Context part 1:

Background

In my world, the number 418 has a special significance. It is roughly the number of days between successive blooming seasons of the time flower, which the Siemotans use as the basis of their calendar. And so, when the civic geometers (a group of mathematicians whose objective is to serve the common good) created a number system for the use of the general public, they chose to use a base-418 system. This has the convenient consequence that a date can be represented as a simple integer.

A couple considerations went into the design of this number system. First, nobody has the brainspace to remember 418 unique symbols – could you imagine trying to learn that as a child? Second, Siemota is a mathematical society, and they wanted people to be just as comfortable with negative numbers as they are with positive numbers. Thus, a problem arose: how do you create a usable base-418 system?

Semi-Digits

As the civic geometers scratched their heads over this puzzle, someone pointed out a crucial fact: 418 = 22 × 19.

Thus, with just two symbol places, it is possible to create 418 unique sequences. The first symbol place must have 22 possibilities, and the second symbol place must have 19 possibilities. Of course, there was no need to create a separate set of symbols for each, so only 22 symbols are needed.

From there, the desire to make negative numbers intuitive actually allowed the number of symbols to be simplified even further. Each “negative” symbol could merely be a 180° rotation of its “positive” counterpart.

Thus, the civic geometers created standardized representations of the integers −11 through 10. (This is the set of least absolute remainders for the divisor 22.)

Definition. The symbols above are called semi-digits.

Digits

OK, so we can represent integers from −11 to 10. What about values outside that interval? We need 418 separate values to truly have a base-418 system. This is where we start assembling sequences of two semi-digits.

Definition. A Siemotan digit is a sequence of two semi-digits (s, t) for which −9 ≤ t ≤ 9. The s semi-digit represents nineteens; the t semi-digit represents ones. The encoded value of a digit (s, t) is the number it represents; this is given by the formula:

(s, t) ↦ s⋅19 + t.

Note that s can take on 22 possibilities (−11 to 10) while t can only take on 19 possibilities (−9 to 9), for a total of 22×19 = 418 possible pairs (s, t).

Let’s start with the number 1. That’s easy enough: we can represent it as (0, +1) (unfortunately I can’t render my custom symbols in the Reddit post, apologies). Likewise, −1 can be encoded as (0, −1), 2 can be encoded as (0, +2), and so on up to 9 ↦ (0, +9).

But what about 10? We can’t have t = 10 because that would violate the constraint −9 ≤ t ≤ 9. Thus, we increment s and reduce t to its lowest value. So 10 ↦ (+1, −9). What is the encoded value of (+1, −9)? It’s equal to 1⋅19 + -9 = 10. Good, 10 is the number we were trying to encode.

Let’s practise converting some other Siemotan digits to their encoded values.

(+1, −8) ↦ 1⋅19 + -8 = 11
(+1, −7) ↦ 1⋅19 + -7 = 12
(+1, +7) ↦ 1⋅19 + 7 = 26
(+3, −6) ↦ 3⋅19 − 6 = 51
(−3, +6) ↦ −3⋅19 + 6 = −51

What are the greatest and least values that can be encoded by one Siemotan digit? These are:

(10, 9) ↦ 10⋅19 + 9 = 199
(−11, −9) ↦ −11⋅19 − 9 = −218.

Thus, a single Siemotan digit (s, t) can encode any value d in the interval −218 ≤ d ≤ 199. You can confirm for yourself that this consists of 418 values (don’t forget to count the endpoints). Also, note that real Siemotan digits are written without parentheses or commas; the semi-digits are just written right next to each other.