Recently added

What's new

Essays arrive in groups rather than one at a time. The most recent group is below in full, and every earlier one after it, newest first.

Essays arrive in groups rather than one at a time, and a group usually opens up a subject not covered before. Between one group and the next nothing changes, so a reader who has seen the most recent group has seen everything.

7 November 2026

8 essays on how it was found, what it costs and temperature

How the word is made. The Prouhet–Thue–Morse word built by doubling. Each row of one, two, four up to thirty-two letters is the row above followed by its swap, which is the same word obtained by replacing A with AB and B with BA.

How it was found

A rule that let the game run for ever

In 1929 German chess drew a game when the same sequence of moves came three times running, and the rule was meant to make chess finite. Max Euwe, six years from the world title and trained as a mathematician, showed that it does not, using a word that had already been found twice for other reasons. Two round trips are enough. The board returns to one position thousands of times, no block of play ever comes three times running, and the rule has nothing to say.

7 figures

How strict a repetition rule can be before it ends every game. For two, three and four available moves, the longest legal game under each repetition exponent from 7/5 to 3. Two moves stop at three letters at exponent two and never stop above it; three moves stop at 38 letters at 7/4; four moves stop at 121 letters at 7/5.

How it was found

How strict a rule can be

A repetition rule can be tightened one notch at a time, and each notch either leaves unending games alive or kills them outright. With two moves a block may not come twice, and every game is over by the fourth letter; relax that by a single letter of overlap and unending games appear. Three moves stop dead at 38 letters under a rule that a whisker's relaxation frees. Four stop at 121. Past each boundary the words go on for ever, and the boundary sits at a fraction nobody would have guessed.

8 figures

The repeater's plan on two moves. Every line of play in which the repeater forces the rule of 1929 on two moves, 4 lines of at most 7 letters, each ending in a block three times running.

How it was found

The opponent makes it repeat

Euwe's unending game needs both players to keep to it. Give the letters to two players in turn, with one of them wanting the draw that a repetition brings, and the rule of 1929 fires within seven letters on a plan of one sentence: always play A. The word that defeats the rule exists and nobody can be held to it. Add a third round trip and the same player can no longer force anything in twenty-six letters. Existence and force turn out to be separate questions, with separate boundaries.

8 figures

A draw by repetition changes no result. Over all labelled games on three and four positions and a sample on five, the result from each start with a draw on the second or third arrival at a position, compared with retrograde labelling of the graph. They agree everywhere.

How it was found

What counting positions buys

A rule that counts arrivals at a position ends every game, which the rule of 1929 could not do, and it needs no proof beyond arithmetic: with L arrivals barred on n positions, no game is longer than (L − 1)n − 1 moves. What it does not do is change a single result. Over every game on three and four positions and a sample on five, drawing on the second arrival or the third gives exactly the labels retrograde analysis had before any rule was written. The rule pays for the game's ending and for nothing else.

7 figures

Refusing a repeat resolves every draw. All games on three and four positions and a sample on five. When the second arrival at a position is refused, every won start stays won and every lost start stays lost, and every drawn start becomes a win or a loss.

What it costs

The number in the rule

Draw on a repeated position and no result moves. Refuse the move instead, as Go does, and nothing is left drawn: over every game on four positions, 10,324 of 16,384 starts are drawn with no rule and every one of them acquires a winner. Which winner depends on the number written into the rule. Bar the third arrival rather than the second and 1,392 of those starts change hands. A question with a linear answer has become one that needs the history, and the extra difficulty lies entirely in the positions the labelling called drawn.

6 figures

Recording the turn doubles an odd cycle. On the left a cycle of three positions, which has no stable set. On the right the same game with the player to move recorded in each position, a cycle of six, with two stable sets, each consisting of all the positions with one player to move.

How it was found

Taking turns removes the hard part

Whether a graph has a stable set is NP-complete, and every hard instance depends on odd cycles. A real game's position graph has none. A position records whose turn it is and every move changes that, so all cycles are even, Richardson's theorem applies, and the set of all drawn positions with one player to move is a stable set of the drawn part, always. Mū tōrere and Three Men's Morris both confirm it. The hard question never reaches a game with turns, and the sets it hands over carry no information at all.

7 figures

The one colour-blind stable set of Mū tōrere. All 33 classes of drawn position in Mū tōrere under rotation, reflection and colour exchange, each drawn as a small board from the viewpoint of the player to move, with the 16 classes of the unique stable set shaded.

How it was found

A rule that cannot see whose turn it is

The stable sets a game hands over work by telling the players apart. Forbid that, and read every board as mine and theirs, and the odd cycles come straight back. On Mū tōrere, with rotations and reflections identified too, the 892 drawn positions fall into 33 kinds, and among those there is exactly one stable set. It names sixteen shapes. A player who always hands the opponent one of them holds the draw against labelled play without knowing a colour, an orientation or an outcome, and the opening itself is one of the sixteen.

6 figures

Where exempting an integer starts to cost. Heating that leaves integers alone, tested for additivity at twelve bonuses from an eighth to two on two pools. It fails on no pair below a bonus of one and on dozens from one upward, while heating with no exemption fails at no bonus on these pools.

Temperature

An exemption that costs nothing below one

Heating that leaves integers alone fails to add at a bonus of one and adds on everything at a bonus of a half, and the natural guess was that a half is where it starts to go wrong, since that is where a fraction's moves become worth fighting over. It is not. Across twelve bonuses nothing fails at any value below one, and dozens of pairs fail at one exactly. The reason is the simplicity rule: below one, the recursion sends every integer to itself, so the exemption is not doing anything at all.

7 figures

Before that

Everything published earlier, newest first. Titles only — the cards are on the full listing.

3 November 2026

10 essays on how it was found, what it costs and out in the world

The solution that was a losing set — how it was found
An odd cycle is the only way to have none — how it was found
What a stable set knows about a draw — how it was found
A stable marriage is a stable set — how it was found
The list that cannot run out — how it was found
A formula drawn as a game — what it costs
A win is not a way of winning — out in the world
Blocking is not enough — out in the world
Three ways to know a draw — out in the world
A win priced in stones — out in the world

30 October 2026

3 essays on what it costs, how it was found and where it stops

Growing is not climbing — what it costs
Two losers that average to a winner — how it was found
A compulsion that points elsewhere — where it stops

25 October 2026

5 essays on temperature, out in the world, how it was found and impartial games

The bonus an integer is owed — temperature
Two reflections and a corner — out in the world
The teacher's game — how it was found
The third take is the other two — impartial games
The rows that cancel — how it was found

14 October 2026

4 essays on sums and comparison, how it was found, out in the world and what it costs

A foot is a menu, not a number — sums and comparison
The weights were not the argument — how it was found
A pairing that changes once — out in the world
A heap that loses under both endings — what it costs

5 October 2026

4 essays on sums and comparison, what it costs, how it was found and out in the world

A universe grows by days, not by sums — sums and comparison
The colour swap pays through a mirror — what it costs
A pairing is checked, not measured — how it was found
Two neighbours keep a pairing, three lose it — out in the world

2 October 2026

4 essays on how it was found, out in the world, temperature and what it costs

A circuit asked once — how it was found
The same rule on a star — out in the world
Why cooling has to add — temperature
A repetition is a guess — what it costs

30 September 2026

4 essays on how it was found, impartial games and what it costs

A machine that knew one theorem — how it was found
A current through the board — how it was found
Two takes, one line — impartial games
Pieces that can only come apart — what it costs

27 September 2026

6 essays on where it stops, impartial games, what it costs, temperature and out in the world

No number bounds it — where it stops
One square for every coin — impartial games
Units modulo three, and one choice — where it stops
Answer where the opponent played — what it costs
Colder exactly when the residues cancel — temperature
Start at four — out in the world

26 September 2026

6 essays on what it costs, where it stops, out in the world, impartial games and sums and comparison

The plays do not say who chose them — what it costs
The price of a tie — what it costs
A coalition counts to three — where it stops
Two rows still reward the move — out in the world
The odd values keep coming — impartial games
The pairs stand still and the order spreads — sums and comparison

25 September 2026

7 essays on sums and comparison, where it stops, impartial games, how it was found, temperature, what it costs and out in the world

What a part would have to report — sums and comparison
The table closes, until heaps of four — where it stops
The pairs are read from the bottom bits — impartial games
Composition at every size — how it was found
The hotter residue survives — temperature
A proof inside a budget — what it costs
The seven were the order — out in the world

24 September 2026

9 essays on what it costs, impartial games, sums and comparison, out in the world, where it stops and how it was found

The cheapest fight is not the yardstick — what it costs
Cram with the last move losing — impartial games
The same parts, different wholes — sums and comparison
A different question at every depth — out in the world
Three pairings and no order — out in the world
A rare class exclusive-or respects — impartial games
What survives a coalition — where it stops
Each size is built from the last — how it was found
One value more than Nim — where it stops

23 September 2026

16 essays on impartial games, where it stops, sums and comparison, out in the world, what it costs and how it was found

The proof is sixteen cells — impartial games
One split is enough — impartial games
The formula is a limit — impartial games
Three heaps and a pass — where it stops
What a component would have to carry — where it stops
The follower does the reversing — sums and comparison
The split slips one day deeper — sums and comparison
Twenty draws and a second recipe — sums and comparison
Cancelling is not pairing — out in the world
A cancelling pair is a zero — out in the world
Even rows always reward the move — out in the world
A misère sum is searched, not added — what it costs
Two heaps of testing are enough — what it costs
Twelve classes, seven questions — what it costs
A staircase, not a slope — what it costs
Four positions, sampled — how it was found

19 September 2026

20 essays on what it costs, where it stops, out in the world and how it was found

Twelve turns, and three different prices — what it costs
Eleven moves and one decision — what it costs
Proving a loss means answering everything — what it costs
The opponent stops choosing — what it costs
A turn is not a bit — what it costs
The auction never gets to the money — where it stops
A coin needs no tie-break — where it stops
Left always wins, and loses more often than not — where it stops
Every chance but a certainty — where it stops
The coldest position has the biggest swing — where it stops
The best chance is the wrong move — where it stops
The endgame theory arrives late — out in the world
A thousand positions and no exception — out in the world
A coin with three strings is worth something — out in the world
Four boxes for every chain after the first — out in the world
Two and four are not conventions — out in the world
The step nobody took for thirty-four years — how it was found
The picture Bouton's proof leaves behind — how it was found
Four thousand nine hundred regions with no name — how it was found
The names are not built out of the old ones — how it was found

16 September 2026

9 essays on how it was found, out in the world and what it costs

A difference the rows cannot predict — how it was found
Two things to hold at once, or three — how it was found
A shortlist with nothing at the top — out in the world
The cheap fights make the rule cheaper — what it costs
The restriction that buys the most — out in the world
A key is a code, and two squares come free — what it costs
Search on in pairs of moves — what it costs
A loop is written with two names — how it was found
A move whose every reply is struck — out in the world

15 September 2026

6 essays on out in the world, how it was found and what it costs

The pairing removes moves it cannot name — out in the world
A wall the pawns cannot cross and the rule can — how it was found
The rule decides who has to remember — how it was found
Two graphs a rule cannot tell apart — what it costs
The winning reply is the fourth choice — out in the world
The bound names the hottest part and the cost does not — what it costs

13 September 2026

5 essays on what it costs, how it was found and out in the world

Where a search may stop — what it costs
A check bit halves the average and not the key — what it costs
Using up the edges instead — what it costs
The capture that has to be made — how it was found
A potential that names every move — out in the world

12 September 2026

14 essays on out in the world, how it was found and what it costs

The first move is a link that is not there — out in the world
Cut is Short on another graph — out in the world
A point with three neighbours — out in the world
Where the needle has a sentence — out in the world
A board one column wider — out in the world
Every move closes the largest gap — out in the world
A board is written as a sum — how it was found
What the play keeps coming back to — how it was found
One bit of memory — how it was found
A count that forgets — what it costs
One board, and recency still wins — what it costs
The order a solver tries the moves in — what it costs
A verdict that changes with the depth — what it costs
A key shorter than the position — what it costs

10 September 2026

20 essays on out in the world and how it was found

What has to break before a pawn is worth a number — out in the world
One king, and two files to be in — out in the world
A position with no value, and the rule that gives it one — out in the world
A ko is won somewhere else — out in the world
Two ways to count a finished board — out in the world
The second dimension is not the deep end — out in the world
The two moves that are not captures — out in the world
What a pass is worth to a theory — out in the world
A hypothesis has to hold all the way down — out in the world
Nothing to subtract with — out in the world
The paper was about how long — how it was found
The gap between two answers — how it was found
Every play ends and no round settles — how it was found
What the arithmetic cost in 1956 — how it was found
The convention Dawson actually used — how it was found
Three complete solutions in nine years — how it was found
A set with a short description — how it was found
The sentence that solved the other convention — how it was found
Two names that add to nothing nameable — how it was found
What two numbers cannot tell apart — how it was found

7 September 2026

50 essays on impartial games, what it costs, temperature, values, where it stops, sums and comparison, out in the world, particular games and how it was found

The family with two witnesses — impartial games
A pairing that is not a symmetry — impartial games
The quantity that carried nothing — impartial games
The table that changes its mind — what it costs
A description, and not a detector — temperature
The easy case was not the reason — values
The price of asking what the parts are — where it stops
A quotient that identifies nothing — where it stops
One expression proved, and one withdrawn — sums and comparison
The four paragraphs prove something else — temperature
The dual was the value table — impartial games
One proof, and one wrong lemma — impartial games
The pairing the formula hides — impartial games
A cross in the table — sums and comparison
The proof needs both reductions — sums and comparison
Two measures bounded, and one not — values
A floor, and not a decline — sums and comparison
One number, stated two ways — sums and comparison
A factor, and not an overhead — values
Close calls nothing resolves — values
Three groups, and three yields — values
A set with three descriptions, and a function with none — out in the world
The obvious cut is the wrong one — particular games
A fraction does not reach — particular games
Three distances too many — particular games
The square that cannot be halved — particular games
Where the numeral stops — particular games
A recipe instead of a census — particular games
The bend above the top — values
A reduction that reads a graph — values
A second pool, designed differently — temperature
The residues as a sequence — temperature
An environment instead of a stack — temperature
Twenty-six other values — sums and comparison
Closing the wild side — where it stops
A stopper and how to find one — where it stops
Which games end at which level — where it stops
A wall an amazon can walk through — what it costs
What it costs to notice a repetition — what it costs
The parts are worth nothing and the sum is not — out in the world
A parity with a first exception — out in the world
What computing further has bought — how it was found
Where the braces stop — how it was found
Three bits of rule — how it was found
Four hundred and seventy steps — impartial games
One of four questions — values
Two clauses and a third question — where it stops
The third point on the curve — particular games
A wall that bends — particular games
The second bend is the boundary — temperature

3 September 2026

20 essays on particular games, values, temperature, impartial games, sums and comparison, where it stops and what it costs

The short side is not in the lemma — particular games
Two strips that end the same way — particular games
A bend that never reaches the surface — values
The mirror was the floor — values
A heuristic that becomes a theorem — values
The price of taking the maximum — values
Where the runs meet — particular games
An effect that changes sign — particular games
A pool built to have an answer — temperature
The parameter was the difference — impartial games
Where the value stops mattering — sums and comparison
The case that was supposed to be hard — sums and comparison
A symmetry that is not a pairing — impartial games
A pattern that has not started yet — impartial games
A side about to lose its move — sums and comparison
The clause that turns the class off — where it stops
The licence that weighs nothing — where it stops
Which top is the top — temperature
The ceiling was a plateau — temperature
A catalogue that builds itself — what it costs

1 September 2026

20 essays on particular games, values, temperature, sums and comparison, impartial games, what it costs and where it stops

The short side only says how many — particular games
Read from the back forwards — particular games
The bend is in the stops — values
The rate was the alphabet — values
A threshold is a detection limit — values
Seventy-two of them were not silence — values
One domino every three cells — particular games
Room pulls two ways — particular games
The worst value in its own interval — temperature
Eight squares, and no hotter — temperature
The same number in two currencies — sums and comparison
The parities, in size order — impartial games
A mex with no impartial game in it — sums and comparison
A pairing, and the pairing — impartial games
The catalogue a strong player needs — what it costs
One half multiplies, the other adds — where it stops
Wrong in one direction only — where it stops
Two counters, and one displaced term — impartial games
The premises an induction would need — temperature
Not a domination, in that order — sums and comparison

31 August 2026

20 essays on values, particular games, temperature, sums and comparison, impartial games, where it stops and what it costs

The threshold was a fact about the census — values
A second level of stops — values
The entry fee was the cap — values
Three rules and a tie-break — values
Half the difference in odd runs — particular games
The cliff a cut invents — particular games
Cut small unless you are behind — particular games
The fractions that were not there — particular games
A rule that beats the hottest — temperature
The obstacle was the catalogue — temperature
Which end a sum lands at — sums and comparison
The option nothing names — sums and comparison
What identifies two subsets — sums and comparison
The count of odd heaps — impartial games
The only way to split into three — impartial games
The two numbers at the top — temperature
The check that was not a check — impartial games
The wild side does not close — where it stops
Half a licence is nearly all of it — where it stops
A catalogue that knows what it will meet — what it costs

30 August 2026

20 essays on values, impartial games, temperature, sums and comparison, particular games, what it costs and where it stops

The weight that blunts the count — values
Half a follow-up out — values
Wider costs less — values
The wider move is the easier game — impartial games
Half of the smaller temperature — temperature
What a strategy has to remember — values
One fight makes a board a fight — temperature
A bound with one number too many — sums and comparison
The third digit — impartial games
Two errors that cancel — particular games
At least five hundred and seventy-one — sums and comparison
Where to stop building — what it costs
The size of a cake — particular games
A check in front of a search — impartial games
The quantity that does not order a board — temperature
Add, then reduce again — sums and comparison
A function with no formula — where it stops
A product against a sum — where it stops
A numeral in the empty squares — particular games
A region one player owns — particular games

28 August 2026

20 essays on impartial games, particular games, values, temperature, sums and comparison, where it stops and what it costs

What the numerals knew — impartial games
A code that climbs by three — impartial games
The rule a smaller move breaks — impartial games
The moves a player can be talked out of — particular games
A game with nothing at stake — particular games
The condition that survived the wider sweep — particular games
The criterion that cannot exist — particular games
The birthday is a floor — values
The bend is the condition — values
How many moves are worth making — values
How big the answer is — temperature
A self-negative value costs a day — sums and comparison
The closure that picks the nimbers — where it stops
When the catalogue starts paying — what it costs
What a fight does to a fight — sums and comparison
No fifth value — sums and comparison
What a game actually produces — temperature
The margin a count needs — values
A subtraction, not a factor — temperature
What the class does not buy — where it stops

27 August 2026

15 essays on values, particular games, impartial games, where it stops, sums and comparison, what it costs and temperature

Fifty-two errors and seven sizes — values
The cheapest way to show a value — values
Which option the reduction keeps — values
Counting the moves each side has — particular games
The rows that are their own mirror — particular games
How thick a wall has to be — particular games
The family the Fibonacci numbers belong to — impartial games
The symmetry one move away — impartial games
The company that is closed — where it stops
What the colon respects — sums and comparison
How often a board falls apart — what it costs
A schedule instead of a number — temperature
What the halving is a function of — temperature
When to leave the environment — temperature
How wrong a nearly-independent split is — sums and comparison

26 August 2026

15 essays on values, temperature, particular games, sums and comparison, impartial games, where it stops and what it costs

Where the order and the sum disagree — values
The same position, written once — values
A fight with no midpoint — values
How hot a real position is — temperature
The answer that starts another fight — temperature
Which shapes are worth fighting over — particular games
The reading that survives too much — particular games
Where the nimbers run out — particular games
A compound of two different games — sums and comparison
The thirty that cancel themselves — sums and comparison
Looking for the symmetry — impartial games
The patch that generalised — impartial games
What restores the theorem — where it stops
The rule the symbols follow — where it stops
The question in the middle — what it costs

24 August 2026

15 essays on values, temperature, where it stops, impartial games, particular games and sums and comparison

What a value costs to write down — values
Which end of the interval is open — values
The birthday of a sum — values
When two thermographs can be added — temperature
How hot a day gets — temperature
A pool built to punish greed — temperature
Nobody comes back — where it stops
What a wider pool rescues — where it stops
What a component has to carry — where it stops
The strategy that is a symmetry — impartial games
How long a row a value needs — particular games
When the regions add — particular games
A sequence with a rule and no period — particular games
The number nobody needs — sums and comparison
How hot a background has to be — sums and comparison

22 August 2026

15 essays on what it costs, where it stops, sums and comparison, temperature, particular games and values

Knowing who wins, and knowing what it is worth — what it costs
Equal in this company — where it stops
Two ways to end with no bound — where it stops
How long it lasts — sums and comparison
The genus of a sum — where it stops
What an infinitesimal does to a fight — sums and comparison
How cold a sum of hot games can be — temperature
What a move nobody makes is worth — temperature
A board that is a sum of its regions — particular games
How much a list of options can lose — values
What is left when the copies pair off — temperature
The numbers it is confused with — values
The strip where every number is a whole one — particular games
How wide a form can get — values
When the bracket decides — particular games

21 August 2026

15 essays on where it stops, impartial games, sums and comparison, temperature, particular games and values

Nobody has to move — where it stops
It ends, and nothing says when — where it stops
Splitting is a move — impartial games
What is left when the small change is thrown away — sums and comparison
Below zero — temperature
The heap is not the position — impartial games
What can be struck out — sums and comparison
A period with a constant added — impartial games
A rule with no promise at all — temperature
The operator that puts the star back — temperature
Topple it from either end — particular games
The other way to move a row — particular games
The reduction that always shrinks — values
Every group must keep breathing — particular games
The values nobody's game produces — values

20 August 2026

15 essays on where it stops, temperature, impartial games, sums and comparison, particular games and values

The one outcome that adds — where it stops
Three players and no answer — where it stops
A pass is not a move — where it stops
A rule with a guarantee — temperature
No two heaps alike — impartial games
What a number does to a fight — sums and comparison
Cooling adds and heating does not — temperature
How rare it is to be bigger — sums and comparison
The values that keep arriving — impartial games
Nothing worth fighting over — particular games
The operator chosen for one game — temperature
Two players, two lists — particular games
A tree is still a number — particular games
Nobody wants to move here — values
The reduction that puts options back — values

19 August 2026

15 essays on where it stops, temperature, impartial games, sums and comparison, particular games and values

A move that must be answered — where it stops
When never ending is a win — where it stops
What a tame heap may be replaced by — where it stops
Cooling by exactly one — temperature
A number and a fight — temperature
A token on a graph — impartial games
The values that are their own negatives — sums and comparison
The code names the move — impartial games
The same strip without the jump — particular games
Two games in one environment — temperature
When the nested sum only sees the value — sums and comparison
Taking from the ends — particular games
One row of Clobber — particular games
An option nobody would take — values
The simplest game above both — values

17 August 2026

15 essays on where it stops, temperature, sums and comparison, particular games, impartial games and values

Tame and wild — where it stops
Two misère outcomes are not enough — where it stops
Double sente is not a property of the position — temperature
An environment made of coupons — temperature
A thermograph with two bends — temperature
Equal in every company — sums and comparison
Maundy Cake — particular games
Taking from several heaps at once — impartial games
Cram — impartial games
The values of every small board — particular games
When the ups add — sums and comparison
Amazons on one line — particular games
How old a value is — values
When a switch is not a switch — values
The fight never runs backwards — values

16 August 2026

15 essays on where it stops, temperature, particular games, sums and comparison, impartial games and values

The condition the recursion rests on — where it stops
One part that never ends — where it stops
Misère play has no negatives — where it stops
Sente is a fact about the rest of the board — temperature
A green edge on a blue one — particular games
A thermograph is built from its options' — temperature
Confused is not the same as unknown — sums and comparison
The losing positions are a code — impartial games
The strip nobody has a formula for — particular games
One rule makes it cold, the other hot — particular games
The other sum, the one that nests — sums and comparison
The period is small and the proof does not say so — impartial games
Where the fight stops — values
What a move is worth to the player making it — values
The switch a player is imagining — values

15 August 2026

15 essays on out in the world, temperature, sums and comparison, impartial games and what it costs

The game in every exercise book — out in the world
The chains decide it before the boxes do — out in the world
The rule that makes Go a finite game — out in the world
A game older than the theory — out in the world
The theorem that names a winner and no move — out in the world
A winning strategy that is a spanning tree — out in the world
The game that is a number system — out in the world
Big is not the same as hot — temperature
Counting at the end changes everything — out in the world
A pawn ending is a sum — out in the world
Independence is a claim — sums and comparison
The digits say which move wins — out in the world
The tartan theorem — impartial games
When a real board falls apart — out in the world
Finding the parts — what it costs

13 August 2026

15 essays on how it was found, impartial games, sums and comparison and values

The first theorem, and the winner it declines to name — how it was found
The theorem that needed none of the theory — how it was found
Two people, four years apart, one theorem — how it was found
A golden ratio thirty years early — how it was found
A chess problem that turned out to be an octal game — how it was found
The sequence nobody has settled — how it was found
The numbers came out of the game — how it was found
The notation was the argument — how it was found
"Hopeless" was a claim about a method — how it was found
The nimbers multiply — impartial games
Where the impartial theory stops — sums and comparison
A conjecture from hand play — how it was found
The first time it told somebody something — how it was found
The recursion this site cannot run — values
What a value leaves out — values

11 August 2026

15 essays on what it costs, where it stops, sums and comparison, impartial games and particular games

A position reached eleven ways is one position — what it costs
An outcome with no value behind it — where it stops
The board falls apart, and the arithmetic changes — what it costs
What counts as the same position, and what that is worth — what it costs
A puzzle asks once, a game asks alternately — what it costs
Comparing two positions means playing a third — sums and comparison
Nim is easy, in binary — what it costs
Take one, three or four — impartial games
"Left wins" has no short proof — what it costs
A game where nobody can be ahead in moves — particular games
The class is named after memory, and that is not an accident — what it costs
Three different claims are all called solved — what it costs
Four values, and the sequence is settled for ever — what it costs
A rule that is never right and cannot be far wrong — what it costs
The cost is in the closure, not in the positions — what it costs

10 August 2026

12 essays on what it costs, where it stops, sums and comparison, impartial games, temperature, particular games and values

The game with the shortest rule is the hard one — what it costs
Start at the end and work backwards — where it stops
Turn the board through a right angle — sums and comparison
The move that gives counters back — impartial games
Two hot fights that add to a cold number — temperature
The same fight, eight times over — temperature
Three ways to add the same games — sums and comparison
A row of coins is already a sum — impartial games
Squash every loop to a point — particular games
Two hundred and fifty-six ways to write twenty-two things — values
Tiny, miny, and the sizes below every size — values
The class where nobody runs out first — values

3–7 August 2026

34 essays on particular games, what it costs, where it stops, impartial games, sums and comparison, temperature and values

Hackenbush is a numeral — particular games
How hard is it — what it costs
Misère play — where it stops
Nim, and the nim-sum — impartial games
The sum is the object — sums and comparison
What is at stake — temperature
Who moves last — values
Comparing positions — sums and comparison
Domineering — particular games
Loopy games — where it stops
Hard, proved — what it costs
Reading a thermograph — temperature
Every impartial game is a Nim heap — impartial games
The simplicity rule — values
Canonical form — values
Grundy sequences, and where they stop being predictable — impartial games
What survives misère play — where it stops
Outcomes do not add — sums and comparison
Playing the hottest — temperature
Toads and Frogs — particular games
Cooling — temperature
Infinitesimals — values
Naming a game with a number — impartial games
Sprouts, and the game that is not one — particular games
Which part to move in — sums and comparison
How many ups — sums and comparison
One board, two rules — particular games
The endgame, accounted for — temperature
The day a number is born — values
Wythoff's game, and the ratio nobody put there — impartial games
Cutcake, where every value is a whole number — particular games
Numbers avoid numbers — values
Amazons, and when a position becomes a sum — particular games
Worth nothing, and worth fighting for — values

Every essay, by subject · by subject · by thread · search