# 1000000000000000000000000000000000000000000000000000000000000000000001000000000000116

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KeywordsContent

Every number in the puzzle actually represents an English word. There are a lot of 0's - perhaps it's a good idea to find where the non-zero place values are to determine the mapping from numbers to words.

The flavortext also suggests that MATH is represented by 1001003. In general, most words correspond to a unique number, though a number, especially smaller ones, can represent multiple words. (Some words cannot be represented by a number)

It will be useful to write out the numbers as words (100 --> "one hundred").

These written-out numbers are closely related to the words that they represent.

1001003 is written out as ONE MILLION ONE THOUSAND THREE. "MATH" is represented by this number.

giving away the system

It's very relevant that "MATH" is a subsequence (i.e. you can pick letters and read them out in order) of ONE [M]ILLION ONE THOUS[A]ND [TH]REE.

giving away the system

In particular, 1001003 is the smallest possible positive integer where "MATH" is a subsequence. No smaller number works, so therefore MATH corresponds to the number 1001003. Note that we are using short scale notation, and we do not write the word "and" between numbers (or else ONE MILLION AND THREE would be smaller).

knows system, needs help with words

When computing the minimal numbers of various words, it's useful to keep the following facts in mind:

J and K never appear in numbers. Z only appears in ZERO (and thus effectively, never appears) and B only appears in BILLION (and thus can only appear once). A, Q, C, P, and M are also in quite short supply, so the presence of numbers like QUADRILLION, OCTILLION and SEPTILLION are likely indicators that such letters appear.

knows system

It's also helpful to note that the answers to the clues on the right hand side of the equation will have the same initial letter as the variable on the left hand side.

Now that you have the words that answer the clues, you'll need to plug them into the expressions at the bottom, though now you'll need numbers again. You'll need to perform the reverse operation now.

extract

Solve for the expressions and you've got three numbers. Time to find their corresponding words yet again!

extract

Once you have three words, use the first two words to derive yet another number. The definition is given by the third word.

I've written out these notes for MATH 1001003 to be as small as possible. Now to figure out what they mean...

A = 61298 1000000000000000000001002 114 400 9 1000000000000000000000146206
B = 175000000000000000000000000001 1100 2 6014 1000000000003026
C = 1000000013000001000000001 175000000000000000000000000001 1002 1000000000000000000000001000 1000000000001000000000000000 1106 3020098 1 1236 2 111020 618000000000000006
F = 400 1000000000000000000000001003 1000111 1002 1000000000000061 1001006 2 1000001000000000000000000000023003
G = 1000003021 44 3 516 1000003
N = 3000000001000000 9 1800 6026 1001000000000000000000001000000000000000000001011 1000000001000000000020098 211000 796 1000000000000000000000013
S = 163000000000001200 4 1000000000000000000064000100 1000 164000000000000000000000000001, 1000000000000000000000000000000000000001000000000000020 1000020
T = 4000000000000000000000000001 201 1000 31000000000000000000000001 4 1298 36 1000000000000000004000100
W = 1001011 136000000000000000000000000000 1100 2 100635000000014 3 11000 44 1000 3100 4000000000001293

A·B+N, G+T+WC+FS