What the hell are...

One way to look at triangular numbers is using, what else, a triangle. Create a triangle by starting with 1 at the top, and continue with a simple sequence as you fill in each row, adding one more number to the row below it. In your triangle, the triangular numbers run down the right side:

Row

 1               1    <- 1st triangular number = 1

 2             2   3    <-   ...

 3           4   5   6    <-   ...           

 4         7   8   9  10    <-    ...      

 5      11  12   13  14  15   <-    ...

 6    16   17  18  19   20  21  <- 6th triangular number = 21

Triangular numbers are positive integers such that the given number of dots can be arranged to make an equilateral triangle. In the following example, you can see that the 1st triangular number (the row) is 1 (the number of dots), the second triangular number is 3, etc.

Rows in the triangle

 1    *        *          *            *             *

 2            * *        * *          * *           * *

 3                      * * *        * * *         * * *

 4                                  * * * *       * * * *

 5                                               * * * * *



TN =  1        3          6            10           15

Reiner Knizia's new game Amun-Re uses triangular numbers in two important ways:

Item cost	When purchasing in Amun-Re, the number of rows is the NUMBER OF ITEMS you buy. The triangular number is the TOTAL COST for those items.
Bidding increments	The bidding increments on each region card is a triangular number, with one exception: 0.

Triangular Numbers in your strategies

Since the cost of goods and region bidding in Amun-Re use triangular numbers, consider the following strategies:

• Build a little at a time each round. The way the cost is calculated results in an increase of the price per item as the number of items you buy increases. Traditionally, you think of buying in bulk to save money, but bricks, cards, and slaves ain't diapers from BJ's. The game requires that you buy a little at a time, over the course of the game, or else break the bank trying to make a big purchase at one time.
• Don't save up for a big explosive building round at the end-game.
• By as much DIFFERENT stuff as you can. Instead of spending 10 on 4 bricks, buy 3 bricks for 6, 2 power cards for 3, and 1 slave for 1, for the same total cost of 10.
• Try to convice other players to buy everything at once. Maybe they are still laboring under a "buy in bulk" mentality.
• Bid low at first. You should plan on bidding low for a region, so that you get overbid. Then you can target other players and force their bids up even more than yours. Each time you outbid someone else, you increase the cost of winning that region significantly.

Two simple formulas

One way to calculate triangular numbers without drawing pictures is to sum up all the whole numbers that come before a certain number. By this method, the 5th triangular number can be calculated with the following formula:

1 + 2 + 3 + 4 + 5 = 15

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Another way to get the triangular number of x (x is any number bigger than zero) is to multiply x by (x + 1) and then divide by 2. By this method, the 5th triangular number can be calculated with the following formula:

5 * (5 + 1) / 2 = 15

Pyramidal Numbers

Don't confuse triangular numbers with pyramidal numbers, although the latter would have been more thematically appropriate in Amun-Re.

According to mathworld.wolfram.com, a pyramidal number is "a figurate number corresponding to a configuration of points which forms a pyramid with r-sided regular polygon bases." Here's the formulaic representation:

Um... Yeah.

To think visually of pyramidal numbers, build a fictitious pyramid of tennis balls. Put down 16 balls as the base. Put 9 balls on top of that. Put 4 balls on top of that. And finally, put a single ball on the top level of the pyramid.

# Rows		Pyramidal Number

 1      o			1



 2     o o		5

       o o



 3    o o o		14

      o o o

      o o o



 4   o o o o		30

     o o o o

     o o o o

     o o o o

So the pyramidal number of 4 is 30; it would take 30 balls to support 4 rows in a pyramid.

Triangular numbers work the same way, though the base of the pyramid is a triangle (having 3 sides) and not a square or pyramid shape (having 4 sides).

Fun Triangular Number Facts

• Everyone in a group of people shakes hands with everyone else. The total number of handshakes will always be a triangular number. For instance, five people will make ten handshakes. Hopefully, everyone washed their hands before they start this experiment.
• If you're pee-shy at the public john, calculating triangular numbers helps release bladder muscles.
• The triangular number of 4 is the number and arrangement of bowling pins.
• In The 12 Days of Christmas, on every day, you are given a total number of items equal to that day's triangular number. For example, on the 3rd day of Christmas, you get 3 hens, 2 turtle doves, and a partridge (6 items). The triangular number of 3 is 6.

Dull Triangular Number Facts

• A Triangular number can never end in 2, 4, 7 or 9.
• All perfect numbers are triangular numbers.
• The numbers in the sequence 1, 11, 111, 1111, 11111,...etc. are all triangular numbers in base 9.
• The only triangular number which is also a prime is 3.
• The only Fibonacci Numbers that are also triangular are 1, 3, 21 and 55.

Other Games that use Triangular Numbers

	Marracash by Stefan Dora	The value of shoppers can be calculated using triangular numbers. It is most evident on a turn when you move multiple shoppers into a shop. If you put the first 4 shoppers in a shop, you get 1,000 dinar, or the triangular number of 4 (in hundreds). The maximum you can score is on the fifth shopper, who alone is worth 500, but added with the rest is worth 1500, or the 5th triangular number.
	Taj Mahal by Reiner Knizia	The running total for scoring series of commodities can be calculated using triangular numbers. For example, when you get your first tea, you get 1 point. When you get your second, you get 2 points (now, it has scored 3 total). When you get your third, you get 3 more points (for a running total of 6 points, the third triangular number).
	Roads and Boats by Jeroen Doumen & Joris Wiersinga	The cost of buying multiple bricks in the wonder on the same turn is a triangular number. For example, if you buy 2 bricks, it costs 3 goods. If you buy 3, the cost is 6, and 4 bricks costs you 10.
	Coloretto by Michael Schacht	Triangular numbers are used for set scoring. A set of 4 red cards, for example, is worth 10 points.
	Hare and Tortoise by David Parlett	The cost in carrots is the triangular number of the number of spaces you want to move. If you want to move 8 spaces, you must pay 36 carrots.
	Corner (Aladdin's Intersection) by Sid Sackson	Triangular numbers are used for set scoring. The value eventually levels off.
	New England by Alan Moon and Aaron Weissblum	VPs for the development cards are the triangular number of the number of squares on the card. Huh? In other words, a card with 4 squares on it is worth 10 points.

Sources