Library · 2. Core Claims · mark I

The 1,157

Claim

Sixteen arrangements under one exclusion rule permit exactly 1,157 combinations.

Construction

Fix a centre tile. Around it are the places another tile can dock, each a choice of circle together with a diamond adjacent to it. There are sixteen of them, lettered A to P, and they are served by eight distinct circles and eight distinct diamonds.

Two arrangements cannot both be present if they would need the same circle, or the same diamond. That single rule — shares a circle or shares a diamond — is the whole combinatorial engine.

Fig. 1 — one arrangement: a circle fused to an adjacent diamond.

Fig. 2 — the badge: eight circles, the centre tile, and the sixteen arrangements each marked at the midpoint of its own circle-to-diamond span, with arrangement A drawn in full. The twenty-two exclusions are drawn between the midpoints — sixteen round the ring, six across it in the accent colour

Proof

Build a graph on sixteen vertices, one per arrangement, with an edge wherever two arrangements share a circle or a diamond. The rule produces 22 edges, and a legal combination is precisely an independent set in that graph.

Sixteen vertices is small enough to settle exhaustively: 65,536 subsets tested, 1,158 survive, and one of those is the empty set.

16 + 98 + 288 + 417 + 274 + 64 = 1,157

No legal combination holds seven tiles. Six is the ceiling, and those six numbers are the shelf capacities this library was originally filed by.

Fig. 3 — the 1,157 by tier, recomputed from the exclusion key. No legal combination holds seven tiles

Two corrections on the record

The interactive model carries an audit note against the source spreadsheet: ENP was wrongly marked excluded when it is valid, and GJLNP was mislabelled as a duplicate of the unrelated CJLNP. Both are fixed in the data the model runs on.

The model also computes the total live rather than quoting it — its own comment says the enumeration is “computed live from KEY (not copied from any spreadsheet)”.

Note

The book says “over a thousand combinations” and leaves it there. What was it like to have the exact number come back?

Open questions

The 24-point badge's count is quoted variously as 25,000+ and as 22,501 up to tier 10. Neither figure has been re-derived here.

Checks

What	How	Source
Count, recomputed	Independent sets over the exclusion key: 16 · 98 · 288 · 417 · 274 · 64, total 1,157	_src/core.py
Exclusion edges	22, from the same key	_src/core.py
Count, independently	Live enumeration in the model, grand total 1,157	nuna-badge-interactive (1).html
The rule itself	Two arrangements exclude each other iff they share a circle or a diamond; no exceptions found in either badge	nuna_master_reference.md
Prior assertion	1,157 stated and spreadsheet-verified before this derivation	Nuna-Starting-Kit (1).md