Cycle — where it appears
Named by 9 essays across 6 fields — each of them below, with the objects they name alongside it.
A ring that no pairing can break
Put everybody in one pool and a stable pairing may not exist. Allow rings as well as pairs and something stable always exists — and the pairs-only answer fails exactly when that stable arrangement contains a ring of odd length. Two sides make every ring even, which is the whole reason the two-sided theorem holds.
A walk that splices in its own detours
Euler proved that a walk crossing every bridge once needs every landmass to have an even number of bridges, and then stated, without proof, that this was enough. The missing half took 137 years, and it is not an argument but a procedure: walk until stuck, notice that stuck can only mean home, and splice in a detour from anywhere with edges left. The procedure never fails, and the reason fits in one sentence about arriving and leaving.
The cheapest way to send
Put a price on every road as well as a capacity and ask for the cheapest way to send four units. Twenty-eight ways exist and one is cheapest, and two certificates prove it without comparing it with the other twenty-seven: no cycle of roads it leaves unused costs less than nothing to push round, and there are prices at the places that every usable road fails to beat.
Every edge on exactly two cycles
Draw a graph on paper without crossings and its faces are cycles, each edge on exactly two of them. George Szekeres and Paul Seymour asked in the 1970s whether every graph without a bridge has such a set of cycles, drawn on paper or not. The Petersen graph needs five, the flower snark on twenty points five as well, every random graph tried has one, and nobody has proved it — though a minimal counterexample would have to be a snark.
Every function is a tree with two marks
There are n to the n functions from n points to themselves, and n to the n − 2 trees on those points. André Joyal noticed in 1981 that the missing factor of n² is a choice of two points — a head and a tail — and that a function, read the right way, simply is a tree with a head and a tail. The reading turns Cayley's formula into one line and hands over a fact about random trees from the birthday problem.
Two random shuffles reach every shuffle
Pick two ways of rearranging n letters at random and allow them to be repeated in any order. Almost always the two together reach every rearrangement there is, or every even one — and when they fail, the usual reason is a coincidence about a single letter that both happen to leave alone. Eugen Netto guessed it in 1882 and John Dixon proved it in 1969.
A perfect coin and a biased shuffle
Give a program a perfect source of random numbers and ask it to shuffle a deck by swapping each card with a card chosen from anywhere in the deck. The result is biased, provably and for every deck of three or more cards: there are nⁿ equally likely runs and n! orders, and the first number is never a multiple of the second. In a deck of 52 some card-and-place pairs come up a third more often than they should. Swapping only with cards not yet placed fixes it; swapping only with later cards makes nothing but single cycles.
The largest prime in a typical number
Count a number's prime factors and the answer is about log log n. Ask instead how big the largest one is, and the answer is a power of n with a law of its own: half of all large numbers have a prime factor above n^0.6065, and the share whose primes all stay below n^(1/u) is Dickman's ρ(u), a function defined by its own past. The same law governs the longest cycle of a random shuffle, and there it arrives at once; for the numbers themselves, at ten million, it is still visibly on its way.
A hundred prisoners and one shuffle
A hundred prisoners each search fifty of a hundred boxes for their own number, one at a time, with no communication once it starts. If any fails, all are executed. Choosing at random gives them a chance of 8 × 10⁻³¹. There is a rule that gives them 31%, and no rule can do better. Every prisoner's individual chance is still one half: what the rule changes is not how often each succeeds but how those successes line up.
Named alongside it
The objects these essays reach for when they reach for this one.
PermutationExhaustive searchParityExistence proofRandom permutationAlternating groupAsymptoticsBijectionBirthday problemBlocking pairBridgeCapacity