13 into 12
pigeonhole 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: "scale"
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.
- each case really does overflow its boxes ×1
- every item is in a box ×1
- more items than boxes forces a box with two ×1
- the fullest box holds ⌈items/holes⌉ ×1
- there are at least as many things as boxes ×1
- there is at least one box ×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.
How close a fraction can get
Drop eight points into seven boxes and two of them share. That one line, applied to the multiples of an irrational number, proves that every irrational has infinitely many astonishingly good rational approximations — and no construction is needed anywhere.
DiscreteMore things than boxes
If there are more objects than containers, some container holds two. That is the entire principle, it is impossible to disagree with, and it settles questions that look nothing like it.
DiscreteOne bottleneck and nothing else
A set of jobs can be filled by distinct people unless some group of jobs has too few candidates between them — and that single obstruction is the only one there is, which is what makes the theorem worth having.
DiscreteSix people at a party
Among any six people, three are mutual acquaintances or three are mutual strangers. Five is not enough, and the arrangement that saves five is a pentagon. Beyond that the numbers become unknowable.
DiscreteThree in a row on the number line
Colour the numbers one to eight in two colours and it can be arranged that no three equally spaced numbers agree. Add the ninth and it cannot. The structure being forced is arithmetic rather than graphical, and the proof is a different proof.
ProbabilityTwenty-three people
A room needs 253 people before someone probably shares a birthday with you. It needs 23 before two of them probably share one with each other. The gap between those numbers is the whole problem.