Generator

One cake, one halving cut at 4/9, and two measures of it

A generator in the applied library, called 24 times across 4 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

divide 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

One cake, one halving cut at 4/9, and two measures of it. A cake as a bar with two step valuations above and below it, the cutter's halving cut marked, and a table of both people's exact value of each piece.

`show: "items"`

Every allocation of 3 indivisible items, and not one of them envy-free. A value matrix for indivisible goods with the round-robin allocation shaded, the exhaustive counts of envy-free and EF1 allocations, and a control matrix on which envy-free allocations do exist.

`show: "cutchoose"`

`show: "trim"`

`show: "moving"`

A knife swept once, 4 proportional pieces, and 3 people who would rather have another. The successive cuts of a last-diminisher division along a cake, with each person's calling point marked and a matrix of every person's value of every piece.

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.

person 1 values their own share at least as highly as person 1's ×9
the value matrix, person 1's row is a list of 3 to 3 numbers ×7
person 1's valuation is a list of 6 to 6 numbers ×6
trimming piece 1 is worth exactly a third of the trimming to person 3 ×6
the control matrix, person 1's row is a list of 3 to 3 numbers ×5
each entry of person 1's valuation is a whole number between 0 and 100 ×4
no allocation of these goods is envy-free — all 8 were tested and each failed ×4
person 1 strictly prefers person 3's piece to their own ×4
person 1's own piece is worth at least 25 to person 1 ×4
person 1's valuation of the whole cake is 100 ×4
person 1's values of the 4 pieces account for the whole cake ×4
person 2's calling mark stays over the bar it belongs to ×4
each entry of the control matrix, person 1's row is a whole number between 0 and 100 ×3
each entry of the value matrix, person 1's row is a whole number between 0 and 100 ×3
person 1's values of the three shares account for the whole cake ×3
person 2 called the knife at exactly 25 ×3
piece 1 is worth exactly a third of the cake to person 1 ×3
the control matrix: person 1 values the whole set at 100 ×3
the valuations are given for 2 to 2 people ×3
the value matrix: person 1 values the whole set at 100 ×3
a knife is asked to pass a non-negative amount of value ×1
a rational is a whole numerator over a non-zero whole denominator ×1
a rational is never divided by zero ×1
every envy-free allocation is also envy-free up to one item ×1
every person still waiting gets a calling mark ×1
every way of giving each item to one of the people was formed ×1
person 2 does not envy person 1's piece before the trimming is divided ×1
person 4, who never called, is left with at least 25 ×1
the chooser envies nobody, having taken the piece it valued higher ×1
the chooser gains strictly more than half — the promise is not tight ×1
the chooser's piece is worth at least half the cake to the chooser ×1
the chooser's values of the two pieces sum to the whole cake ×1
the claim that somebody is left envious is either made or not made ×1
the claim that the chooser strictly gains is either made or not made ×1
the control matrix gives a row of values for two, three or four people ×1
the control was searched over every allocation too ×1
the control: a chooser with the cutter's own measure gains exactly nothing ×1
the control: the same search finds an envy-free allocation where one exists ×1
the cutter's envy is exactly zero: the piece left behind is worth what the other is ×1
the division being shown is one of cutchoose, trim, items, moving ×1
the division is proportional and not envy-free: somebody prefers another's piece ×1
the exact arithmetic stays inside the safe integer range ×1
the exact value of the left piece and the decimal one agree ×1
the number of items in the control matrix is a whole number between 2 and 6 ×1
the number of items in the value matrix is a whole number between 2 and 6 ×1
the number of segments the cake is cut into is a whole number between 2 and 12 ×1
the round-robin allocation is envy-free up to one item ×1
the three pieces go to three different people ×1
the trimmed piece ties person 2's second largest exactly ×1
the value asked for is still ahead of the knife ×1
the value matrix gives a row of values for two, three or four people ×1
two calling marks never print on top of one another ×1
under the cutter's own measure the two pieces are exactly equal ×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.

Applied

Envy-free, up to one item

A cake can be cut anywhere, and every guarantee in this anchor was bought with that freedom. Take the knife away and the exhaustive search over every allocation of three objects returns nothing envy-free at all — so the subject weakened the word until taking turns was enough to reach it.

Applied

One cuts and the other chooses

The oldest rule in fair division promises each of two people at least half the cake by their own measure, and it keeps that promise exactly. It does not promise what the word "fair" is usually asked to carry, and the gap opens the moment the two measures disagree across the cut.

Topology

One line that halves them both

Two shapes lying anywhere on a page, of any sizes and any shapes at all. There is always a single straight line that cuts both of them into two equal halves at once — and finding it needs no cleverness, only the observation that a quantity which reverses sign has to pass through zero.

Applied

Three people and a trimmed piece

For two people, one cut and one choice deliver a division nobody would swap out of. For three, the same promise costs a trimming, a residue and a choosing order contrived so that an advantage once given cannot be taken back — and the verdict is not three numbers but a whole three-by-three matrix.

The whole library · What the figures prove