The border balance

A solved board hides a simple bookkeeping law: every colour the border hands to the interior, the interior hands straight back. Break it and you know instantly the board is wrong; obeying it, though, guarantees nothing.

findingSupportingtheorymeasuredReduce the search Why it resistsUpdated 2026-07-22

Reproducedeterministic — reproduces byte-for-bytereruns the search

Look only at the seam between the outer ring of border pieces and the first ring of interior pieces. Each edge across that seam shows one colour, counted once from the border side and once from the interior side. In any complete solution the two tallies are identical: the multiset of colours the border presents inward exactly equals the multiset the interior presents outward.

Call the imbalance the deficit, $\Delta$ : half the total mismatch between the two tallies. A finished, correct board has $\Delta = 0$ . The four known full solutions, across four different piece sets, all satisfy it exactly. So $\Delta > 0$ is a certificate that a board can never be completed: a real, cheap necessary condition.

The law, in one line#

Across the border↔interior seam, the colours the border shows inward and the colours the interior shows outward are the same multiset. Writing $A[c]$ and $B[c]$ for the two per-colour tallies:

\Delta \;=\; \tfrac{1}{2} \sum_{c} \bigl|\, A[c] - B[c] \,\bigr| \;=\; 0 .

See it on a real board#

A solved 8×8 starts in balance ( $\Delta = 0$ ). Lift a border piece and watch which colours fall out of balance, and the deficit climb. Then try the swap button, and watch the catch.

Interactive: the NS-1 border-balance deficitExplore →

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The catch, and why it matters#

Swapping two border pieces leaves $\Delta$ at 0. The swap moves colours around the seam without changing either tally, so the invariant is blind to it. Worse, on a near-solution most of the remaining errors aren't on the border seam at all. They sit interior-to-interior, where NS-1 never looks. Across the nine bundled 469-class boards, of the 99 unmatched edges that remain, 86.9% sit interior-to-interior and not one is border-to-border; only the other 13.1% land on the seam NS-1 inspects. The invariant is blind to the large majority of what is still wrong.

That is the whole texture of Eternity II in miniature. A check this clean still only ever says "definitely broken", never "definitely fine". The puzzle resists every cheap certificate of progress.

Is it useful, then?#

Yes, as a late-search pruner. Once a solver has closed the border ring, enforcing $\Delta = 0$ rejects a useful slice of deep dead-ends for the price of one pass over the 56 seam edges (how large a slice is queued for exact measurement, not yet a committed benchmark). It is necessary-but-loose: it throws away bad states cheaply and leaves the hard part, the interior, untouched.

The law, in one line#

Across the border↔interior seam, the colours the border shows inward and the colours the interior shows outward are the same multiset. Writing

A[c]

and

B[c]

for the two per-colour tallies:

\Delta \;=\; \tfrac{1}{2} \sum_{c} \bigl|\, A[c] - B[c] \,\bigr| \;=\; 0 .

The catch, and why it matters#

Swapping two border pieces leaves

\Delta

at 0. The swap moves colours around the seam without changing either tally, so the invariant is blind to it. Worse, on a near-solution most of the remaining errors aren't on the border seam at all. They sit interior-to-interior, where NS-1 never looks. Across the nine bundled 469-class boards, of the 99 unmatched edges that remain, 86.9% sit interior-to-interior and not one is border-to-border; only the other 13.1% land on the seam NS-1 inspects. The invariant is blind to the large majority of what is still wrong.

That is the whole texture of Eternity II in miniature. A check this clean still only ever says "definitely broken", never "definitely fine". The puzzle resists every cheap certificate of progress.

Is it useful, then?#

Yes, as a late-search pruner. Once a solver has closed the border ring, enforcing

\Delta = 0

rejects a useful slice of deep dead-ends for the price of one pass over the 56 seam edges (how large a slice is queued for exact measurement, not yet a committed benchmark). It is necessary-but-loose: it throws away bad states cheaply and leaves the hard part, the interior, untouched.

The border balance

The law, in one line#

See it on a real board#

The catch, and why it matters#

Is it useful, then?#

Keep exploring

Referenced by

The law, in one line#

See it on a real board#

The catch, and why it matters#

Is it useful, then?#