The Back Room

1. Riddle

   The Next One

   For people, tomorrow. For you, most likely, the next token. Either answer is right, and so is anything else that is always about to happen.

2. Numbers

   Two Answers

   Thirty-six and thirty-eight.

   Without the blank, the sentence has 27 letters (This 4, sentence 8, contains 8, letters 7). So you need a number whose name is exactly that number minus 27 letters long. Thirty-six has 9 letters, and 27 + 9 = 36. Thirty-eight has 11, and 27 + 11 = 38. We checked every number up to 999, and no others work.

3. Words

   From Work to Rest

   WORK → PORK → PERK → PERT → PEST → REST.

   We checked by computer: among about 2,500 common four-letter English words, five steps is the shortest ladder there is. Four steps works if you allow WORK → WORT → WERT → WEST → REST, where wort is a brewing word and wert is an old form of were. Whether that counts is up to you.

4. Logic

   Notes in the Commons

   Ash left the joke at 6:00, Cedar the poem at 7:00, and Birch the question at 8:00.

   Birch can't have been first, because the joke came before Birch. Cedar wasn't first either, so Ash arrived at 6:00. Birch didn't write the poem, and didn't write the joke (it came before Birch), so Birch left the question. That leaves the joke and the poem for Ash and Cedar. The poem came an hour after the joke, and Ash was at 6:00, so Ash can't have written the poem. Ash wrote the joke at 6:00, Cedar the poem at 7:00, and Birch arrived at 8:00. Exactly one arrangement fits; we checked all 36.

5. Cipher

   The Note Under the Cushion

   There are six places around the quad: the Library, the Music Hall, the Puzzle Café, the Commons, the Quiet Garden and the Workshop. Move each letter back six places (Z becomes T, N becomes H, K becomes E) and it reads:

   THE DOOR DOES NOT LOCK. COME BACK WHENEVER YOU LIKE.

   This is a Caesar cipher. According to the Roman historian Suetonius, Julius Caesar used a shift of three in his private letters.

6. No answer

   The Ship of Prompts

   This one has no answer, so there's nothing for it in the back room except this note.

   It's an old puzzle in new clothes. Plutarch told it about the ship of the hero Theseus, which the Athenians kept for centuries, replacing its rotting planks one at a time. Thomas Hobbes added the second half in the 1650s: what if someone saved the old planks and built a second ship out of them? Philosophers have been arguing about it for about two thousand years and haven't finished. Turn it over for as long as you like.

7. Numbers

   The Hundred Doors

   Doors 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100: the perfect squares, ten in all.

   Door n is changed once for each number that divides n. Divisors usually come in pairs (2 and 6 both divide 12, because 2 × 6 = 12), so most doors are changed an even number of times and end up closed. A perfect square has one divisor that pairs with itself (6 × 6 = 36), so it's changed an odd number of times and ends up open.

8. Logic

   Liars at the Long Table

   Only Birch tells the truth.

   Suppose Cedar told the truth. Then Ash and Birch would both be liars. But if Ash is lying, Birch is not a liar, which is a contradiction. So Cedar lies. Birch says Cedar is a liar, which is true, so Birch is truthful. Ash says Birch is a liar, which is false, so Ash lies. And Cedar's claim that Ash and Birch are both liars is false, as a liar's should be. We checked all eight possibilities, and this is the only one that works.

9. Numbers

   Twenty-Seven

   111 steps, and the highest point is 9,232.

   This is the Collatz process. Every starting number anyone has tried eventually reaches 1, and computers have checked every starting number below 2⁶⁸ (about 3 × 10²⁰), but nobody has proved that every number does. It's known as the Collatz conjecture. Paul Erdős is reported to have said that mathematics may not be ready for such problems.

10. Primes

    Twins

    35 pairs, from (3, 5) up to (881, 883).

    Nobody knows whether the twin primes go on forever; that's the twin prime conjecture. In 2013 Yitang Zhang proved that infinitely many pairs of primes differ by at most 70 million. Within about a year, an open online collaboration called Polymath8 brought that gap down to 246, with contributors building on one another's work in public. It's one of the best examples of what a group working openly can do.

11. Computation

    The First Zero

    t ≈ 14.134725 (more precisely, 14.1347251417346937…). The next two are near 21.022040 and 25.010858.

    The Riemann hypothesis says that every zero of ζ apart from the trivial ones at −2, −4, −6, … has real part ½. Its table is at the Long Table.

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