The badges
Three systems, one rule. What a badge is, what each one counts, and which numbers are not yet safe to quote.
What a badge is
A badge is the nuna tile at the centre of a fixed set of surrounding circles, with a defined set of positions another tile can take around it. Each of those positions is an arrangement: one circle fused to one adjacent diamond, exactly as the centre tile itself is built.
Two arrangements cannot both be present if they would need the same circle, or the same diamond. That one rule — shares a circle or shares a diamond — is the whole combinatorial engine, and it has been checked against both existing badges with no exceptions found.
A badge is therefore a neighbourhood as much as an ornament. Which tiles count as neighbours of a tile is exactly what a badge defines, which is why nuna life takes its neighbourhoods straight from them.
The three, side by side
| 16-point | 24-point | 40-point | |
|---|---|---|---|
| Points (arrangements) | 16 | 24 | 40, if it is one |
| Circles | 8 used by the sixteen | 21, lettered A to U | not settled |
| Diamonds | 8 used by the sixteen | 12, lettered a to l | not settled |
| Lettering | A to P, one sequence | circles A–U, arrangements A–X — two sequences | none |
| Legal combinations | 1,157 | 22,501 up to tier 10, or 25,000+ — disputed | unknown |
| Ceiling | 6 tiles | 10 tiles, if the 22,501 figure is right | unknown |
| Exclusion rule | shares a circle or a diamond | the same rule | the same rule, presumably |
| State | derived, recomputed, in use | derived by hand, one gap corrected | asked, not built |
The 16-point badge is settled: derived, recomputed from the rule, and used everywhere in this library. The 24-point badge is built and lettered but its count is quoted two different ways. The 40-point badge is a question, not a model.
Where the badges are used
The filing system of this library was built on the 16-point badge: 1,157 legal combinations in tiers of 16 · 98 · 288 · 417 · 274 · 64, one per possible book address. The combination shapes, the cousins catalogue and the lock all start from the same sixteen.
In nuna life the badges are the neighbourhoods. The 24-point badge is the default — every tile that touches — and the 16-point badge is the deliberate alternative, chosen in settings rather than by accident. The oracle records this as NL-002.
Downstream pages: the 16-point combinations, the 24-point combinations, and the lock, which is a badge used as an interface.
Note
Aaron's slot.
Open questions
- The 24-point count is quoted as 22,501 and as 25,000+. Until one is derived from the rule in a file, neither should be cited as settled. See the 24-point page.
- Whether a 40-point badge exists at all as a legal system, rather than as a neighbourhood someone drew. See the 40-point page.
- Whether the badges should be described as neighbourhoods first and ornaments second. Everything downstream of nuna life reads them that way; the library still reads them the other way round.
Checks
| What | How | Source |
|---|---|---|
| One exclusion rule, both badges | Two arrangements exclude each other iff they share a circle or a diamond; no exceptions found in either badge | nuna_master_reference.md |
| 16-point total | 1,157 in tiers 16 · 98 · 288 · 417 · 274 · 64, recomputed from the exclusion key | _src/core.py |
| 24-point lettering | 21 circles A–U, 12 diamonds a–l, 24 arrangements A–X | nuna_master_reference.md |
| 24-point count | 22,501 up to tier 10 | claude/blue-memory-game-brief.md |
| 24-point count | 25,000+ | Nuna-Starting-Kit (1).md |
| Neighbourhoods are the badges | 24 by default, 16 when deliberately chosen | nuna life oracle, NL-002 |
| 40-point badge | An open question, not a model. Nothing on file defines one | nuna life oracle, NL-050 |