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**Author** : Eric Pietrocupo

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Numeric Clairvoyance and Structural Numbers

Mathematics has a lot of applications in board games. I have been playing with numbers a lot without realizing that I was doing a lot of math behind the process. I'll attempt to share my experience with number inside this article.

In the early design of a game, you need to improvise a lot rules and components. Even if you improvise, you generally want to design something reliable that you will not have to change too often. So a good idea is to use some mathematics to design something more solid. The most common use of math for improvising is to use number sequences.

There are various kind of number sequence where some has a named according to their author. Not all of them could be used for board game, especially if there are very large numbers in the sequence. Here are a couple of popular sequence:

**Multiplication**: X x Y

You select a constant value X and multiply by value Y which is a value that will increment from 1 to N.

3 x 1 | 3 x 2 | 3 x 3 | 3 x 4 | 3 x 5 | 3 x 6 | 3 x 7 | 3 x 8 |

3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 |

The number sequence is: 3, 6, 9, 12, 15, 18, 21, 24 ...

This sequence creates multiples of X, it's a very common number sequence.

**Exponential**: X^{Y}

You select a constant value X and you add exponent Y which is a value you increment from 1 to N.

2^{1} | 2^{2} | 2^{3} | 2^{4} | 2^{5} | 2^{6} | 2^{7} | 2^{8} |

2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 |

the number sequence is: 2, 4, 8, 16, 32, 64, 128, 256 ...

**Division**: X / Y

You select a constant value X and divide it by Y which is a value ranging from 1 to N. In this case, X should be a larger value if you want to have integers.

100 / 1 | 100 / 2 | 100 / 3 | 100 / 4 | 100 / 5 | 100 / 6 | 100 / 7 | 100 / 8 |

100 | 50 | 33 | 25 | 20 | 16 | 14 | 12 |

The number sequence is: 100, 50, 33, 25, 20, 16, 14, 12, ...

I have taken the number 100 as a constant because it is often used in percentages and probabilities.

Polygonal numbers are a bit harder to visualize, so you need to look at the pictures below. The most common polygonal sequence is triangular numbers which is found in many board games. But you can generate sequences with other regular shape than a triangle.

The basic idea of a polygonal number is that you create a shape with balls and count the number of balls in the same. The position in the sequence is determined by the number of balls each side of the shape should have. If you want a triangle with 2 balls on each side you need 3 balls, if you want a triangle with 3 balls on each side, you need 6 balls. Look at the picture below to see the idea.

**Triangular Numbers**

So each time you grow your triangle, you add a new number to your sequence. The concept is exactly the same for all other shapes, Here are the number sequence for the basic regular shapes.

Shape | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

Triangle | 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 |

Square | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 |

Pentagon | 1 | 5 | 12 | 22 | 35 | 51 | 70 | 92 |

Hexagon | 1 | 6 | 15 | 28 | 45 | 66 | 91 | 120 |

**Square Numbers**

**Hexagonal Numbers**

**Triangular numbers** can easily by calculated by adding numbers together.
For example:

2 | 3 | 4 | 5 | 6 | 7 | 8 |

1+2=3 | 3+3=6 | 6+4=10 | 10+5=15 | 15+6=21 | 21+7=28 | 28+8=36 |

**Square numbers** can easily be calculated by adding exponent 2. For example:

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

1^{2}=1 | 2^{2}=4 | 3^{2}=9 | 4^{2}=16 | 5^{2}=25 | 6^{2}=36 | 7^{2}=49 | 8^{2}=64 |

There are many other number sequences for various types of application. I found a list of number different kind of number sequence on Wikipedia which were named by their author. You can find it here

They are not all useful for board game design because some sequence rapidly create very large numbers. Here a a few sequence that could be useful

**Prime Numbers**: these are numbers that can only be divided by 1 and themselves. The beginning sequence is :

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 ...

**Fibonacci sequence**: This is an interesting number sequence that I discovered by myself and used in my Fallen Kingdoms board game. The concept is simple, in a sequence, you take the previous number of the sequence and add it to the current number of the sequence to get the next number of the sequence. For example, at some point in the sequence, the values are 2, 3, 5, 8. The value 5, is created by adding 2 and 3 together, and the next value 8 is creates by adding 3 and 5 together. Here is the beginning of the sequence.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 ...

The sequence start to be interesting at 2. One of the wonder of this sequence is that the next numbers always increment in almost the same proportions. Here is the proportion ratio between the each number step.

Step | A of B Proportion |

2-3 | 66% |

3-5 | 60% |

5-8 | 62% |

8-13 | 61% |

13-21 | 61% |

21-34 | 61% |

34-55 | 61% |

It means that in a game, if player increase their power according to this sequence, they will always get the same percentage bonus.

As you can see, there are various sequence of numbers possible, but how do we use them in the game. There are various areas you could use sequence.

A classic example is to combat unit strength. In a game like civilization, each time you uncover new technology, your units get more advanced and would be more efficient in combat, but how efficient? This is where you select a sequence, if your sequence incrementation is too low, you will end with a game like civilization revolution where an army of archers is capable of destroying a tank.

Since you want high tech unit to be not match for very low tech unit, and exponential increment seems like the best solution: 1, 2, 4, 8, 16, etc. So if your tank is 8 and your archer 1, he really have no chance. But if the riffle men are 4, it's not so bad, they are only at half the strength. But that should be OK for a 1 level lower unit to be able to defeat the tank.

Another application could be the price you must pay to acquire object or to reach a next level. Again, in civilization revolution, you have a series of milestones that determine the amount of gold required to reach the next level. This gold level has been determined with a sequence of numbers.

Let's take 2 very simple sequence: 1,2,3 VS 2,3,4. Each sequence has 3 number with a +1 increment for each step where the second sequence start at 2. They look like very similar sequence but there is a huge difference if you consider the proportions of the incrementation. Each step is a regular +1 increment, but the proportion is not.

Step | Proportion | Step | Proportion | |

1-2 | +100% | 2-3 | +50% | |

2-3 | +50% | 3-4 | +33% |

So with sequence 2-3-4, the incrementation proportion of the values is much smoother than 1-2-3. But you can also consider proportion by comparing ratio. The ratio between value 1 and 3 is 1:3, which means 3 is 3 times more powerful than 1 while the ratio between 2 and 4 is only 2:1. So if you want extreme values in your sequence not to have a huge proportion difference, you could change the proportion by simply increasing the start of the sequence. If a 2:1 ratio is too much, you could use sequence 3,4,5 where the proportion is even lower.

This is one of the reason why I do not like triangular numbers at the beginning of the sequence. If you look at the sequence proportion, you'll see that the proportions start really strong and then lower. If you look at the 2 first step, from 1 to 6, you have an incrementation of 600%.

Step | Proportion |

1-3 | +200% |

3-6 | +100% |

6-10 | +66% |

10-15 | +50% |

15-21 | +40% |

This is the reason why I like Fibonacci sequence since the incrementation proportion is always approximately the same.

This is a tool that I have found recently while searching on the internet for number sequence. It's a triangle of numbers which contains within it various number sequences including some sequence we have listed above.

A pascal triangle is similar to a binary tree where the branches get mixed together. The triangle is composed to that the child number placed under their parent is equal to the sum of their parents. Here is how it looks.

**Pascal Triangle**

You can extract number sequences in various ways, normally you select a diagonal as a sequence. That is the case for triangular numbers. But you can also have hollow diagonals which are traced through numbers, the Fibonacci sequence can be found this way.

**Triangular numbers in a Pascal triangle**

**Fibonacci numbers in a Pascal triangle**

So you could use this triangle to create various sequence. You do not even have to have a straight line to have a sequence, you can use your imagination to create any shape you like.

For more information about pascal triangle, you can check the internet or wikipedia. It's a very popular topic.

This section is a mix of many other ways to play with numbers. Feel free to use anything you like.

In many of my games, I try to assign it a number. That might look very weird to say, but yes, I can say that for example my game Fallen Kingdoms has been designed around the number 3. Here are some example of how they appear in the game:

- 3 type of resources
- 3 production track with length in multiples of 3
- 3 type of buildings
- 3 type of trophies
- 3 cards for each of the 6 categories of technology
- 3 paths are used for invasion

So as you can see, the whole game is based on the number 3 or on multiples of 3. Now you might ask, how defining a game around a number could be useful? Here are the reasons:

- The game is more symmetric. Symmetry makes the game more elegant, but is not more important than asymmetry which must both be present in a game.
- It makes it easier for the players to remember how the game works. (ex: How many action could I do?)
- It makes it easier to improvise in early design. (Ex: How many resources should there be in he game?)

It might seems like a trivial concept, but I particularly find it useful. If might sometimes add restriction to your design because you could need a 5th element to keep the symmetry, but most of the time, when you find that 5th element, you are much more satisfied with the results than if you would had only used 4.

If it possible to combine numbers together. Some numbers fit well with each other. For example, the number 2 and 4 fit well because, one number is half the other. Sometimes, it depends on the shaped it is placed in.

It reminds me of a situation where we had to place statues on 5 library columns. How many statues do we need to place and where do we need to place them to make sure it is the most elegant possible. The answer is 3, and they must be arranged in the following specific pattern:

The next section has 6 columns, now it was a bit problematic because even if the number 3 fit in 6, it does not show well in line.

so instead, we decided to place 4 statues like this:

**Update note (August 26th 2012)**: It seems that this is related to the property of "Local Symmetry" defined by Christopher Alexander who made a similar experiment using various strips of pattern. The more local symmetry a pattern has, the more likely it is be beautiful.

But if you used a circular pattern, 3 would fit pretty well in 6, it has even been used for a popular icon:

But 3 in a circular shape does not fit well:

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