutive triangular numbers results in a natural number series 1, 2, 3, 4, 5 , 6 , and so on. That is, the n^{th} triangular number is obtained by adding “n” to the (n - 1)^{th} triangular number. Ex : 5^{th} Triangular Number = 5 + 4^{th} Triangular Number = 5 + 10 = 15 . In other words, the n th triangular number is the sum of all numbers from 1 to n . Ex : 4^{th} Triangular Number = Sum of all number s from 1 to 4 = 1 + 2 + 3 + 4 = 10

Summation of numbers from 1 to n Thus the series of triangular numbers is nothing but the series of summation of numbers from 1 to n. Did you spot t his when you looked at Figure 2? Property #2: Let us now find the sum of consecutive triangular numbers.

**0 1 3 6 10 15**

0 + 1 1 + 3 3 + 6 6 + 10 10 + 15

1 4 9 16 25

Figure 5 : Sum of consecutive Triangular Numbers

If you observe, the sum of consecutive triangular numbers results in a series of square numbers 1, 4, 9, 16, 25, 36, and so on. That is, the sum of the n^{th }triangular number and the (n+1)^{th} triangular number is nothing but the (n +1) th square number. Ex : 3^{rd} Triangular Number + 4^{th} Triangular Number = 6 + 10 = 16 = 4^{2} = 4^{th} Square Number.

Figure 6: Two consecutive Triangular Numbers make a Square Number.

Let us now look at some practical applications of Triangular Numbers.

Handshake Puzzle – Consider the well - known handshake puzzle. If there are 5 folks who meet up for a party, and if each one of them shakes hand with everyone else, then how many handshakes happen in total? There are different ways of solving this puzzle. Let us look at one simple way . Out of the 5 folks, one of them shakes hand with the other 4 folks , and goes for a coffee drink. So 4 handshakes happen. Let us represent each handshake as a circle.

Figure 7 : 4 Handshakes in total Now, out of the 4 folks, one of them shakes hand with t he other 3 folks, and goes for a coffee drink . So 3 handshakes happen. Again, represent each handshake as a circle and add it to the previous set of circles.

Figure 8 : 4 + 3 Handshakes in total Now , out of the 3 folks, one of them shakes hand with the r emaining 2 folks, and goes for a coffee drink. So 2 handshakes happen. Again, represent each handshake as a circle and add it to the previous set of circles.

Figure 9 : 4 + 3 + 2 Handshakes in total Do you see the pattern building up? Now the remaining 2 folks shake hand s with each other, and go together for a coffee drink. So 1 handshake happen s . And we get the final set of circles as below.

Figure 10 : 4 + 3 + 2 + 1 Handshakes in total What have we got here? A Triangular Number! Yes, the 4^{th} Triangular Number = 10 . This represent s the number of handshakes that happen in a party of 5 folks where each one shakes hand with all others. In general, in a party of “n” folks, the total number of handshakes would be the (n -1)^{th} Triangular Number.

For more such properties read the full article attached below.