## Arithmetic Techniques Part 1: Techniques for calculating n/36

#### Created by Sho Sengoku, 2002

Divisions (fractions) n/36, like 5/36 or 21/36, are inevitable in backgammon, but there is a method to calculate all 36 fractions (from 1/36 to 36/36) with simple addition processes by memorizing only 12 of them. You don't have to memorize all 36 fractions, from 1/36 to 36/36.  Here's how to do it.

 1 2.77 10 27.77 19 52.77 28 77.77 2 5.55 11 30.55 20 55.55 29 80.55 3 8.33 12 33.33 21 58.33 30 83.33 4 11.11 13 36.11 22 61.11 31 86.11 5 13.88 14 38.88 23 63.88 32 88.88 6 16.66 15 41.66 24 66.66 33 91.66 7 19.44 16 44.44 25 69.44 34 94.44 8 22.22 17 47.22 26 72.22 35 97.22 9 25 18 50 27 75 36 100

### Numbers you must memorize

In the table above, a number in the left of every two columns represents n (from 1 to 36) of n/36, and a number in the right on n shows the calculation result of n/36 in % to two decimal places, omitting the figures below the second decimal place.  By memorizing all columns with color in the table, you can calculate all other n/36 with only addition.

First of all, you probably don't need much effort to memorize 9/36, 18/36, 27/36, and 36/36, because you already know % number of 1/4 (=9/36), 1/2 (=18/36), 3/4 (=27/36), and 1 (=36/36.)  Numbers you really have to memorize are eight at most, from 1/36 to 8/36.

### How to calculate n/36 without division process

Example:   23/36

Since 23 is 18 + 5, the fraction 23/36 is also 18/36 + 5/36.  We have already known % value of 18/36 and 5/36, 50.00% and 13.88% respectively, and 23/36 can be calculated as 63.88% by addition of them.

In general , a numerator of any "not colored" columns in the table is an addition of % value of 9/36, 18/36 or 27/36 and 1/36~8/36, and you can get % value of n/36 by just adding numbers you have memorized.

### Other useful techniques

 1 2.77 10 27.77 19 52.77 28 77.77 2 5.55 11 30.55 20 55.55 29 80.55 3 8.33 12 33.33 21 58.33 30 83.33 4 11.11 13 36.11 22 61.11 31 86.11 5 13.88 14 38.88 23 63.88 32 88.88 6 16.66 15 41.66 24 66.66 33 91.66 7 19.44 16 44.44 25 69.44 34 94.44 8 22.22 17 47.22 26 72.22 35 97.22 9 25 18 50 27 75 36 100

Because 4/36 is 11.11% that is very easy number to multiply,  you can get the result quickly by using this characteristic when n of n/36 is an integral multiple of 4 (8, 12, 16, 20, 24, 28, and 32.)

Example: 24/36

Since 24 is 6 times 4, 24/36 is 11.11% times 6, that is 66.66%.