Field

Particular games

Hackenbush, Nim, Domineering, Toads and Frogs — the specific games the general theory was built to explain.

The picture is the numeral. Blue-red Hackenbush strings and their values. Left may cut a blue edge, Right a red one, and everything above the cut falls. The value of each string is a number, and reading the string from the ground upward gives the binary expansion of exactly that number.

Hackenbush is a numeral

Draw a stalk of coloured edges. Read it as a string, blue for one and red for zero, and the string is the binary expansion of what the position is worth. Not approximately — exactly, and the site computes it both ways and refuses to build if they disagree.

Domineering

One player places dominoes vertically, the other horizontally, on a shared grid. The rules take one line, the values are a mess, and that mess is the point — this is what the theory looks like applied to a game nobody designed for it.

Toads and Frogs

Toads shuffle right, frogs shuffle left, and either may jump over one of the other. A strip six cells long is worth exactly up. Another six-cell strip is worth exactly down. Nobody has a formula for which.

One of these is a game. Every length that came out of 40 random games from each starting position. Brussels Sprouts always ends after exactly five crosses less two moves, so whoever is to move at that point was decided before the first curve was drawn. Sprouts ends at different lengths depending on how it is played, which is what makes it worth playing.

Sprouts, and the game that is not one

Two games played with dots and curves, invented in the same room, all but indistinguishable on paper. One is unsolved past forty spots. The other has no decisions in it at all — the winner is fixed before the first curve is drawn.

Col and Snort on a path of four. One graph, two games, and one word of difference between the rules. Col forbids painting next to your own colour, which makes every move a small self-harm and drives the values towards numbers. Snort forbids painting next to your opponent's, which makes every move a land grab and drives them towards fights. Both values are computed from the same recursion.

One board, two rules

Col forbids painting next to your own colour. Snort forbids painting next to your opponent's. One word differs, the boards are identical, and the values that come out are not the same kind of object.

Cutcake: every value an integer. The value of an m by n cake, for every small m and n. Left cuts down, Right cuts across, and neither player ever gains by moving — so nothing is ever at stake, every value is a whole number, and the number says exactly how many spare moves one player has.

Cutcake, where every value is a whole number

A partizan game in which no position is ever worth a fraction, a star or a fight. Every value is an integer, the integer is a count of spare moves, and the pattern it follows is decided by binary digits.

Amazons, and when a position becomes a sum

Every technique on this site starts from a position already broken into independent parts. Amazons does not begin that way — the board is one fight until the arrows cut it, and the moment of cutting is something the play produces rather than the analyst assumes.

a triangle on a stalk, worth ∗2. A Hackenbush position in which every edge is green, so either player may cut any of them and the position is impartial. Its value is a single Nim heap. Two principles find which one: fusion, which collapses every cycle to a point and leaves that many loops behind, and the colon principle, which replaces a branch by a stalk as long as the branch's own value.

Squash every loop to a point

Colour every Hackenbush edge green and the game becomes impartial, so the whole picture is worth a single Nim heap. Two principles find which one without playing anything — fuse the cycles, then run one pass up the tree — and a nine-vertex lattice that costs 1,283 positions to solve costs twelve steps to read.

Clobber: every value smaller than every number. Blue and red stones on a small board. A move takes one of your own stones onto an orthogonally adjacent enemy stone, which is removed. Because adjacency is symmetric, a player has a move exactly when the opponent does — so no position can ever be worth a whole move to anybody, and every value that comes out is an infinitesimal.

A game where nobody can be ahead in moves

A blue stone beside a red one is a move for both players at once. So neither player can run out while the other still has something to do — and every value the game produces is smaller than every positive number, by the shape of the rule rather than by inspection.

A green edge is not a number. Green edges may be cut by either player, which makes the position impartial in that part. A single green edge is worth ∗ — a value that is neither positive, negative nor zero, and which no number can equal.

A green edge on a blue one

Blue over green and green over blue are the same two edges in the other order. One is worth 1∗ and the other ↑∗ — a number with a star on it against something smaller than every positive number — so a stalk with all three colours in it stops being a numeral and starts being a position whose value depends on what is underneath.

The strip nobody has a formula for

Some toads, a gap, some frogs. Two counts and a spacing is the whole description, and the values that come out of it are integers, stars, switches with eighth-point options and a down — four classes inside one two-parameter family, which is why nobody has written the formula.

Every position of both games, and where the fights are. Every colouring of every graph in the library, under both rules, with the temperature of each computed. A position with a positive temperature is one both players want to move in. Col has none anywhere; Snort has them on every graph, and the hottest grows with the graph.

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.

Maundy Cake

Cutcake with one word added: a cut must divide the piece into equal parts. The values are still whole numbers, and the rule this site has been repeating about them is false — over all 1,296 cakes to 36×36 the largest-odd-divisor account has 946 counterexamples. What survives is a count of prime factors, and it says who wins without saying by how much.

Small Domineering boards and what they are worth. Every value here was computed from the moves rather than looked up. Even on boards this small the values are switches and infinitesimals rather than numbers, which is the ordinary situation for a partizan game and the reason the theory needs more than arithmetic.

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.

Amazons on one line

A board one square high is small enough to evaluate completely: every strip from two to ten squares with one amazon a side is 37,886 positions taking 81 distinct values, and every one of them is an integer, a switch, a number plus a star, or a bare star. Not one is a fraction — and forcing the arrow onto the square just vacated, which takes a freedom away rather than adding one, produces 1,196 that are.

The same strips, with no jumping. Elephants and Rhinos: toads move right and frogs move left, one square into an empty one, and nothing may hop over anything. The pieces keep their order for ever, and the values are computed by the same recursion as the game with the jump in it.

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.

End-Nim: a player at each end of the row. Rows of heaps in which Left may take from the leftmost heap and Right from the rightmost. The value beside each row was computed by the game recursion and reduced to canonical form; the outcome beside it says who wins. A single heap is a Nim heap, because both players may take from it — and that is the last thing about this game that looks like Nim.

Taking from the ends

End-Nim is Nim's board with a player at each end, and it takes one sentence to state. Not one of its 5,460 small positions is worth a non-zero number — the game is all-small, so zero is the only number any of them can reach — and there are 2,693 distinct values between them. The outcome says a great deal more: 4,738 of those positions are won by the same player whoever moves, and on two heaps the rule is that the larger end wins.

One row of Clobber

Every string of blue, red and empty squares up to eight long — 9,840 rows — carries one of only 111 values, and every one of them is infinitesimal. A third of the rows are worth exactly zero. Six alternating stones are worth zero and eight are worth a form that takes four lines to print, so the values do not simplify as the row grows: they explode, while the row stays trivial to describe.

Shove strips, and what each is worth. A shelf of positions with the value the recursion returns beside each. Every one is a number: Shove has no hot positions at all, which is unusual for a partizan game and is the first of the essay's three claims.

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.

What each heap is worth. The value of a single heap of each size. Nothing here repeats: the forms grow deeper as the heap grows, which is what stops the impartial theory's periodic table from having an analogue.

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.

Trees, and what each is worth. A row of blue-red Hackenbush trees with the value the recursion returns under each. Every one is a number, and none of them is the binary reading of anything a reader can see in the picture.

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.

The same row, cut and toppled. Rows of blue and red drawn once and evaluated twice: as a Hackenbush string, where a player cuts an edge of their own colour and everything above it falls, and as Toppling Dominoes, where a player knocks one over and everything on the chosen side falls. Both values are computed by the same recursion from the two rulesets.

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.

Push and Shove over every strip up to 6 squares. The same strips under both rules. A cliff lets coins fall off and a wall does not, and the census says what that one clause is worth: both games are entirely made of numbers, they never agree on a value, and the obvious board-reading is right far more often under the wall than under the cliff.

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.

Every group must keep breathing

NoGo is Go with no captures at all: a stone may be placed only if, afterwards, every group on the board still has a liberty. That makes a move's legality a fact about the whole board rather than about the squares it occupies — and a board therefore almost never breaks into independent parts. Of 117 boards here whose empty points fall into two regions, 24 are the sum of their regions and 93 are not.

Every Domineering shape up to four squares. The pieces a partly played board falls into, each with the value the recursion gives it. Left plays vertically and Right horizontally, so a tall shape is worth something positive and a wide one something negative, and the quarter turn is not a symmetry of the game.

A board that is a sum of its regions

A table of rectangles is a table about the openings. A partly played Domineering board is not a rectangle, and evaluating one means splitting it into pieces no domino can straddle, looking each piece up and adding. The catalogue of 104 shapes does it correctly on every one of the 3,227 positions of a 3×4 board it covers — and among the shapes are two worth an up and a down, which no rectangle ever is.

$Every strip, without the hop. Toads and Frogs with the jump deleted, over every strip up to eight squares. The fourth column is the argument: whenever the value is a number it is a whole number, without exception, so the fractions the ordinary game produces are made by the hop and by nothing else.$

The strip where every number is a whole one

Delete the hop from Toads and Frogs and the halves, quarters and ups vanish completely: over 9,801 strips, every value that is a number is an integer, without a single exception. The guess that the hopless game therefore has a formula reading the gaps is half right and exactly wrong — 1,460 strips of eight squares are switches, and three strips with the same counts of toads, frogs and empty squares are worth 1, {2 | 1} and 2.

What an approximation is worth. Every pair of Clobber rows up to six squares, judged twice: by their up-brackets and by the comparison itself. The bracket is never wrong where it speaks, and most of what it declines to answer has no answer.

When the bracket decides

A Clobber row's value is an all-small game nobody can hold in their head, so the practical answer is the up-bracket: a pair of integers between which its atomic weight must lie. As an approximation it is worth exactly what it settles — 1,585 of 7,875 pairs of rows are ordered by it, every one of those orders is right, and of the 6,290 it declines, 4,222 have no answer either.

What the third colour reaches. Every row of Toppling Dominoes up to 7 long, over two colours and over three, with the number of distinct values each set of rows carries. Each value was computed by the recursion; the last column is the count of values three colours reach that two do not, cumulatively.

How long a row a value needs

Add a third colour that either player may topple and a row of seven dominoes reaches 1,047 distinct values where two colours reach 149. That makes the length of the shortest row worth a value into a measure of the value's complexity — one a reader can hold in their hand — and it is not the birthday: 1↑ is born on day three and needs seven dominoes.

When the regions add

The rung below described the NoGo boards whose regions add as the ones with symmetric walls, and said the description was a guess made from six examples. It is wrong: fourteen symmetric boards do not add and sixteen that add are not symmetric. What replaces it is a criterion about liberties — sound on all 117 boards, provable in a line, and complete on only nine of the twenty-four.

A sequence with a rule and no period

The values of the subtraction game where Left takes one or two and Right takes one or three never repeat — thirty-one heaps, thirty-one different values. They are nonetheless completely described: three seeds and the rule v(k + 3) = {0 | v(k)} generate every one of them, which is what a pattern without a period looks like.

What a Domineering region is worth, by size. Every connected shape of at most six free squares, sorted by whether its value is a number, an infinitesimal distance from a number, or hot. Hot shapes do not appear at all until four squares, and the hottest shape of six is the two-by-three rectangle.

Which shapes are worth fighting over

Forty-four of the 104 Domineering regions of at most six squares are worth numbers and the rest are not, and the rung below said no visible property of a shape predicts which. Half of that is wrong: a region only one orientation fits in is a whole number, on all eleven of them, for a reason a reader can supply in a sentence. The other half stands, and thirty-three shapes are what makes it stand.

When counting the free squares gets Push right. Every Push strip of at most seven squares, split by whether any line of play can bring two coins of opposite colour together. Where none can, the count of free squares in front of each coin is the value, without exception; where one can, the count is right more often than not.

The reading that survives too much

Counting the empty squares in front of each coin gets a Push position right half the time, and the rung below said the failures were exactly the positions with two coins of opposite colour side by side. Sixty-six of the 1,072 failures have no such pair, the smallest is five squares long, and the condition that does decide it is not about the board at all — it is about every position the board can reach.

Where the nimbers run out

A single End-Nim heap is a Nim heap and every palindromic row is worth a nimber, so the impartial theory looks as though it might get a long way into a partizan game. It gets one row in thirteen. Five nimbers occur in five and a half thousand rows, the palindromes account for two fifths of them, and the rows worth something else run to 2,693 distinct values.

How much a domino count knows about a region. The difference between the two players' largest domino packings, set against the value of the region. On the regions worth numbers the count is the value under half the time; on the rest it lands somewhere between the two stops on 85 per cent of them.

Counting the moves each side has

How many dominoes could each player still place? Subtract, and there is a whole number computable from the drawing with no game theory in it. Over 1,042 regions it is the value on 141 of the 315 worth numbers, lands between the stops on 619 of the other 727, and its failures are two different kinds — one of which was inevitable and one of which is a fact about the game.

Three counts, and two of them are the same number. Rows of End-Nim that read the same backwards, rows equal to their own negative, and rows worth a nimber. The second and third counts are identical over every row in the sweep, so being its own negative is exactly the condition for being worth a nimber in this game.

The rows that are their own mirror

Four hundred and ten End-Nim rows are worth nimbers and 168 of them are palindromes, so a condition covering the other 242 was outstanding. It is that the row is equal to its own negative — and on this game that condition is not merely sufficient but exact, which is more than the group law promises and is a fact about End-Nim rather than about games.

Thickness is not the variable; the wall's own groups are. The same strips split by whether the wall is all one colour. A wall of one colour is a single group with liberties on both sides and it never separates them, at any thickness. A wall of two colours is two groups breathing in opposite directions and it nearly always does.

How thick a wall has to be

A single stone between two empty stretches of a NoGo board couples them, and the obvious repair is a thicker wall. Over 590 walled strips a thicker wall does help — and splitting the same 590 by the colour of the stones shows that thickness was never the variable. A wall of four one colour couples the sides exactly as one stone does.

What the correction buys. The packing count as a point against the packing count as an interval, on the regions worth numbers. The point is right on 141 of 315; the interval contains the value on 209, at a mean width of about half a move.

The moves a player can be talked out of

The difference of the two players' largest domino packings is the value of a Domineering region on 141 of the 315 worth numbers. The count is optimistic for its owner and pessimistic for the other, and one number cannot be both — so it becomes an interval, from what a player can be reduced to against what the opponent can achieve. The interval contains the value on 209, is a single point on 505 of the 1,042 regions, and never exceeds two moves wide.

Two thousand values and no temperature. Every End-Nim row of up to six heaps, with its temperature. Not one of the 5,460 is hot, every one of them is worth an infinitesimal, and the 361 worth numbers are 361 rows worth nought.

A game with nothing at stake

The rung below explained a coincidence with a claim it did not compute: that End-Nim carries no hot self-negative values. The census says something stronger. Not one of 10,919 rows across three shapes of board is hot, every one of them is worth an infinitesimal, and the 361 rows the earlier census called numbers are 361 rows worth nought. The reason is one line of the rule, and 2,693 distinct values sit underneath it.

Two conditions, one of which survives. Two candidate conditions on a pair of subtraction lists, scored over all 961 pairs drawn from one to five. Translation holds on 83 pairs and every one of them repeats; all-odd holds on 49 and four of them do not.

The condition that survived the wider sweep

Which pairs of subtraction lists have a value sequence that repeats? Over the 49 pairs drawn from one, two and three, two conditions answer it identically — a translation and all-odd — and both are exactly right. Over the 961 pairs drawn from one to five, all 83 translations still repeat with no exception and four all-odd pairs do not, at heap ninety with a period as long as forty-two. Neither condition is necessary: 104 pairs repeat that satisfy neither.

Three strips a criterion cannot tell apart. Three Push strips identical in length, reading, coin counts and run structure, whose readings are wrong by a quarter, a half and a quarter more than one. The order of the colours inside the run is the only thing separating them.

The criterion that cannot exist

The rung below asked for a quantitative version of its condition — turn 'the reading survives mixing three quarters of the time' into a statement about the strip. Three strips of four squares settle it. `.LLR`, `.LRL` and `.RLL` have the same length, the same reading, the same coins and the same single run, and their readings are wrong by 1¼, ¼ and ½. The error is a fact about the order of the colours, and 207 of 805 statistical classes carry more than one of them.

What survives being added. The packing count and the packing interval on boards of one to four regions. The count is exact on 45 per cent of single regions and 11 per cent of four-region boards; the interval contains the value 67 per cent and 74 per cent of the time.

Two errors that cancel

Replacing the packing count with an interval left a doubt that the pessimistic half would add across a board. It adds, for a one-line reason. What is worth measuring is what the reading is then worth: over boards of one to four regions the count decays from exact on 45 per cent to exact on 11, and the interval's containment does not decay at all — it rises from 67 per cent to 74, because the interval's width adds and its error does not.

Seven, ten, thirteen and sixteen. The four values the rung below named, each read off the prime factorisation: one plus the largest prime plus the product of the two largest.

The size of a cake

Ω gives the sign of a Maundy Cake and says nothing about the size, and the rung below left four values — 7, 10, 13 and 16 — unaccounted for. For a one-row cake they are a formula: write the prime factors largest first and add up their running products. The rule behind it is greedy — cut by the largest prime — and it is exact on every one-row cake to two hundred and wrong on a fifth of the two-sided ones.

$A numeral in the empty squares. Runs of coins with one to five empty squares in front, and the value of each. Every row is a binary expansion converging on a fraction the colours determine.$

A numeral in the empty squares

The rung below ruled out a quantitative criterion for Push and asked for a numeral over the coins combined with a count over the gaps. The two ingredients are the right way round: the colours pick a fraction — −1, −1/3, −1/7, −1/15 — and the empty squares give the binary precision, so a run of k coins before one of the other colour with g gaps is worth exactly (1 − 2^(−kg)) ÷ (2^k − 1). And it does not compose: a strip of two runs is not the sum of them, on any pair tried.

Still a count of squares. One-sided Amazons regions on two-dimensional boards, swept exhaustively. Every one is worth exactly the number of free squares in it.

A region one player owns

On a strip, a region containing only one player's amazons is worth exactly its free-square count, on all 45,057 positions of the rung below's sweep — and it predicted the exactness would fail in two dimensions, where an amazon can be short of room in one direction and not another. It does not fail. Over 2,412 two-dimensional regions there is no exception, and the reason is one clause: an amazon may shoot back at the square it has just left.

The count is a count of odd runs. The largest packing of dominoes a player can hope for in a region, written as a formula in the region's own lines. Every run of odd length wastes one cell, so the packing is half of what is left, and the count is half the difference between the two directions' odd runs.

Half the difference in odd runs

The rung below asked what the regions the packing reading fails on have in common, and whether it is something a player could see. It is: the reading itself. The count has a closed form — half the difference between the region's odd horizontal runs and its odd vertical runs — and it is exact seven times in ten when it claims one move of advantage, on none of the largest regions where it claims two, and it exaggerates four times in five when it is wrong at all.

Moving them apart does not make them independent. The value of a two-run Push strip as the gap between the runs widens. Each row converges, and none of them converges to the sum of its two runs.

The cliff a cut invents

The rung below asked for a correction term in the gap between two Push runs. There is none, because the gap's contribution vanishes: widen it and the strip's value converges geometrically, at a rate set by the back run's length alone, to a limit that is not the sum. And Shove — whose reading is exact everywhere — fails at the same cut, which says the broken thing is the cut and not the game.

Cut small unless you are behind

The rung below found the greedy rule — cut at the largest prime — wrong on 104 of 552 Maundy Cakes and asked for a description of them. On all 104 the best cut is at the smallest prime, the exact opposite. A middle divisor is never needed on any cake in a sixty by sixty grid, and which of the two extremes wins is decided by Ω alone: cut small when Ω(m) + 1 ≥ Ω(n), large otherwise, and that is exact on all 3,540.

$Almost none of them is a number. Every three by three Amazons region holding one amazon of each colour, sorted by what kind of value it carries. Fifty-six of the two thousand are fractions and the great majority are hot positions.$

The fractions that were not there

The rung below counted 1,452 fractions among the shared Amazons regions and asked which fractions they are. Fifty-six of them are fractions. The other 1,396 are hot positions with a fraction somewhere inside their options, counted by a regular expression looking for a slash — and the quantity the separation of the two amazons actually sets is not a denominator but a temperature.

Running products, and where to stop. Six Maundy Cakes with the prime factors of the longer side, the running products those primes make, and the value the sum of them gives.

The short side only says how many

The rung below settled which cut to make in a Maundy Cake and left the value open. With the cut settled the recursion is a walk, the walk unrolls, and what it unrolls into is the running products of the long side's prime factors, largest first. The short side never enters the products at all — it decides how many of them there are and nothing else, so sixty-two different short sides give one value.

Every gap dies at the last run's rate. Three-run Push strips with each gap widened in turn, and which of four candidate rates the convergence matches. The rearmost run's rate wins every family and the compound rate wins none.

Read from the back forwards

The rung below found a two-run Push strip converging at a rate set by the back run and asked what a third run does — whether the rate is still the rearmost run's, or whether the rates compound. It is the rearmost run's, and for every gap: widen the front gap of a three-run strip, two whole runs away, and the value still dies at the last run's rate. Shove, the game one clause away, compounds.

Four counts and an interval. One Domineering region with its run lengths in each direction and both packing counts written as sums over the runs.

One domino every three cells

The rung below gave the optimistic packing count as a formula in odd runs and asked for the other end of the interval, expecting a formula in the even ones. Parity is the wrong arithmetic: the smallest maximal packing is a sum of ⌈(len−1)/3⌉ over the runs, exact on all 1,042 shapes. That makes the whole interval readable off a drawing — and shows it can never reach the value, because regions with the same runs have different values.

The same distance, different room. Two shared Amazons regions with the amazons the same distance apart, differing in how many squares both of them can still reach. The one sharing more is the colder.

Room pulls two ways

The rung below found the distance between two amazons setting a shared region's temperature and asked for something finer — the squares each can reach, or the squares both can. Neither beats the distance on its own. Together they beat it by half as much again, and the reason is that they pull opposite ways: further apart is hotter, and sharing more reachable squares is colder.

Term by term. Every cut of one long side, with its value written as running products beside the terms of the largest-prime cut.

The short side is not in the lemma

The closed form for a two-sided Maundy Cake rested on one unproved statement: that no divisor beats the largest prime. Written out, that statement never mentions the short side — it is an inequality between a multiset of primes and a term count — and once it is stated that way it has a two-line proof, term by term. The ladder ends in a theorem rather than a grid.

Hold the tail, vary everything in front. The grouping test: every tail against every prefix, asking whether the rate at which a widening gap stops mattering is a function of the tail alone.

Two strips that end the same way

If a Push strip's sensitivity is governed by its last run, then two strips agreeing at the far end should behave the same however different their fronts. On 78 of 80 tails they do, exactly. On two of them a single empty square in the prefix reaches across a gap that grows without bound and halves the rate — and the run reading turns out to be sound in one direction only.

Same runs, different values. Three groups of Domineering regions sharing a run-length multiset, with the value of each member.

Where the runs meet

A Domineering region's value interval is a function of its run lengths and its value is not — twenty-one groups of shapes share a multiset and disagree. The crossing count separates none of them, and neither do ten other local statistics, fifteen sets of which agree on everything and differ in value. What separates nineteen of the twenty-one is where along its runs each crossing sits.

What each stage buys. The account built up one quantity at a time, with the random control priced beside the last row.

An effect that changes sign

Which squares two Amazons share turns out to matter about as much as how many — three shared squares in a line run at 0.63 where three scattered run at 2.51. But the effect of clumping is hotter at one distance and colder at the next, so the arrangement predicts well and describes nothing, which is not what the four rungs below it produced.

The two proofs, beside each other. Maundy Cake's rule was proved by restating its lemma so the short side vanished into a multiset of primes and a term count. Cutcake's rule takes the same five steps, with the multiset replaced by a binary length — one integer instead of a multiset — and the closing argument correspondingly shorter. The one line where they differ is which cut a reader would guess.

The obvious cut is the wrong one

Maundy Cake's rule was proved by restating its lemma so the short side vanished. Cutcake's collapses the same way — into a binary length instead of a multiset of primes — but the cut the argument needs is not the one the ladder predicted. Halving is wrong on a third of all cakes, and the smallest counterexample is six squares by two.

$The threshold is one whole move. The prefixes grouped by their own value, against the rate they produce behind LRRL. No value class splits, and the boundary falls exactly at one move: three quarters of a move is not enough and one move is, with nothing between them. A fractional advantage does not reach across a gap that grows without bound and a whole move does.$

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.

The offsets, and what they separate. The junction descriptor of each member of two split groups, beside the value each holds.

Three distances too many

The junction descriptor records how far a crossing sits from four ends, and the rung below asked what the value does when one crossing slides along its run. It reads one bit — the offset's parity — and only when the run has odd length. The other three distances reach the value not at all.

Five kinds of empty square. Every empty square in a hopless Toads and Frogs strip falls into one of five kinds, and the value follows from which. Three of them are free moves for one player or the other, one of them is where a position stops being a number, and one is a wall that splits the strip into independent pieces.

The square that cannot be halved

Every number in hopless Toads and Frogs is a whole number, which the rung below measured on seven thousand strips and could not explain. The reason is that every empty square is either one player's alone or split evenly between them — except one, and that one is where the numbers stop.

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.

Every domino Left can topple in LRRL. A row of dominoes, blue for Left and red for Right, and each of the mover's options below it. Toppling a domino leftward removes it and everything to its left; rightward removes it and everything to its right. The value under each option is what the game recursion returns for the row that survives.

A recipe instead of a census

Counting a thousand values in seven dominoes suggests Toppling Dominoes reaches every short game, and a count is not a construction. The obvious construction — lay the two options either side of a Left domino and a Right one — is exact on day one, right on a third of day three, and cannot be applied to nine in ten values at all.

The third point. Amazons regions on a 4 × 4 board holding one amazon of each colour, by the Chebyshev distance between them. A 3 × 3 board reaches distance two and this one reaches three, and the mean temperature rises across all three.

The third point on the curve

Four rungs below this one measure a shared Amazons region's temperature against how far apart its two amazons are, and every one does it on a 3 × 3 board, where the distance can only be 1 or 2. Two points make a direction, not a curve. A 4 × 4 board reaches distance 3 — and cannot be evaluated at all until the regions are cut to five free squares. Restricted that far, the answer is neither a sign that flips nor an oscillation: the rise continues and it is running out, the second step being 36 per cent of the first.

A criterion that does not lift. The NoGo separation criterion run on strips and on three-row boards. On strips it holds on nine boards and explains all nine independent ones. On 227 three-row boards it holds on none at all.

A wall that bends

On a NoGo strip, two empty stretches add when no group breathes into both — a wall of two stones of different colours does it, and the criterion explains nine of ninety-three boards and all nine that it covers. On a three-row board it explains none of 227, and not because it is less accurate. A wall across a board has to bend, a stone at the bend sees empty squares on both sides by itself, and every one of the 227 has a group breathing into both regions. The condition is unsatisfiable.

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