This article contains tons of spoilers for the puzzle-filled book, 'The Master Theorem: A Book of Puzzles, Intrigue, and Wit'.

If you haven't heard of it - or you know of it, but haven't solved all the puzzles yet - I strongly encourage you to stop reading and start solving!

What follows is a summary of my journey, puzzle-solving techniques I discovered, and hints for the last puzzle that are not available outside of unorganized posts in the discussion forums.

Pages from my fully-annotated copy of the book

I needed a ton of hints

I worked through puzzles in a separate notebook, too

Recalling how I found the clues

The tutorial theorems taught me the basics

• Think A-Z when you see 1-26. And vice versa.
• Try to create groups when spotting identical or similar symbols...
• ...and start by putting them in order
• If it seems like gibberish, you probably need to use a cipher
• Numbers in the low single and double-digits may indicate a need to look within a word
• It's ok to Google quotes, ciphers, and other tools or knowledge that may seem required to solve the puzzle
• There is always a leading indicator in the puzzle's left-hand-side exposition

And from there, a slew of other puzzle-solving techniques

• Simple arithmetic, including fractions and prime numbers
• Puzzle title
• Repetition throughout the puzzle exposition
• Talking aloud - sounding out seemingly foreign elements of the puzzle
• Avoid anything that may seem like a distraction, especially for easy-seeming puzzles (those marked green or even yellow)
• A character that seems to have no mapping to a letter or number is likely a space character, or something that designates a separation between the part to the left and part to the right
• Look for repeated elements with portions that differ, then isolate those portions
• Look for similar amounts of one item and another. There's probably a hidden but important relationship there.
• Try Googling the phrase with the word cipher appended
• Direction is important, and it may not be the most obvious way
• Any time the author refers to themself in the first person - and continues to list a preference - a clue is likely to follow
• Numbers can be odd and even. But so can circumstances, and probability.
• The order of an item could indicate how far into it the correct letter is
• Keep an eye out for repeated digits or letters
• Metaphors and puns are often a major clue
• Binary numbers can manifest as off/on, up/down, really any occurrence of polarity, not just 0s and 1s
• If a font seems different, that's probably a big part of the puzzle
• There are no typos in the book. If it seems like a typo, it's almost certainly a clue.
• It may be a literal logic puzzle. Find a natural way in and continue following the path.
• Look for references to clocks and the position of the hands
• Remember that keyboards have numbers and letters, phones have numbers and letters, and other things also probably have numbers and letters
• If two items are closer to each other than to other items, they probably form a group
• I'll repeat: try not to let one aspect distract you (color or count, when the important thing is number of sides)
• Emptiness often represents 0
• Use your multiplication tables: 2 -> 4 -> 8 -> 16 -> 32 -> ...
• Try thinking in 3D, or bottom-up instead of top-down or 2D
• Explore any and all references to real-world things in the exposition
• Try not to get distracted by parts of the puzzle clearly not highlighted
• The numbers may reference page numbers, and elements on those pages

The elusive 42nd puzzle

• It's not in the book
• Just three hints that seem irrelevant at first
• So, what is one to do next?

Since the book offers cryptic hints and no solution...

...and because the discussion forum for this puzzle feels all over the place...

...I'm going to walk through the puzzle below.

For two other reasons, actually:

1. I want to write the guide I wish I had [found on the forums]
2. I absolutely loved this puzzle. I may be the best in the book...even though it's not in the book

SPOILERS AHEAD
I REPEAT: SPOILERS AHEAD
YOU READ THAT RIGHT: SPOILERS AHEAD
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SPOILERS START HERE

Finding the entrance

Hint #2 advices you to Prove you have solved all of the theorems.

As mentioned in the Effin' Web Services puzzle, there's a website for the book where puzzlers can log in and enter their proposed solution to see if it's correct.

Entering the correct solution for all 41 theorems triggers the reveal of the 42nd theorem.

There's no correct solution you could enter, though.

Every attempt seems to generate a Wrong response.

Except...each response is different.

Some responses begin to reference earlier puzzles.

Eventually, the response asks a question and offers a link!

The scroll - and hunt - begins!

A secret page on the website!

A few sentences placed at random spots down the page.

A text input field.

A lock icon.

A button.

Clicking it opens a new page.

Puzzle 1 of 9: Your Last Chance

A reference to The Matrix.

Red pill or blue pill?

Hovering over one casts a shadow on the page.

Hovering over the other does nothing.

Clicking either does nothing.

We want to take the red pill, though, right?

Back to the scrolling page.

Type red into the box.

Unlocked!

The page grows vertically. The proverbial rabbit hole grows deeper.

More text appears.

Scrolling down reveals another text input field, lock icon and button.

Puzzle 2 of 9: The Very First Thing

This looks like the first tutorial puzzle!

Recall that anything in red is usually the answer.

What should be circled?

Referencing the puzzle:
E-X-C-E-L-L-E-N-T

Back to the scrolling page.

Type excellent into the box.

Unlocked!

The rabbit hole grows deeper.

Another puzzle!

Puzzle 3 of 9: Common Experience

• A globe
• People encircling it
• They're holding flags
• There are speech bubbles on the globe
• Clicking them causes sound to play
• It is someone uttering that character in some alphabet
• The speech bubbles can be dragged
• They snap into place when dragged over a flag

So, identify which bubbled character belongs over which flag.

This may take a bit of Googling.

...

Boy, did it ever!

When properly assigned, the page auto-repeats playing each sound in a clockwise order starting with the one at the 12 o'clock position.

When combined, the word uttered is:
E-s-p-er-a-n-to

That is the name of a supposed international language.

Back to the scrolling page.

Type esperanto into the box.

Unlocked!

The rabbit hole grows deeper.

Another puzzle!

Puzzle 4 of 9: Any Amount of Emotion

Flying emojis!

Once there's a gap, you can see the static and transparent blank-face emoji in the middle.

There are five transparent dots below it.

Clicking an emoji stops it.

Then it slides quickly offscreen.

Clicking two or more emojis stops them.

A few of the dots light up.

Clicking five emojis stops them.

All the dots light up.

That emoji is no longer one of the moving emojis.

Perhaps we're meant to click on five of each emoji type to completely clear the screen!

After doing so, only the blank-face emoji remains.

Along with the answer, flat.

Back to the scrolling page.

Type flat into the box.

Unlocked!

The rabbit hole grows deeper.

Another puzzle!

Puzzle 5 of 9: Mindless Workers

• A blank screen
• Multi-colored buttons
• Well, three colors to be exact
• Arranged in a grid
• Clicking toggles each button
• Except the space bar
• Clicking it reveals a message
• One character at a time
• One 6-character line at a time
• All characters are a 0 or 1
• The characters flash on the screen at different speeds

Are we meant to compose a message in binary that is five characters long, where each character corresponds to a number 1-26?

Are we meant to arrange the toggles in a way that corresponds to their color?

No, it's the timing that's the key.

Toggle each button such that the characters appear on screen as fast as possible.

After a lot of testing, the binary message no longer shakes.

It fades away to reveal the answer:

tomorrow
  and tomorrow
    and tomorrow

Back to the scrolling page.

Type tomorrow and tomorrow and tomorrow into the box.

Unlocked!

The rabbit hole grows deeper.

Another puzzle!

Puzzle 6 of 9: What is all of this

A word game!

Determine the rubric by which a word would be considered valid or invalid.

First round:

• Double-letter words

As in tree, like the background image.

Second round:

• Words that start with a vowel

Third round:

• Each new word starts with the last letter from the previous word

Finishing the third round reveals the answer.

Back to the scrolling page.

Type absurdism into the box.

Unlocked!

The rabbit hole grows deeper.

Another puzzle!

Puzzle 7 of 9: Why is there something rather than nothing?

• A rectangular grid of letters
• Eight tetronimo buttons
• Pressing each one toggles the single tetronimo in the second row filled or bordered
• A Go button that attempts to color in the grid of letters
• And a link to a TED talk!

Watching the TED talk eventually reveals the correct state of all eight tetronimo toggles to generate the speaker's surprising pattern:
00011110

Then, reading the newly filled in letter squares from top-to-bottom, left-to-right in each row, reveals a repeating phrase that servers as the answer.

Back to the scrolling page.

Type emergent behavior into the box.

Unlocked!

The rabbit hole grows deeper.

Another puzzle!

Puzzle 8 of 9: In Lumine Tuo Videbimus Lumen

A black page with the message in latin.

A reference to light.

A handy shortcut to highlight all text on the page is CMD- or CTRL-A.

Doing this reveals a pasta-rific answer.

Back to the scrolling page.

Type ramen into the box.

Unlocked!

The rabbit hole grows deeper.

Another puzzle!

Puzzle 9 of 9: Fancy suits and shined shoes

A black page with a sound clip.

Pressing play starts a broadway musical song.

It seems to be about people being two-faced.

Near the end, a photo slides onto the page.

The singers chant Hypocrite!

The song ends.

Is that the answer? Hypocrite?

Nope.

After a lot of Googling:

• The song is called Facade
• That's not the answer
• It's from Jekyll & Hyde The Musical
• No combination of those is the answer

After more Googling:

• The photo is of Butler Library
• That's not the answer
• It's also called the Rare Books & Manuscripts Library at Columbia University
• No combination of those is the answer

After even more Googling:

• A user on a forum mentioned that the photo is interactive and there's a map on the other side

Dragging left on the photo flips it over horizontally.

This reveals a key difference between the sides:

• On one side are names written on the facade of the library
• On the other side are pairs of numbers separated by a slash

The numbers must relate to the names, but how?

Does this require indexing into each name?

The problem with that:

• Some names don't have the number of letters needed to meet the number dictated by one side of the pair

Maybe both sides are in reverse order?

Maybe only one of the numbers in each pair is meant to be the index?

My head hurts.

...

It seems like the numbers are centered horizontally over the names, but I should confirm.

Using my design tool, I made three rows:

• Names appearing left-to-right
• Numbers appearing left-to-right
• Names appearing flipped, right-to-left

A secret revealed!

The numbers are perfectly horizontally centered over the mirrored names read right to left.

But what else about the numbers am I meant to understand?

Can I make any groups?

The first numbers often repeat: 2, 3, 4, 9

The second numbers never repeat: 1, 2, 3, 4, 5, 6, 7, 8

Woah! Those must be the ordering!

And the first numbers are the index!

But from the end of the word, or the beginning?

Starting with right-to-left and with the first ordered word:

• P
• R
• E

This feels like I'm on to something!

Finishing all eight letters reveals the answer!

Back to the scrolling page.

Type pretense into the box.

Unlocked!

The rabbit hole grows deeper.

The solution, at last!

And it gets darker.

Until there's complete darkness.

And a video appears.

It's a clip from The Matrix!

The clip's name is the same as the third hint: Why do you persist?

Eventually Mr. Anderson asks that of Neo.

And Neo replies, "Because I choose to".

The screen fades to white.

And repeats Neo's reply.

I'm redirected back to the page full of 42 puzzles.

I type because I choose to in the solution box for 42.

Still invalid.

Sneakily, the the in the puzzle name is now clickable.

Clicking it changes the to your.

Now typing anything in the solution box is valid.

Because any reason is good enough to keep puzzling!

In case you couldn't tell by now, I highly recommend this book to anyone interested in stretching their brain

It's packed full of puzzling boss battles.

The search for answers will reveal new knowledge about the world, historic figures, message encryption techniques, and more!

You're likely to remember each puzzle. The more you struggle, the more likely that puzzle with stick with you long after.

Hopefully - like for me - you will solve some of the harder puzzles without needing any hints. What a rewarding feeling that is.

And once you're done with the puzzles in the book, you can try that doozy of a 42nd puzzle.

It serves as an incredible demonstration of how interactivity, motion and sound adds new depth to these already brain-twisting puzzles.

Onward, to Elite!

There's a second book in the series.

It is said to contain even harder puzzles.

I couldn't be more excited and intimidated to crack it open.

Meet you there, fellow puzzlers!