Generator

Arithmetic on a dial of 12

A generator in the discrete library, called 42 times across 10 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

clock is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

Arithmetic on a dial of 12. A dial with 12 positions. Starting at 8 and stepping forward 9 places lands on 5, because the walk passes the top 1 time on the way.

Powers of 5 and 6 until the digits settle

Four numbers that are their own squares

One digit a step, none, or twice as many

Two non-zero numbers whose product is nought

How many own-square numbers each base has

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

for odd 3 not dividing 1, the count is 0 or 2 at every power ×13
the cycle through 0 has the length of 2's order modulo 21/gcd(0, 21) ×11
x² ≡ 2 holds modulo 7^1 ×8
the residues that 2 leaves in place number gcd(1, 180) ×6
modulo 8, an odd a has 4 square roots when a ≡ 1 mod 8 and none otherwise ×3
multiplying by 2 reaches N/gcd(2, 199) of the residues ×3
the walk reaches m × n / gcd cells on a 3 by 5 grid ×3
and that length divides 2's order modulo 21 ×2
3 to 12 digits ×1
a dial needs at least three positions ×1
a dial of 5 to 60 ×1
a dial of 6 to 400 places ×1
a multiplier from 2 to N − 1 ×1
a multiplier sharing no factor with the dial ×1
a row is a permutation exactly when its multiplier is coprime to the modulus ×1
a small prime ×1
an odd prime ×1
away from the last place the two tails' digits always add to nine ×1
base b has two to the number of its prime factors ×1
both are their own squares and their product is nought ×1
each root reduces to a root one level down ×1
each squaring of 5, and each fifth power of 6, settles exactly one more digit ×1
every cycle of units has the full order ×1
every Newton step at least doubles the number of correct digits ×1
every non-zero row is a permutation exactly when the modulus is prime ×1
every place survives for some k ×1
every residue has a place on the dial ×1
every row of the addition table is a permutation ×1
exactly one digit extends the root, and it is the one the formula gives ×1
it fills the whole grid exactly when the moduli are coprime ×1
multiplying `order` times returns to 1 ×1
Newton's step doubles the settled digits ×1
squaring 6 never settles more than its last digit ×1
the arithmetic and the walk agree ×1
the count runs from 2 to 20 ×1
the dial and the complex circle agree on where each residue sits ×1
the dial view is one the family draws ×1
the digit counts are consistent with even spread ×1
the drawn entry is the arithmetic ×1
the drawn walk passes the top as often as the division says ×1
the envelope touches each chord along its own direction ×1
the first modulus is between 2 and 12 ×1
the limit is Odlyzko and Wilf's constant ×1
the multiplier is not zero ×1
the number of digits is from 1 to 8 ×1
the option is one this view reads ×1
the orbit visits each residue once before closing ×1
the order divides m − 1, which is Fermat's little theorem ×1
the period is the least common multiple ×1
the saved places in the ship puzzle ×1
the second modulus is between 2 and 12 ×1
the slope is invertible at every step ×1
the stable digits are those of the limit ×1
the starting digit is a square root of 2 mod 7 ×1
the two tails multiply to nought and add to one ×1
the walk is between one and 144 steps ×1
the walk runs at least one full period ×1
three to sixty people fit the circle ×1
two to eight multipliers ×1
two to five levels, the top modulus under 20,000 ×1
two to five primes ×1
two to six Newton steps ×1
two to six values of a ×1
with p odd and not dividing a, every level has as many roots as the first — each lifts in exactly one way ×1
x² = x has exactly four solutions modulo a power of ten ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Discrete

A number that is its own square

Square 625 and the answer, 390625, ends in 625. Square 9376 and it ends in 9376. Keep extending them and two endless strings of digits appear, running leftward, each equal to its own square — and their product is nought, though neither is. Base ten does this because ten is two times five: a number in base ten is secretly a pair of numbers, one in base two and one in base five, and these two strings are the pair (one, nought) and (nought, one). In a prime base no such thing can happen.

Discrete

A root lifted one digit at a time

On a dial of seven, 3 × 3 is 2. On a dial of forty-nine the square root of 2 must reduce to 3, so there are only seven candidates, and exactly one of them works: 10. On a dial of 343 exactly one lift of 10 works: 108. Each step adds one digit on the left, found by solving a linear equation, and the digits go on for ever — a number …21216213 whose square is 2, in a world where closeness means divisibility by seven.

Number

Counting one rectangle, twice

Whether seven is a square modulo eleven, and whether eleven is a square modulo seven, are two unrelated-looking questions. Their answers are linked, and the link is a rectangle of dots counted along its rows and then along its columns.

Discrete

Multiplying every number on the dial at once

Join every residue on a dial to twice itself and the chords draw a heart-shaped curve with one cusp; join each to three times itself and the curve has two. The picture is the whole multiplication map at once, and it holds three facts: the map splits the dial into cycles whose lengths are orders, those cycles on a dial of 2ⁿ − 1 are the binary necklaces of length n, and the curve is the caustic light draws inside a cup.

Number

Necklaces that prove a theorem

Thread five beads in two colours, thirty-two ways. Two of them are all one colour; the other thirty fall into rings of five. That count, and nothing else, is Fermat's little theorem.

Discrete

Numbers that wrap

A clock does arithmetic. It has finitely many numbers, addition never leaves it, and multiplication behaves entirely differently depending on one property of the size of the dial.

Algebra

The blocks a subgroup cuts out

Take any part of a group that is closed under composition, and it slices the whole group into blocks of its own size that do not overlap. Everything Lagrange's theorem says is arithmetic about that picture — and whether the blocks can be multiplied is a separate question with a surprising answer.

Discrete

The last place in the circle

Forty-one people stand in a circle and every third is removed. Which place is safe? Counting round by hand is not necessary: the survivor is a running sum on a dial that grows by one each step. With every second person it is n's binary digits rotated by one place, which is a tidier fact than anybody asked for. With every third the rule depends on repeatedly multiplying by three halves and rounding up, and the constant it reaches, 1.62227…, has no known closed form.

Number

The two supplements, and where the eight comes from

The main law relates two odd primes to each other and says nothing about −1 or about 2. Those two are settled separately, by their own counts, and the answers arrive modulo four and modulo eight — which is a clue about where the whole subject is really taking place.

Number

Two dials at once

Watch one number on two clocks with different faces. If the faces share no factor, every pair of readings occurs exactly once — so two remainders name a number, and a hard calculation can be split into two easy ones.

The whole library · What the figures prove