The 24-point badge
Twenty-one circles, twelve diamonds, twenty-four arrangements — and a count this project quotes two different ways.
What it is
The 24-point badge is the next neighbourhood out. It was built by expanding the circle grid, keeping 21 circles and relettering them A to U, discarding the rest, and lettering the twelve diamond gaps between them a to l. Its 24 arrangements are lettered A to X.
The rule is unchanged: two arrangements exclude each other when they share a circle or share a diamond. The reference records no exceptions in either badge.
The naming collision
Circles run A to U. Arrangements run A to X. They are two different sequences and the same letters mean different things in each, which the reference flags explicitly rather than quietly fixing.
This is worth carrying forward into anything that names a 24-point combination: a code like AD is only unambiguous once you say whether its letters are circles or arrangements. The 16-point badge has no such problem, since its sixteen circles and sixteen arrangements were lettered together.
The count, which is disputed
Two figures circulate, and no file on hand derives either from the rule.
22,501, up to tier 10, recorded in the memory game's brief as verified against the engine that also returns 1,157 for the 16-point badge.
25,000+, recorded in the standing context block of the starting kit.
They are not reconcilable as roundings of each other, and the tier-10 qualification on the first suggests the two may be answering different questions — one capped, one not. Until a file carries the enumeration tier by tier, as the 16-point badge's is carried, this page quotes both and settles neither.
One exclusion was missing
The 24-point exclusion table was worked out by hand and then checked systematically against the rule. One gap — a missing mutual exclusion — was found and corrected.
The library's index names that pair as B–X. The reference records the correction but does not name the pair, so the specific letters are uncited here. Either the index knows something the reference does not, or the letters came from somewhere that should be found and cited.
Shapes and families
The Shapes tab gives 8,933 combination shapes and 4,459 families for this badge. Those figures come from that tab's own work and no file on hand carries them, so they are recorded here as reported rather than checked.
For comparison, the 16-point badge gives 717 connected shapes in 362 families, and that family count is itself disputed — 317 also circulates, and the likely cause is whether point joins count as contact. Whatever is decided there has to be decided the same way here, or the two tabs will be counting different things.
Note
Aaron's slot.
Open questions
- 22,501 or 25,000+. Neither is derived from the rule in any file on hand.
- Does 22,501 now qualify? Book 4.6 — the 24-point badge dataset — is reserved until the enumerator returns a count worth trusting. A figure recorded in a brief, with no tier table and no re-derivation, is not yet that; the same engine returning 1,157 for the smaller badge is evidence, not proof. The cheapest route to lifting the reservation is to run the 24-point enumeration the way the 16-point one was run, and publish the tiers.
- Whether tier 10 is the ceiling or a cap. If no legal combination holds eleven, say so and the badge is closed the way the 16-point badge is at six.
- Which pair the corrected exclusion was. The index says B–X; nothing on file names it.
- Whether the 8,933 and 4,459 figures use the same contact rule as the 16-point counts.
Checks
| What | How | Source |
|---|---|---|
| Construction | Circle grid expanded, 21 circles kept and relettered A–U, the rest discarded, 12 diamond gaps lettered a–l | nuna_master_reference.md |
| 24 arrangements | Lettered A–X, a separate sequence from the circles — flagged as a naming collision to watch for | nuna_master_reference.md |
| Exclusion rule | Shares a circle or a diamond; no exceptions found in either badge | nuna_master_reference.md |
| Missing exclusion | Table hand-built, systematically rechecked, one gap found and corrected — the pair is not named | nuna_master_reference.md |
| Count: 22,501 | Up to tier 10, verified in the same engine that returns 1,157 for the 16-point badge | claude/blue-memory-game-brief.md |
| Count: 25,000+ | Stated in the standing context block | Nuna-Starting-Kit (1).md |
| Silhouette | True badge shape derived the same exact-arc way as the 16-point version | nuna_master_reference.md |
| Artwork on file | The badge drawn with its centre tile filled and its surrounding circles in outline | 24 Point Badge.png |
| Shapes and families | 8,933 shapes, 4,459 families — reported by the Shapes tab, not on file | — |