User blog:Ubersketch/BMS for small children.

Primitive Sequence System
PrSS is the most basic form of BMS, having only 1 row. A matrix looks like this: (a)(b)(c)... where a, b, c... are all non-negative integers. A column is an area surrounded by two parentheses including only numbers or commas. Matrices may be followed by [n] where n is a non-negative integer. (If you are familiar with the fast-growing hierarchy, or other similar notations, [n] is exactly the fundamental sequence function except for cases where the matrix is followed by a column of 0s.) I will provide some example matrices. First, we have to find the bad part, which will be marked in bold. We just find the first column smaller than the last column, called the bad root, and mark it as the bad part, as well as every column following it, save for the final column. Now, on to evaluation. If these cases don't apply then do the following:
 * (0)(1)(2)
 * (0)(1)(2)(3)
 * (0)(1)(1)
 * (0)(1)(2)
 * (0)(1)(2)(3)
 * (0)(1)(1)
 * (empty string)=0
 * 1) (0)[n]=(#[n])+1 where # is any string.

Remove the last column, remove the bad part, and copy it n times. (Note that when n is 0, the bad part is not copied at all.) That's it for PrSS.
 * (0)(1)(2)[n] evaluates to (0)(1)(1)... with n (1)s.
 * (0)(1)(2)(3)[n] evaluates to (0)(1)(2)(2)... with n (2)s.
 * (0)(1)(1)[n] evaluates to (0)(1)(0)(1)... with n (0)(1)s.

Pair Sequence System
PSS is a bit more complex but is as easy to learn as PrSS. This time a matrix looks like (a,b)(c,d)(e,f)... where a, b, c, d, e, f... are all non-negative integers. The nth row is an array of the nth entries of each column. Everything else, except when otherwise stated, is the same. I will provide some example matrices. Finding the bad part is a bit more complex. We look at the first row. If the entry immediately before the last element is greater than or equal to the last element, we cross it out. We repeat this until we have a strictly increasing sequence, meaning every entry is larger than the last. The column of the second entry to the right that isn't crossed out is the bad root, and the bad part is defined the same as PrSS. Now we just take the last column and the bad root and subtract their entries, this is called the difference, and replace the last entry with 0. Expansion is the same as in PrSS, except, the difference is added zero times to the first copy of the bad part, one time to the second copy, two times to the third copy, etc. Using this same process on PrSS yields the same results as if we evaluated PrSS as I told you. That's it for PSS.
 * (0,0)(1,1)(2,0)(1,1)
 * (0,0)(1,1)(2,2)
 * (0)(1)(2)(1)
 * (0)(1)(2)
 * (0)(1) (2) (1) -> (0) (1)(2) (1)
 * (0) (1) (2)
 * (0,0)(1,1)(2,0)(1,1)
 * (0,0)(1,1)(2,2)
 * (1,1)-(0,0)=(1,1)
 * (2,2)-(0,0)=(2,2)
 * (1,0)
 * (2,0)
 * (0,0)(1,1)(2,0)(1,1)[n]=(0,0)(1,1)(2,0)(1,0)(2,1)(3,0)... n times.
 * (0,0)(1,1)(2,2)[n]=(0,0)(1,1)(2,0)(3,1)... n times.

Bashicu Matrix System
It's finally time to learn the entirety of BMS. A matrix looks like this: (a,b,c...)(d,e,f...)(g,h,i...)... where a, b, c, d, e, f, g, h, i... are all non-negative integers. The columns can be any finite length.

TBD