The rows that are their own mirror

A row of End-Nim is worth a nimber one time in thirteen. Where the nimbers run out counted 410 of 5,460 and found the palindromes among them — 168 rows that read the same backwards — and closed on the remainder:

A condition that is satisfied by every palindrome and by 242 other rows is a condition nobody has stated, and the rows are in hand. The obvious first guess — that they are the rows equal to their own reverses as values rather than as sequences — is checkable in one pass and is not checked here.

It is checked here. The guess is right, it is right for a reason the guess did not supply, and the reason turns a sufficient condition into an exact one.

Two conditions that are the same condition #

Reversing an End-Nim row negates it. That is not an observation, it is the convention: Left takes from the left-hand heap and Right from the right-hand one, so turning the board round hands each player the other’s moves, and a position turned round is a position with the players exchanged — which is what the negative of a game is.

So G(row) = −G(reverse of row) on every row, always, and the census asserts it rather than assuming it.

Two consequences follow immediately and they are worth separating.

A row equal to its own reverse as a value is equal to its own negative. Substitute: if G(row) = G(reverse) and G(reverse) = −G(row), then G = −G. The guess the rung below made, and the condition its own negative, are the same condition written twice.

A palindrome is such a row. A row that reads the same backwards is its reverse, so it certainly has the same value as its reverse. That is the mirror argument the rung below gave, and it explains 168 rows.

The 242 are rows that read differently backwards and come out worth the same thing anyway. 1,2,1,3 reverses to 3,1,2,1 and both are worth nought. 4,2,3,1 reverses to 1,3,2,4 and both are worth nought. Nothing about the sequences says so.

What the group law promises, and what it does not #

Self-negative values form a subgroup — the two-torsion of the group of values, the elements of order at most two — and every nimber is in it, because ∗n + ∗n = 0.

That gives the implication the rung below used: palindrome ⟹ self-negative ⟹ in the two-torsion. What it does not give is the last step, because the two-torsion is strictly larger than the nimbers. The values that are their own negatives counts thirty of them among the values born by day three, and only four of the thirty are nimbers; the other twenty-six are things like {1 | −1} and {1∗ | −1∗} — switches with symmetric stops, self-negative and nothing like a Nim heap.

So the honest statement of the rung below’s argument is: a palindrome is its own negative, and the site’s evaluator reports that all of them are nimbers, which the argument does not explain. The rung below said exactly that and left it there.

So on day three the two-torsion is more than seven times the size of the nimbers inside it. On End-Nim the two coincide exactly. Four hundred and ten rows are self-negative; 410 rows are worth nimbers; and they are the same 410, with no exception in 5,460.

That is the finding, and it is asserted twice over. A row that was its own negative and worth something else would stop the build; so would a row worth a nimber and not its own negative.

Why a whole class of values is missing #

The coincidence needs an explanation and there is one available, though it is a sketch rather than a proof.

The self-negative values that are not nimbers are hot — they have a fight in them, which is what makes their two stops symmetric about nought rather than equal to it. End-Nim has no hot positions of that kind at these sizes, and the reason is structural: the game has no way to bank an advantage. Every move takes counters from an end, every position with counters in it offers both players a move, and no row is ever worth a positive number.

A value like {1 | −1} needs a position where moving is worth a whole point to whoever does it. End-Nim’s moves are worth what the rest of the row is worth, and the rest of the row is another End-Nim row, so the fight never resolves into a number to be won.

That is a claim about the game rather than about the census, and it is the sort of claim a wider sweep would test rather than confirm: rows of eight heaps have more room in them, and whether the coincidence survives is exactly what a larger enumeration would say.

The same question asked of the other games here #

The coincidence looks less like luck once it is asked of the site’s other rulesets, and the comparison is short because the answer is different in every one of them.

Clobber rows are all-small, so their values are infinitesimals and a self-negative infinitesimal need not be a nimber at all — ↑ + ↓ = 0 makes neither of them self-negative, but the class is full of forms that are their own negatives without being Nim heaps. Toads and Frogs produces switches, and a symmetric switch like {1 | −1} is the exact counterexample End-Nim declines to produce. Shove is cold everywhere and its values are numbers, of which the only self-negative one is nought.

So the three neighbours split three ways: one where the two-torsion is much larger than the nimbers, one where it is much smaller, and one where the whole question collapses. End-Nim is the one where the two sets are equal, and being the odd case out of four is a reason to expect the equality to be small-size rather than structural.

That is the reading this page settles on, and it is deliberately weaker than the census. The equality is exact on 5,460 rows and it is not offered as a theorem, because the mechanism that would make it one — that End-Nim has no hot self-negative values — is argued in a paragraph and computed nowhere.

The condition is exact and is not a shortcut #

The rung below asked two questions and the second one has a discouraging answer.

What the 242 have in common, and whether the property is checkable without evaluating.

The property is checkable. Checking it means computing the value of the row and the value of its reverse and comparing them, which is twice the work the shortcut was supposed to save.

So the census tries every condition that can be applied by looking at the row instead, and scores each as a test for worth a nimber.

Every readable condition fails, and they fail in both directions.

The two end heaps are equal catches 200 of the 410 and wrongly claims 1,168 rows that are not nimbers. The heaps nim-add to nought — the obvious import from Nim — catches 102 and wrongly claims 612. The two halves have the same heaps in some order is the closest of the six and still claims 236 rows wrongly while missing 226. An even number of heaps catches 294 and claims 4,074.

None of them is close. That is worth reading as a positive result rather than as a failure of imagination: the property is not a property of the sequence of heaps. It is a property of the value, and the value is not a function of the multiset, the parity, the ends or the halves.

Where the 242 come from #

The other reason the rung below could not see the condition is that the whole phenomenon is invisible at the sizes a person checks by hand.

Not one row of three heaps or fewer is worth a nimber without being a palindrome. At one heap every row is a nimber and every row is a palindrome. At two and three the two counts agree exactly. The first non-palindromic nimber appears at four heaps, and there are eight of them.

Then it takes over. At five heaps there are 32; at six there are 202 against 64 palindromes, so the palindromes have fallen to under a quarter of the answer.

That is the shape of a great many findings on this site and it is worth naming. A description checked on the cases small enough to enumerate by hand agrees perfectly, and the population it was checked on is the population where the description and the truth cannot be told apart. Three heaps is not a small version of six heaps; it is the size at which the question does not yet exist.

What this leaves of the impartial reading #

The rung below’s finding was that the impartial theory reaches one End-Nim row in thirteen. This page does not change that count and it changes what the count is a count of.

The 410 are not a residue left over from some impartial core. They are the fixed points of an involution — the rows that the board-reversing map sends to values equal to their own — and the number of them is a fact about how many rows have that coincidence rather than about how much of the game is Nim-like.

That reframing has one practical consequence. A solver cannot use the condition, for the reason above, but it can use the mirror: G(reverse) = −G(row) means a table of evaluated rows answers twice as many rows as it holds. That is a genuine saving of half the work and it is available on every row rather than on the 410, and it is the thing the symmetry is actually worth.

What a reader should take from a failed shortcut #

Six readable conditions failing is a negative result, and negative results have a habit of reading as nobody has found one yet. This one says more than that, and it is worth stating in the form it takes.

Each of the six is a function of the row’s combinatorial data — the multiset of heaps, their order, their parity, their sum, their nim-sum. Between them they exhaust the descriptions a person forms while looking at a row of heaps, and each of them is a function that a relabelling of the row would not disturb in the way the value is disturbed.

The value is not such a function. 1,2,1,3 and 1,3,1,2 have the same heaps in the same multiset and one is worth nought while the other is not — which is the whole content of the rung below’s finding that the ordering is the position. Once the order matters, no symmetric function of the heaps can decide anything.

So the failure is forced rather than accidental, and the shape of a condition that could work is now clear: it would have to be a function of the sequence that respects reversal, be sensitive to order, and not be the value. Whether such a thing exists is open and this page has not narrowed it; what the six failures establish is that it is not any of the things a reader tries first.

There is one consolation and it is worth having. Being told that recognition is as dear as evaluation is a useful thing to be told, because it stops a solver author looking. Nothing worth fighting over reached the same conclusion about Shove from the other end — a game whose values are entirely readable in principle and whose reading fails — and the two together are a fair sample of how often a small partizan game admits a shortcut.

More than the group law promises #

The condition is exact here and the group law only ever promised half of it, and the gap between those two statements is the whole of what this page adds.

Every nimber is its own negative. That is arithmetic — $\\ast n + \\ast n = 0$ — and it holds in every game there is. So worth a nimber implies its own negative, always, everywhere, with nothing to check.

The converse is false in general. The two-torsion of the value group contains games that are not nimbers: switches like $\\{1 \\mid -1\\}$, and twenty-five more on day three alone. So its own negative does not imply a nimber, and any game whose self-negative rows included one of those would break the equivalence.

End-Nim’s rows never reach one. That is the finding, and it is a fact about this game rather than about games: the values a row can take happen to intersect the two-torsion only in the nimbers, so the converse holds here and would fail on a ruleset whose values reached further into the subgroup.

Which is why the result cannot be transported. A reader carrying self-negative means nimber to another game is carrying half a theorem and half an accident, and the accident is the half that does the work. The safe statement is the group law’s: self-negativity is necessary, and whether it is sufficient is a question about which values that particular ruleset produces — a question the realisability census is the right instrument for and this page does not ask.

What the census does not say #

Four limits, and the first bounds everything.

Six heaps of four counters. Every count is over 5,460 rows and the coincidence between the nimbers and the two-torsion is checked on exactly those. A row of eight heaps has room for structure this sweep has never seen, and the honest expectation is that the coincidence is a small-size phenomenon rather than a theorem — because the two-torsion of the value group is very much larger than the nimbers and there is no reason End-Nim should avoid all of it for ever.

The explanation is a sketch. End-Nim has no hot self-negative values because it cannot bank an advantage is an argument about what the game produces and it has not been made into a proof. What would make it one is a statement about End-Nim’s temperatures, and this page has not computed them.

Six readable conditions is not all readable conditions. The six here are the ones a reader would try; a condition built from the row’s Zeckendorf representation, or from the parity of the number of heaps above the median, has not been tried. The census does not claim no readable condition exists; it claims none of the obvious ones works, and that the exact one is not readable.

And worth a nimber is a property of the canonical form. A row is counted as a nimber when its reduced form is that of a Nim heap, read off the name the reduction produces. That coincides with playing like a Nim heap in every company, since equal values are interchangeable — so the distinction does not bite, and it is why the test is a form comparison and not a play-out.

The convention, named #

Normal play. A row is a sequence of positive heaps; a move takes any positive number of counters from the leftmost or rightmost heap, and emptying a heap exposes the next. Left takes from the left end and Right from the right, which is the convention that makes the game partizan and without which nothing on this page has a subject.

Every value is computed by the recursion and reduced to canonical form. Its own negative is tested as G = −G with the negation computed and the comparison run on canonical forms, not inferred from the reversal; the identity between that test and equal to its own reverse is asserted separately, so the two are checked against each other rather than assumed equal.

Where the ladder goes next #

The endnim anchor has three rungs: the game with its two-heap rule, how little of it the impartial theory reaches, and now the condition that says which part.

The rung above is the temperature. The argument that End-Nim contains no hot self-negative values is the whole explanation of this page’s coincidence, and it is stated and not computed. A census of End-Nim temperatures would settle it in one pass at this size and would say something the site does not know: whether End-Nim is cold everywhere, which would make it a very unusual partizan game, or merely cold in the two-torsion.

Two neighbours are worth the trip. The values that are their own negatives is the subgroup this page lands in, counted on day three, where thirty values are self-negative and only five are nimbers — the gap this game does not have. And where the impartial theory stops is the general account of what one number per position can carry, of which End-Nim remains the sharpest example on the site.