# Solution to 1000000000000000000000000000000000000000000000000000000000000000000001000000000000116

The main conceit in this puzzle is converting numbers into words. But how is this done? There are some hints in the flavortext, as well as the somewhat unusual structure of the numbers — very sparse digits and lots of zeroes. If we consider the place values of the nonzero digits, we find that many of these correspond to powers of thousands, which suggests that certain properties of these powers of thousands are required by the words. The flavortext does also suggest writing out the numbers in full. Doing so with, say, MATH 1001003 gives ONE MILLION ONE THOUSAND THREE. MATH is a subsequence of this string, and moreover this is the smallest number such that this property holds! Note that the M forces the MILLION, the A forces the THOUSAND, and TH can be satisfied by THREE.

Note that there are a few rules that should be followed in this puzzle, notably that the word "and" should be omitted from the spelling of all numbers (including the word "and" would allow us to represent MATH with ONE MILLION AND THREE), and that we are using short scale notation.

Armed with this knowledge, we can determine the clues. Of course, while each word has a unique number under this scheme, some numbers will have multiple words mapping to them. We can use surrounding context to resolve these ambiguities, or at least enough to solve the clues. As an additional hint, the letter of each variable happens to be the starting letter of the answer.

We should then convert these words into numbers, as the mathematical operations should suggest. This step can be done with some intuition with which letters will likely require larger numbers in order to satisfy their requirements.

Letter | Clue | Answer | Number |
---|---|---|---|

A | Soothing plant often found in crosswords | ALOE | 1000000000000001 |

B | Device used to shoot arrows | BOW | 1000000002 |

C | Portable device that can call others relying on towers to relay signals | CELLPHONE | 1000001001000000000000000000000031 |

F | Ford car model that also means to concentrate | FOCUS | 4000000000000000000000000106 |

G | Birthstone of the first month | GARNET | 8119 |

N | Team in United States capital playing World Series sport | NATIONALS | 1000000000000000000000000000001000000000000006 |

S | Exhausted or consumed a resource, especially money | SPENT | 1000000000000000000000017 |

T | Force when a rope or string is pulled | TENSION | 361 |

W | Metal disc used to distribute the load of a threaded fastener | WASHER | 2603 |

We can now compute the expressions:

A·B+N = 1000000000000001·1000000002+1000000000000000000000000000001000000000000006 = 1000000000000000000001000000003000001000000008G+T+W = 8119+361+2603 = 11083

C+F−S = 1000001001000000000000000000000031+4000000000000000000000000106−1000000000000000000000017 = 1000005000000000000000000000000120

Finally, we can reconvert these back to words: the most compelling completion is ALPHABET LENGTH → CONFECTIONERY. The length of the alphabet is 26; a confectionery with value 26 is TWIX, the answer.