by Matt Gruskin
Answer: MUSTACHES
Problem: New Year’s Town/​Patriots’ Day Town

This puzzle is presented as a set of 30 video clips from television shows. The clips have been converted to grayscale. The solver can switch between clips by rotating a Gray code rotary encoder. This allows the solver to associate a 5-bit Gray code number with each video clip.

The video clips can be grouped into thematic sets of five by key words or references in their content. These same thematic sets of five are all also used by the Grayble episodes of the animated series Adventure Time. The title of this puzzle (“Mathematical!”) is a reference to Adventure Time, and some phrases in the flavortext (all sorts, past adventures) point to Adventure Time and Graybles (including the Graybles Allsorts series of shorts).

Clip Number Gray Code Show Thematic Content Theme Theme Element Adventure Time Episode
1 00001 Seinfeld George believes a driver is making an “obscene gesture” at him. Fingers Middle Five More Short Graybles
2 00011 The X-Files Mulder eats slice after slice of “sweet” potato pie. Tastes Sweet Five More Short Graybles
3 00010 South Park Coffee tastes bitter. Tastes Bitter Five More Short Graybles
4 00110 Futurama Zapp says Amy is “weakening our bargaining position.” Stages of Grief Bargaining Another Five More Short Graybles
5 00111 The Goldbergs “You smell like a gym sock’s butt” etc. Senses Smell Five Short Graybles
6 00101 The West Wing “What I’m saying is, it’s physiological.” Maslow’s Hierarchy of Needs Physiological Graybles 1000+
7 00100 Arrested Development “Taste my sad, Michael.” Senses Taste Five Short Graybles
8 01100 The Good Place “We love each other.” Maslow’s Hierarchy of Needs Love Graybles 1000+
9 01101 Family Matters “You knew the queen of England?” Royal Straight Queen Five Short Tables
10 01111 The Office “Keep your acceptance speeches short” Stages of Grief Acceptance Another Five More Short Graybles
11 01110 Cheers “Jack up the price” Royal Straight Jack Five Short Tables
12 01010 The League “I gotta touch pinkies with you?” Fingers Pinky Five More Short Graybles
13 01011 CSI “From ten years ago” Royal Straight Ten Five Short Tables
14 01001 It’s Always Sunny in Philadelphia “You got a lot of anger in you, kid.” Stages of Grief Anger Another Five More Short Graybles
15 01000 Bob’s Burgers “She’ll wear down your self-esteem over a period of years.” Maslow’s Hierarchy of Needs Self-Esteem Graybles 1000+
16 11000 Full House “Focus on the king” Royal Straight King Five Short Tables
17 11001 The Simpsons Mr. Burns has a novelty foam hand with an extended index finger. Fingers Index Five More Short Graybles
18 11011 The Golden Girls “Denying you’re in denial” Stages of Grief Denial Another Five More Short Graybles
19 11010 Fawlty Towers “Depression is a very bad thing” Stages of Grief Depression Another Five More Short Graybles
20 11110 Brooklyn Nine-Nine “I walk into the light of self-actualization.” Maslow’s Hierarchy of Needs Self-Actualization Graybles 1000+
21 11111 Star Trek: The Next Generation “You are personally responsible for the captain’s safety.” Maslow’s Hierarchy of Needs Safety Graybles 1000+
22 11101 Broad City “You guys sound so good” Senses Sound Five Short Graybles
23 11100 Community Annie can’t see the board from this far back. Senses Sight Five Short Graybles
24 10100 Friends Phoebe finds a thumb in a can of soda. Fingers Thumb Five More Short Graybles
25 10101 Twin Peaks “Your enthusiasm brings a salty tear to the eye.” Tastes Salty Five More Short Graybles
26 10111 Scrubs Trying to put an engagement ring on a ring finger. Fingers Ring Five More Short Graybles
27 10110 Rick and Morty “All my nerves, they don’t allow for the sensation of touch” Senses Touch Five Short Graybles
28 10010 ALF “Sour cream” Tastes Sour Five More Short Graybles
29 10011 Desperate Housewives “Savory chicken” Tastes Savory Five More Short Graybles
30 10001 How I Met Your Mother “We’re gonna ace this final” Royal Straight Ace Five Short Tables

We can re-order this data and use it to fill the Gray codes into the blanks at the bottom of this puzzle. The required order is first by Adventure Time episode air date, then by main theme followed by secret theme, then by the order the theme elements appeared in the Adventure Time episode. This ordering is clued in the blanks by slashes separating episodes, and light gray blanks instead of black blanks cluing secret theme rather than main theme.

1 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 1 1 0 1 / 0 0 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 0 1 1 0 0 1 0 0 0 0 1 1 0 1 1 1 0 1 0 1 0 / 0 1 1 1 1 1 1 0 1 1 0 0 1 1 0 0 1 0 0 1 1 1 0 1 0 / 0 0 1 0 1 1 1 1 1 1 0 1 1 0 0 0 1 0 0 0 1 1 1 1 0 / 1 0 0 0 1 1 1 0 0 0 0 1 0 1 1 0 1 1 1 0 0 1 1 0 1

The resulting boxed Gray codes can be converted into letters (using A=1, B=2, etc.) to spell the cluephrase WHAT BELONG IN THE CUPBOARD. This is a reference to a song that BMO sings in the Graybles Allsorts short All’s Well That Rats Swell entitled “Mustaches Belong In The Cupboard.” The answer to the puzzle is MUSTACHES.