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

The mathematical definition of a triangular number n as t_n is follows.

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.

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

Triangular number 1 is represented using just one dot.

Triangular number 2 is still not very interesting.

Triangular number 3 has a value of 6.

Triangular number 4 has a value of 10.

The 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.

• 1 "μονάδα" (mo-NA-tha) ≈ "unit, monad" and refers to the supreme being and to unity.
• 2 "δύναμη" (THEE-na-mee) ≈ "power" or dyad, requiring two parts, such as action-reaction. The English word "dynamic" comes from this Greek word.
• 3 "αρμονία" (ar-mo-NEEA) ≈ "harmony". The English word "harmony" comes from this Greek word.
• 4. "κόσμος" (KO-smos) ≈ "world, people, universe" or people in the world

The tetractys also symbolized the four classical elements.

• 1. "φωτιά" (fo-tee-A) ≈ "fire".
• 2. "αέρας" (a-EH-ras) ≈ "air".
• 3. "υγρά" (ee-GRA) ≈ "water". The English word "hydro" in "hydrogen" comes from this Greek word.
• 4. "γη" (ghee) ≈ "earth, land" . The "ge" in "geometry" comes from this Greek word.

The tetractys also symbolized the four dimensions of space.

• 1. point (one point, zero dimensions, point)
• 2. line (two points, one dimension, line)
• 3. plane (three points, two dimensions, triangle)
• 4. space (four points, three dimensions, tetrahedron)

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

Triangular number 5 has a value of 15.

The 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.

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

Triangular number 7 has a value of 21.

If 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.

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

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

The verses keep adding another line, so the song can be quite long when sung in entirety. How many gifts total?

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.


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.

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]
local g = 0 for v=1,12 do for i=v,1,-1 do g = g + i end end print(string.format("g(12) = %d",g))
Here is the output of the Lua code.
g(12) = 364

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

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]
for n=1,30 do local g = 0 for v=1,n do for i=v,1,-1 do g = g + i end end print(string.format("g(%d) = %d",n,g)) end
Here is the output of the Lua code.
g(1) = 1 g(2) = 4 g(3) = 10 g(4) = 20 g(5) = 35 g(6) = 56 g(7) = 84 g(8) = 120 g(9) = 165 g(10) = 220 g(11) = 286 g(12) = 364 g(13) = 455 g(14) = 560 g(15) = 680 g(16) = 816 g(17) = 969 g(18) = 1140 g(19) = 1330 g(20) = 1540 g(21) = 1771 g(22) = 2024 g(23) = 2300 g(24) = 2600 g(25) = 2925 g(26) = 3276 g(27) = 3654 g(28) = 4060 g(29) = 4495 g(30) = 4960
The Portuguese word "lua" ≈ "moon". Lua was developed in Brazil and has a syntax based on the Pascal (and Modula-2) programming languages.

Triangular 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.

• Number 153: number of the fish
• Non-rational number approximation
• Height of a regular triangle
• Pythagorean theorem