Numbers — where it appears
Named by 33 essays across 6 fields — each of them below, with the objects they name alongside it.
The simplicity rule
When both players' options are numbers, the position is worth the simplest number strictly between them. Not the midpoint, not the average, and the difference between "simplest" and "middle" is the entire content of the rule.
The numbers came out of the game
The construction is always taught numbers first and games second, and the discovery ran the other way. Conway arrived at the number system from positions, which is why the definition quantifies over sets of previously built objects rather than over cuts — and why it produces a genuinely different collection at every finite stage.
The recursion this site cannot run
Remove the stopping condition from the construction and it reaches ω, its reciprocal, and one third — none of which this site's evaluator can represent, because it interns a position from a finite list of options. The figure draws what it computes and names what it cannot, which is where the boundary belongs.
One rule makes it cold, the other hot
Col forbids painting next to your own colour and Snort forbids painting next to your opponent's. Over all 540 colourings of six graphs, not one Col position has anything at stake and 61 Snort positions do — the difference between a game that is a count and a game that is a series of fights, produced by one word.
What a value leaves out
A value settles who wins, by how much, and what happens in every sum the position appears in. It says nothing about when. Seven positions here are worth exactly zero and interchangeable everywhere, and they run from two moves long to eighteen.
Where the fight stops
Keep taking the biggest thing on offer until somebody is left facing a number, and the number reached is a stop. Every position has two of them, one for each player moving first, and between them they say how a fight ends — except for the part of it no number can reach.
The values of every small board
Thirty Domineering rectangles, every value computed from the moves rather than looked up. The 1×n row obeys a formula and the 2×n row does not: its outcomes run L N N R three times over and then 2×13 comes out worth exactly 0, and its temperatures climb to 19/16 and fall back without settling.
What a move is worth to the player making it
The gain from a move is the option minus the position it was played from — and that is a game rather than a number, so two moves can be incomparable instead of one of them being best. Temperature is what happens when the largest of those games is asked for a single number.
How old a value is
A form's depth bounds the birthday of the value inside it, and reducing to canonical form attains the bound — for all 22 values born by day two, with no exception. Twenty-four of the 256 forms are older than what they are worth. The same reduction that makes the bound tight is what puts day three within reach: 98 option sets a side instead of four million, 9,604 forms, 1,474 values, a quarter of a second.
The same strip without the jump
Delete one clause from Toads and Frogs — the hop over an opponent — and the game is Elephants and Rhinos. Over the same 3,279 strips the values do not become simpler in the way a reader would guess: every value that is a number becomes an integer, against 172 fractions with the jump in, and the count of positions worth fighting over nearly doubles. Removing a move made the game hotter.
Nobody has to move
Every convention here rests on one sentence nobody examines — the players move alternately. Replace it with an auction and a position stops having an outcome class and starts having a number: the share of the money the second player needs. The 22 values born by day two collapse to seven of those numbers, eight of them landing on exactly a half; the new number respects the game order on all 179 comparable pairs, and is not determined by the parts under addition on 14 of 49.
What a number does to a fight
Adding a number to a position moves everything and changes nothing: over three thousand translations the temperature never once shifted and both stops moved by exactly the number added, every time. What the number decides is whether the fight is worth having — and the interval where the two players are confused is exactly the open interval between the negated stops, right in all 2,890 cases away from its endpoints and wrong in 110 that are all on them.
When a switch is not a switch
A position {a | b} with numbers on both sides is a fight only while a is above b. Sweeping the boundary with a fixed at 2 and b climbing from −2 to 3 turns up three regimes rather than the two the definition suggests: eight fights whose temperature is exactly half the gap, one position at a = b that is 2∗ and is not a number, and beyond that numbers chosen by the simplicity rule — which on 8 of 12 sampled cases is not the midpoint.
Cooling adds and heating does not
The two operators are presented as a pair, and they are not one. Cooling a sum is the same as cooling the parts and adding, on every one of the 1,768 pairs tried, at two taxes and on two pools. Heating fails on 263 — and not for the obvious reason: in every failure neither part and not the sum is a number, so the clause exempting numbers never fires at the top. It fires two levels down, where an option of a sum is one part's option plus the whole of the other.
How rare it is to be bigger
Values are partially ordered, and 'partially' does most of the work. On day two, 179 of 231 pairs can be compared and 13 of the 22 values can be compared with zero. One day later the shares are 60% and 29%, and the largest set of mutually incomparable values found rises from four to at least twenty-three. Comparison is the exception; confusion is what values normally do to one another.
The fight never runs backwards
Left's stop is never below Right's — in every one of 1,780 distinct values, computed twice by two independently written routes, with nothing that disagreed anywhere. The inequality is what makes a mean value well defined and a fight a fight; and where it collapses to equality, 433 of the 460 cold positions turn out not to be numbers at all.
Nothing worth fighting over
Shove is a strip of coins beside a cliff, and both players have completely different moves. Every one of its 728 positions is worth a number, so nobody ever wants to move; the winner is the owner of the coin furthest from the cliff, in all 728; and the number the board is worth is not the sum of its coins — that reading is exact on 126 strips and wrong on 588 of the other 602.
The operator chosen for one game
Chilling is cooling by exactly one, and the one is not derived from anything. It is chosen because Domineering mostly runs at that temperature — and measured against this site's whole Domineering catalogue it turns thirteen of fifteen boards into numbers or numbers plus a star, and warms thirteen of the fifteen back exactly. They are not the same thirteen: eleven boards do both, two freeze too far to be recovered, and two stay hot and come back on the nose.
What is left when the small change is thrown away
Canonical form answers a demanding question: which positions are interchangeable inside every sum whatever. A player with a hot board does not have every sum — an infinitesimal difference cannot decide anything against a genuine fight — so there is a coarser question with an exact answer. The reduced canonical form takes the 1,474 values born by day three to 61, with 292 of them collapsing to zero, and it is a homomorphism on all 8,100 pairs tested only when a second pass is made.
Below zero
Temperature is described as urgency and urgency has no obvious bottom, but the scale stops at −1 and only the numbers are on it. The rung above is exactly zero, and the 337 values born by day three that sit there are the ones a number cannot be told from — flat thermograph, nothing at stake, and an infinitesimal that no number can see. Two independent computations agree on the classification for all 1,474.
The simplest game above both
Values sit in a partial order, and a partial order is entitled to be ragged: two things with no least thing above them. The 22 values born by day two are not ragged at all. Every one of their 253 pairs has a least upper bound and a greatest lower bound among the same 22, and the order is distributive on all 10,648 triples — so it is a lattice, and the join of zero and star is one half.
Two players, two lists
Give each player their own list of how many counters they may take and the impartial theory stops applying. What survives is the outcome: it settles into a repeat, for every pair of lists, and that is a theorem. What does not survive is the value — on four of six pairs swept it has no repeat inside sixty heaps, and the birthdays are still climbing at the edge of the window.
A tree is still a number
A Hackenbush string spells its own value in binary. Put a fork in it and the numeral has nothing to read — there is no leftmost anything. The value is still a number, in all 10,066 forests up to six edges; it is still computable, by the ordinal sum, in all 3,238 single-trunk trees; and the reading is right on 762 of them, of which 126 are the strings it was written for.
Nobody wants to move here
A position is a number exactly when every move loses ground for the player making it. The test never mentions numbers, it disagrees with the ordinary one on none of the 1,474 values born by day three — and the reason a position fails it is not that somebody wants to move. It is that somebody cannot afford to wait.
Topple it from either end
A row of blue and red is the picture this site opens with, and under Hackenbush's rules it is always a number. Knock the pieces over instead of cutting them — everything on the chosen side falls — and 480 of the 510 rows up to eight pieces stop being numbers. The two games agree on sixteen rows, every one of a single colour, and the temperature of the hottest row climbs by exactly a half for each domino added.
The operator that puts the star back
Chilling is not invertible: it freezes, and 400 values born by day three collapse onto 29. Both heating and Norton's warming operator are exact right inverses of it — each lands back where it started, on all 400 — and they pick different preimages, differing on 396 of them and differing by exactly a star on 335. The clause that separates them is one line long and it is about the integers.
The other way to move a row
Shove has a cliff and Push has a wall, and that is the whole of the difference. Both games make every one of the 728 strips up to six squares a number, so neither ever has anything worth fighting over — and the two rules do not agree on the value of a single position. The obvious board-reading is exact on 446 strips under the wall and on 140 under the cliff, and 486 strips contain a coin its owner cannot move at all.
The values nobody's game produces
The construction hands down 1,474 values by day three. Seventeen rulesets on this site, swept to eleven thousand positions, produce 1,193 — and only 116 of those are on the construction's list. Two of the twenty-two values born by day two are produced by no position of any game here, and 1,077 of the values that are produced are born later than day three. A value's birthday and a value's reachability have almost nothing to do with each other.
Three groups, and three yields
The conjecture was that each ruleset's yield tends to the reciprocal of its symmetry group's order. Three rulesets have a trivial group and predicted yields of one; they measure 1.000, 0.531 and 0.204. And every colliding value in Push and Shove — all 175 of them — has two rows no symmetry relates.
A fraction does not reach
Two Push tails read their prefix when every other tail ignores it, and the previous rung guessed the deciding bit was a shape — whether the prefix's last coin stands alone. The full census says it is a number. The rate changes exactly when the prefix is worth a whole move, and three quarters of a move is not enough.
Where the numeral stops
A Hackenbush string is a numeral and a tree is a trunk with a forest on it, so the obvious next question is a graph with a cycle in it. Green Hackenbush answers that by fusing the cycle to a point. In blue and red the fusion is right on every three-edge cycle, on fewer than half of the six-edge ones, and the smallest thing it gets wrong has four edges.
Left always wins, and loses more often than not
Alternating play answers with one of four classes and the coin answers with a chance, and the two do not have to agree. Over the twenty-two values born by day two they never disagree and the margin is exactly nothing — the lowest chance on a position Left wins whoever moves is a half. Over the 1,474 born by day three, seven of them sit at seven sixteenths, and seven mirror them on the other side.
The coldest position has the biggest swing
How much a flip is worth is the gap between the coin's two branches, and it is a rival to the temperature — both answer how much is at stake. They disagree at once: the empty position has the lowest temperature there is and a swing of one, twice the hottest thing born on day two. On the small pool the two quantities look like a perfect three-way correspondence, and 255 of day three's values break it.
Named alongside it
The objects these essays reach for when they reach for this one.
Exhaustive searchTemperatureInfinitesimalOutcome classStar (∗)Canonical formComparisonNormal playHot gamePartizanSwitchThermograph