clock
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.
At its defaults
show: "table"
show: "orbit"
show: "crt"
What it checks while it draws
Collected by running the family and listening to lib/verify.js, not written
here. The count is how many separate times this build put that claim to the test.
- the walk reaches m × n / gcd cells on a 3 by 5 grid ×3
- a dial needs at least three positions ×1
- a row is a permutation exactly when its multiplier is coprime to the modulus ×1
- every non-zero row is a permutation exactly when the modulus is prime ×1
- every residue has a place on the dial ×1
- every row of the addition table is a permutation ×1
- it fills the whole grid exactly when the moduli are coprime ×1
- multiplying `order` times returns to 1 ×1
- the arithmetic and the walk agree ×1
- the drawn entry is the arithmetic ×1
- the drawn walk passes the top as often as the division says ×1
- the first modulus is between 2 and 12 ×1
- the multiplier is not zero ×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 second modulus is between 2 and 12 ×1
- the walk is between one and 144 steps ×1
- the walk runs at least one full period ×1
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
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.
NumberNecklaces 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.
DiscreteNumbers 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.
NumberTwo 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.