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Triangular numbers
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1. Triangular numbers
A triangular number is the number of objects arranged in a equilateral triangle, as in the examples below.

2. Mathematical formula
The mathematical definition of a triangular number n as tn is follows.

Triangular number equation

3. Combinations
The following mathematical equation is called a binomial coefficient and represents the number of distinct pairs of objects that can be selected from n+1 objects. That is, combinations.

Binomial coefficient

4. Visualizations
Below are a series of visualizations of triangular numbers starting at 1.

5. Triangular number 1
Triangular numbersTriangular number 1 is represented using just one dot.

6. Triangular number 2
Triangular numbersTriangular number 2 is still not very interesting.

7. Triangular number 3
Triangular numbersTriangular number 3 has a value of 6.

8. Triangular number 4
Triangular numbersTriangular number 4 has a value of 10.

9. Tetractys
Triangular numbersThe modern Greek word "τετρακτύς" (teh-tra-KTEES) ≈ "tetractys", also called the tetractys of the decad, is a triangle shape of triangular number 4.

This pattern and number was considered special (mystical) by Pythagoras and his followers for their "secret" worship. Each row and number had a special meaning.

10. Elements
The tetractys also symbolized the four classical elements.

11. Dimensions
Dimensions 0 to 3
The tetractys also symbolized the four dimensions of space.

12. Bowling
The pattern of triangular number 4 is used for the setup of 10 bowling pins.

13. Triangular number 5
Triangular numbersTriangular number 5 has a value of 15.

14. Billiard balls
Billiard ballsThe pattern of triangular number 5 is used for the setup of 15 billiard balls. Note that the balls are shown in order and not following the rules for a starting configuration of any particular game of billiards.

15. Triangle peg puzzle
Peg boardIt is also the pattern of the 15 pin triangle peg puzzle board. Peg board with pinsThe game starts with 14 pegs and jumps are to be made until only one peg remains.

16. Triangular number 6
Triangular numbersTriangular number 7 has a value of 21.

17. Dice
Unique rollsIf two dice are rolled, and the number of unique ways in which the two dice can land, can be arranged in a triangle formed by triangular number 6.

18. Triangular number 7
Triangular numbersTriangular number 7 has a value of 28 which is a perfect number.

19. Dominoes
DominosEach of 28 different with dots from 0 to 6 can be arranged in a triangle formed by triangular number 7.

20. Triangular number 12
Twelve days of Christmas Twelve days of Christmas

The Christmas song "The twelve days of Christmas", in the final verse, has 12 lines, or parts, and 78 gifts. The number 78 is an example of a "triangular number". The song is sung over and over.
 
How many total gifts?


Information sign More: Triangular number 12

21. Total number of gifts
Twelve days of ChristmasThe verses keep adding another line, so the song can be quite long when sung in entirety. How many gifts total?
1*12 + 2*11 + 3*10 + 4*9 + 5*8 + 6*7 + 7*6 + 8*5 + 9*4 + 10*3 + 11*2 12*1


22. Mathematical equation
If one adds all the triangular numbers from 1 to 12, when singing all twelve verses, the total number of gifts is 364. In mathematical terms, the number of gifts g is given by the following formula.

Gift total for the twelve days of Christmas
According to some Christian traditions, the last gift, to make a gift for each day of the year (ignoring leap years), is the gift on Christmas of the birth of the baby Jesus.

23. Computer program
Of course, computer scientists or software engineers may prefer to use a program to count the gifts, such is in the following Lua program.

Here is the Lua code [#1]

Here is the output of the Lua code.


24. Lua
The Portuguese word "lua""moon". Lua was developed in Brazil and has a syntax based on the Pascal (and Modula-2) programming languages.

25. Computer program
Here is a list of "gift" numbers for verses from 1 to 30 in the sense of the total gifts in the "12 days of Christmas". Here is a Lua program to compute these value.

Here is the Lua code [#2]

Here is the output of the Lua code.

The Portuguese word "lua""moon". Lua was developed in Brazil and has a syntax based on the Pascal (and Modula-2) programming languages.

26. Triangular number 17
Number of the fish using dotsTriangular number 17 has a value of 153. Triangular number 17 is interesting in the sense that the value, 153, is the "number of the fish". For more information, see the following.

27. End of page

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