Mathematics

One plate, many entries

• 289.8420% of 69

  420 percent of 69 is 69 percent of 420. Both are 289.8. Percentages reverse because multiplication reverses. Seventeen percent of eighty-three does the same. A teaching trick. Not a special 69/420 law. Play, labeled play.

• 42₁₃6 × 9 in base 13

  6 × 9 = 54₁₀ = 4×13 + 2 = 42 in base 13. The arithmetic holds. Adams denied it as his origin. A frame, not a transit.

• 69⁶⁹ ≡ 6969ⁿ mod 420

  Raise 69 to an odd power. Divide by 420. The remainder is 69. Square first and it is 141. The cycle locks. So 69 to the 69th is 69 again, modulo 420. Exact. Not rare: 81 residues out of 420 satisfy x³ ≡ x (mod 420). 69 is one of them. The pretty reason is the consecutive triple 68 × 69 × 70, which already holds every prime-power in 420.

• 42 · 420Balanced rooted trees

  A tree starts at one point and branches. Here every leaf must arrive at the same depth. With ten nodes there are exactly forty-two such shapes, if you forget labels and forget order around a fork. With fifteen nodes there are four hundred twenty. One sequence. Two exact counts. Nobody wrote the rule to find them.

• 1/42Bernoulli B₆

  Jakob Bernoulli was adding up 1² + 2² + 3² + … and looking for a shortcut. The numbers that fall out of that work are named for him. The sixth one is exactly 1/42. Not cake. Not Catalan. A fraction analysis keeps handing back.

• C₆ = 42Cake number C₆

  Six flat cuts through a cake, as many pieces as possible: 42. Named, exact, one term of a sequence. Not Catalan.

• 42Fifth Catalan number

  Picture pairing parentheses so they always close. Or drawing paths that never go below the ground. The number of those ways, for a given size, is a Catalan number. The fifth one is 42. The one before it is 14. That pairing was noticed afterward.

• 420LCM(1…7)

  The smallest number that 1, 2, 3, 4, 5, 6 and 7 all divide is 420. It has exactly 24 positive divisors, and 42 is one of them. Seven factorial is 5,040, twelve times larger. Same family of sevens. Not Plato’s tribes.

• moa(6) = 42moa(6) = 42

  Count the different kinds of group that have a given size. Size 42 has exactly six. No smaller size has exactly six. One of the six is a circle of forty-two. The other five are not.

• Z = 42Molybdenum

  Molybdenum is element 42. The integer is the proton count. JPL later nicknamed it the Douglas Adams element. That nickname is a joke on the integer, not a second proton count.

• 42Order-3 magic cube

  A magic cube stacks numbers so every straight line through the block adds to the same total. The smallest one that works uses 1 through 27. Those lines add to 42. Order-3 perfect cubes do not exist. Semiperfect, or Andrews, cubes do: rows, columns, pillars, and the four space diagonals.

• 42p(10) = 42

  Catalan 42 asks how relations can be legally arranged: parentheses that close, paths that stay above the ground. Partition 42 asks how a whole can be split when order is forgotten. Ten written as a sum of positive integers, disregarding order, in forty-two ways. 6+4 is the same pile as 4+6.

• 42Primary pseudoperfect 42

  Half plus a third plus a seventh plus one forty-second equals one. That is 42 itself. The same 42 sits in a tiny closed list: 1, 2, 6, 42, 1806.

• 6×7 · 20×21Pronic · 6×7 and 20×21

  A pronic number is two consecutive integers multiplied. Six times seven is 42. Twenty times twenty-one is 420. They sit in the same sequence. Not 42 × 10 in a trenchcoat. The same theorem says 42 is the sum of the first six even numbers, and 420 the sum of the first twenty. The generating rule was there before anyone wanted a 42.

• dim 42Sphere packing in dimension 42

  A 2026 paper improves sphere packing in dimensions 39 and 43. It walks 38 through 43 because those are the old Conway–Sloane records. Dimension 42 sits in the middle. The construction did not choose 42. It has not even beaten 1982 there.

• 42Sum of three cubes

  The last unsolved case below 100. Booker and Sutherland found three cubes in 2019. The researchers noticed Adams. We did not invent that rhyme.

• 3, 7, 42Triangle · heptagon · 42-gon

  An equilateral triangle, a seven-sided tile, and a forty-two-sided tile can close around a point with no gap. Forty-two is the most sides any of the three can have. Turn the same fractions over and the leftover is 1/42. That leftover is why a curved surface can have at most 84(g − 1) symmetries. 84 is twice 42. The tiles did not hire Adams.