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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.
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
Impartial games
The proof is sixteen cells
Lasker's Nim has a four-clause formula that was checked on two thousand heaps and never proved. The proof fits in a four-by-four table: the last two bits of a split's value are fixed by the last two bits of its parts, so no split can land in its own heap's class — except at 3 mod 4, where it lands exactly on the one value the takes leave missing and pushes the answer up by one.
Impartial games
One split is enough
A heap of n in Lasker's Nim offers ⌊n/2⌋ ways to split, and the values use at most one of them. Allow only the split that takes a single counter off and every heap to six hundred keeps its value; of all sixty-three sets of split sizes up to six, a set keeps the formula exactly when it contains 1 or 2. Equal halves alone give back plain Nim, because a split into equal parts is a move to nought.
Impartial games
The formula is a limit
Cap the take in Lasker's Nim at k counters and the game is a finite rule table, 4.33…3, whose Grundy sequence repeats with period k + 1 rounded up to even and follows Lasker's formula until the cap bites. The formula is what those periods converge to. And the same column of codes, with a free split in front, holds Kayles itself: the rule 4.4 on a heap of n + 1 is Kayles on a row of n.
Where it stops
Three heaps and a pass
Nim with a single pass that may not end the game is easy on one heap and on two: a heap swaps each odd size with the even one above it, and two heaps lose exactly at (2k − 1, 2k). On three heaps the losses are known only as a list. Fix the smallest heap and each slice of the list settles into a pattern after an irregular start — period 4, 8, 10, then 160 at a smallest heap of ten, and nothing visible from eleven.
Where it stops
What a component would have to carry
For a held pass to be decided by a summary of each component, the summary must separate every pair of components some company tells apart. The Grundy value does not — Nim 1 and Kayles 8 are equal games that a held pass separates beside a single Nim heap of two. Nor does the Grundy value with the component's own held-pass value: Kayles 3 and Kayles 6 agree on both and are split by a company of two Nim heaps. Over twenty-four components, fifteen classes against fourteen pairs, and the gap widens as the pool grows.
Sums and comparison
The follower does the reversing
The gift-horse theorem for the ordinal sum needs two cases, and the second — the added option is reversible — was counted and not described. Recorded move by move, the reversing answer is always Right's move inside the follower: on all 410 escapes under five followers, and on every one of the 2,628 gift horses under every follower that gives Right a move at all. The case split is by follower, not by horse.
Sums and comparison
The split slips one day deeper
The reversal case of the gift-horse theorem was described in one line — the follower's own move reverses every gift horse, whenever the follower has one — and tested only where it was found. In the mirror it holds exactly, with 1 and −1 trading places. One day deeper it fails: under ↑ and ½, three gift horses on built day-four bases are not reversed by the follower's move. All three are dominated, so the theorem stands; the clean split by follower does not.
Sums and comparison
Twenty draws and a second recipe
Day four's comparability was reported as 60.6 per cent against day three's 59.7, from one built sample and one calibration. Built twenty times with each of two recipes whose biases differ by six points, and calibrated against all 1,474 day-three values rather than a quarter of them, the corrected figure spreads over twelve points from seed to seed and the two recipes agree within one standard error. The floor survives; the decimal was one draw.
Out in the world
Cancelling is not pairing
The coin rows that cancel against their own negatives looked like the rows whose coins pair off as nested equal pairs, and on rows of four they are exactly those. From six coins the description fails in both directions — twenty rows pair off perfectly and do not cancel, and one coin set has a hundred and thirty-six that cancel with no pairing at all — and a row of five coins cancels, though an odd row can never pair off. What does hold, on every row swept, is that the first player in a row plus its negative never finishes behind.
Out in the world
A cancelling pair is a zero
Two cancelling coin rows side by side cancel against their two negatives — all 190 pairs from rows of two, four and five coins, the odd row that cancels without pairing off included. And a cancelling row beside its negative is invisible next to any other row: in 741 tests against every row of one to three coins, neither score of the context moves. A non-cancelling pair moves a score in 452 of 780. Scoring games have no inverses in general; this class has them, and they behave as inverses must.
Out in the world
Even rows always reward the move
Milnor's mean-value theory needs an incentive to move — the player to move must do at least as well as if the opponent moved first. On a coin row with an even number of coins that is not a hypothesis but a theorem: the first player can collect one whole parity class of coins, and one of the two classes holds at least half the total. So the condition excludes no even row whatever the coins, the class the earlier table called 'incentive at the top' was every row of four, and the hereditary condition is a condition on odd intervals alone.
What it costs
A misère sum is searched, not added
Under normal play the outcome of a sum of heaps is a nim-sum of numbers already known: twenty stored values decide every sum of Dawson's chess with heaps up to nine, however many heaps it has. Under misère play each sum is a new position to search. One outcome costs six positions for a single heap, two hundred for four heaps and over five thousand for eight, and a table of every eight-heap outcome costs a hundred thousand. The misère quotient is the only thing that brings the price back down.
What it costs
Two heaps of testing are enough
A misère quotient is computed by testing positions against positions, and the universe used to find twelve classes of Dawson's chess was every position of up to four heaps tested against every other — 511,225 outcomes. Varied one size at a time, the count stops growing at tests of two heaps and positions of three: 12,100 outcomes find the same twelve classes. The narrower universe the earlier essay drew did not merge anything; it held fewer positions. And the corner that is enough moves: for Kayles at heap twelve, two-heap tests miss a class.
What it costs
Twelve classes, seven questions
Twelve misère classes of Dawson's chess were found by testing 715 positions against 715 others. Seven of those tests are enough to tell every class from every other — a greedy choice against a floor of four, since each test is one yes-or-no question. Kayles needs nine of 715 and Nim sixteen. The seven cost almost nothing to use and cannot be found without the whole closure, and they do not carry: the tests found with heaps up to seven tell apart only seven of the twelve classes with heaps up to nine.
What it costs
A staircase, not a slope
With the misère closure cut forty-fold, Dawson's chess can be classified at heaps far beyond nine. The count of classes is a staircase: six from heap three to eight, twelve from nine to twelve, seventeen from thirteen to sixteen. Normal play steps once in that range, from four to eight at heap thirteen, where a Grundy value of four first appears. Misère play steps there too, and once more at heap nine, where normal play does not move at all — the first wild heap. Heaps eleven, fifteen and sixteen are also wild and move nothing.
How it was found
Four positions, sampled
Ten names write both sides of every loopy region of two positions, and forty-eight every region of three. Four positions are over four billion graphs and cannot be counted, but they can be drawn. Three thousand regions at each of three densities: the forty-eight names cover between 95.9 and 99.5 per cent, the thirteen names invented for three positions come back at four almost all of them, and the sparsest sample meets thirty-five sides nothing earlier reproduces — a floor of eighty-three names, and a curve that grows by accretion rather than collapse.
Before that
Everything published earlier, newest first. Titles only — the cards are on the full listing.
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