 | Here is an illustration of person P4 related to person P213 with common ancestors, Gudrun Vinge and Gustav Nøstbakken. There are two rows for each common ancestor. The top row corresponds to P4. The bottom row corresponds to P213. To read the relationship, view from left to right, noting a mother for each pink cell and father for each blue cell. So Gudrun is P4's mother, and Gustav is P4's father. Additionally Gudrun is P213's father's father's mother, and Gustav is P213's father's father's father. P213 is the daughter of P4's nephew. In general, we can quickly assess the relationship by counting the vertical white lines. P4 has 0 white lines, and P213 has 2 white lines in each of their rows, so in a general sense P213 is P4's 0th cousin, 2x removed. This general rule doesn't work well with nieces and nephews because we're not used to talking about 0th cousins. |
 | In this example, the common ancestors are Axel and Marie. Counting the vertical lines, the top rows have 1 line, and the bottom rows have 3 lines. That difference is two, so P7913 in P4's 1st cousin 2 times removed. Axel is P4's mother's father. Marie is P4's mother's mother. Axel is P7913's father's mother's mother's father. Marie is P7913's father's mother's mother's mother. In general, for a married pair of ancestors, each row in the set will be identical except for the rightmost cell, where one is blue and one is pink. That's because the path to each of them is the same except the final step to the common ancestors. |
Now let's discuss calculating the net relationship. To do this, we calculate a number, which describes the relationship. First cousins have a value of 1. Second cousins have a value of 2. Third cousins have a value of 3, etc. We can do the calculations by first setting up a table.
| Value | Value2 | Relationship | Explanation |
|---|---|---|---|
| 1 | 1 | siblings | Each sibling has one shared parent of the same sex. |
| 2 | 4 | first cousins | Each cousin has two shared grandparents of the same sex. |
| 4 | 16 | second cousins | Each 2nd cousin has 4 shared great-grandparents of the same sex. |
| 8 | 64 | third cousins | Each 3rd cousin has 8 shared 2nd-great-grandparents of the same sex. |
| 16 | 256 | fourth cousins | Each 4th cousin has 16 shared 3rd-great-grandparents of the same sex. |
For each shared ancestor, we calculate a number by starting with 1 and multiplying by 2 for each vertical line. So in the last illustration, above, we get: 2, 8, 2, 8.
The numbers for each line for a common ancestor are multiplied together. This gives is a value of 16 for each ancestor in this example.
We're really interested in the reciprocal, through: 1⁄16 and 1⁄16. These are added together to produce a sum of 1⁄8. The reciprocal of this number, 8, is then used to look up the relationship in the above chart, searching in the Value2 column. So P4 and P7913 have the relationship of somewhere between first and second cousins. To calculate a value that is between values in the table, above, or beyond the range of the values listed, we just calculate the log2 of the square root of our sum, or the log4 of the sum. So log4(8) = 1.5. But what if there are multiple paths to a common ancestor? Well, in that case, we just do simple arithmetic.
 | In this example, P4 and P9411 are 4th cousins 2 different ways. Rather than figuring out whether the common ancestors married each other and which married which (you can tell by the colors of the cells in this case that Hellie and Madela may be a pair and Maria and Hendrich may be a pair), it's easier to just calculate half-relationships. So P4 and P9411 are half-4th cousins 4 different ways. Using the above calculations, our numbers for each row are simply 16, 16, 16, 16, 16, 16, 16, 16. Multiplying the numbers in the paired lines together we get, 256, 256, 256, 256, producing our reciprocals 1⁄256, 1⁄256,1⁄256,1⁄256. Adding these 4 numbers together, we get 1⁄64, and looking up 64 in the table, above, or finding log4(64) we get 3. That means P4 and P9411 have a net relationship of 3rd cousins. This makes sense, since they are 4th cousins two different ways. |
Things get a little more complex if there are multiple ways to get to an individual ancestor. However, so far the steps have purposely separated out all the rows from each other to make this complexity simpler. Next we consider the following example.
| This examples starts out being simple, giving us the numbers 512, and 512. But what's happening on the next row for Dortea? The row is divided into 4 smaller lines, representing 4 different ways Dortea is the ancestor of P4. Let's add them in parentheses: (64, 64, 64, 64). Doing this for the whole example we get: 512, 512, (64,64,64,64), 256, 512, 512, 16, 16, 512, 512, (64,64), 256, 16, 16. The first thing to do is to resolve what's in parentheses. These numbers all represent reciprocals, so let's first represent the values that way: 1⁄512, 1⁄512, (1⁄64,1⁄64,1⁄64,1⁄64), 1⁄256, 1⁄512, 1⁄512, 1⁄16, 1⁄16, 1⁄512, 1⁄512, (1⁄64,1⁄64), 1⁄256, 1⁄16, 1⁄16. Resolving the parenthetical terms is done by simply adding them. So we get: 1⁄512, 1⁄512, (4⁄64), 1⁄256, 1⁄512, 1⁄512, 1⁄16, 1⁄16, 1⁄512, 1⁄512, (2⁄64), 1⁄256, 1⁄16, 1⁄16. Now we have one number per line, so we multiply each related pair together: 1/262144, 4/16384, 1/262144, 1/256, 1/262144, 2/16384, 1/256 (one for each ancestor). These are then simply added together. Before adding, lets use a common denominator in the list: 1/262144, 16/262144, 1/262144, 1024/262144, 1/262144, 32/262144, 1024/262144. That gives us a sum of 2099/262144. The log4 of its reciprocal is about 3.482, so P4 and P7345 have a relatedness of 3.482nd cousins. |
