Solution: MATH -34012208 Set Theory
Answer: SEA LANE

Written by Amos Eshel

Each set of numbers solves in a different way to a 3-digit number.

Subpuzzle 1

Pair a positive number with a negative number and add them together. The new numbers are composed of X followed by N zeroes, followed by Y (and maybe some more zeroes). X & Y are all between 1 & 26, single digit numbers preceded by a 0. N is in increasing order (1-5). Order by N and read X/Y as A1-Z26.

SI: 190090000 = 409818765 - 219728765
XO: 240015000 = 501226345 - 261211345
NE: 140000500 = 374201122 - 234200622
FO:  60000150 = 421010971 - 361010821
UR: 210000018 = 626666659 - 416666641

For an answer of 614.

Subpuzzle 2

Written by Jonah Ostroff
These numbers spell words/names with a missing letter located at the decimal point using calculator spelling (aka BEGHILOS). Group by the number of digits, and order by location of the decimal point to extract the missing letters.

T: E.HEL = 734.3
W: HO.LS = 57.04
O: BEZ.S = 5.238
O: IGLO.S = 5.0791
H: SLEIG. = .91375
F: SEL.ISH = 451.735
I: BOOB.ES = 53.8008
V: BELIE.E = 3.31738
E: LEGIBL. = .781937

For an answer of 205.

Subpuzzle 3

Each number is a multiple of the previous number. These are converted A1-Z26 and read in given order.

T = 20           * 1 = 20
H = 8           * 20 = 160
R = 18         * 160 = 2880
E = 5         * 2880 = 14400
E = 5        * 14400 = 72000
O = 15       * 72000 = 1080000
H = 8      * 1080000 = 8640000
O = 15     * 8640000 = 129600000
N = 14   * 129600000 = 1814400000
E = 5   * 1814400000 = 9072000000

For an answer of 301.

Subpuzzle 4

Write each whole #portion out in English. The numbers of characters match the denominators, which are also unique from 5-14; order by this. Then use the numerator as an index into the written out #.

F: FIFTY = 50 3/5
O: GOOGOL = 10^100 5/6
U: HUNDRED = 100 2/7
R: FOURTEEN = 14 4/8
O: NINETYTWO = 92 9/9
N: NINETYNINE = 99 / 3/10
E: EIGHTYSEVEN = 87 10/11
T: SEVENTYSEVEN = 77 6/12
W: TWENTYMILLION = 20,000,000 2/13
O: TWELVETHOUSAND = 12,000 9/14

For an answer of 412.

Subpuzzle 5

Each of these numbers can be written as X^Y, and using logic the Y’s can be set up such that they are uniquely 1-10 and the X’s are between 1 & 26.

O: 15^1 = 15
N: 14^2 = 196
E:  5^3 = 125
O: 15^4 = 50625
N: 14^5 = 537824
E:  5^6 = 15625
N: 14^7 = 105413504
I:  9^8 = 43046721
N: 14^9 = 20661046784
E:  5^10 = 9765625

For an answer of 119.

Subpuzzle 6

These numbers can be interpreted as (consecutive) dates. The dates correspond to holidays in the particular year. Read the first letters in year order.

Martin Luther King Jr. Day: Jan. 15, 2018 = 1152018
April Fools' Day:           Apr.  1, 2019 = 4012019
Yom Kippur:                 Sep. 27-28, 2020 = 92709282020
D-Day:                      Jun.  6, 2021 = 6062021
Ash Wednesday:              Mar.  2, 2022 = 3022022
Yom Kippur:                 Sep. 24-25, 2023 = 92409252023

For MAY DAY which provides an answer of 501 (aka May 1st).

Subpuzzle 7

If one treats these numbers as a grid, one can find the number 7 traced out as a 7, 0 as a 0, and 5 as a 5.

177777776517000837455555550
203496764640734092852092118
384317085401985602956467033
529975264805184206625555534
626700358302569905349782756
827512691710841037152730159
973818943649000237085555546

For an answer of 705.

Bringing it all together

The 7 answers can be ordered by their unique 1-7 1st digits. Convert the other two digits A1-Z26.

S: 1 19
E: 2 05
A: 3 01
L: 4 12
A: 5 01
N: 6 14
E: 7 05

For a final answer of SEA LANE.