The 16-point badge
Sixteen arrangements, one exclusion rule, 1,157 combinations, and six as the ceiling. The system this library is filed by.
What it is
Sixteen arrangements around one centre tile, lettered A to P. Between them they use eight distinct circles and eight distinct diamonds, which is why they conflict at all: three of them want the diamond at (0, −1), three want (−1, 0), and three want (1, 0).
Two arrangements exclude each other when they share a circle or share a diamond. That rule, applied to the sixteen, gives 22 exclusions and 1,157 legal combinations.
The sixteen
Each arrangement is a hub circle, a rotation, and the diamond it reaches for. Nodes are in lattice steps of s = r√2, with the centre tile's circle at the origin and its diamond at (0, 1).
| Mark | Hub | Circle | Rot | Diamond | Excludes | |
|---|---|---|---|---|---|---|
| A | D | (0, -2) | 0° | (0, -1) | B, P | |
| B | G | (1, -1) | 90° | (0, -1) | A, C, P | |
| C | G | (1, -1) | 0° | (1, 0) | B, D, E | |
| D | I | (2, 0) | 90° | (1, 0) | C, E | |
| E | K | (1, 1) | 180° | (1, 0) | C, D, F, G | |
| F | K | (1, 1) | 270° | (2, 1) | E, G | |
| G | K | (1, 1) | 0° | (1, 2) | E, F, H | |
| H | M | (0, 2) | 270° | (1, 2) | G, I, J | |
| I | M | (0, 2) | 0° | (0, 3) | H, J | |
| J | M | (0, 2) | 90° | (-1, 2) | H, I, K | |
| K | J | (-1, 1) | 0° | (-1, 2) | J, L, M | |
| L | J | (-1, 1) | 90° | (-2, 1) | K, M | |
| M | J | (-1, 1) | 180° | (-1, 0) | K, L, N, O | |
| N | H | (-2, 0) | 270° | (-1, 0) | M, O | |
| O | F | (-1, -1) | 0° | (-1, 0) | M, N, P | |
| P | F | (-1, -1) | 270° | (0, -1) | A, B, O |
The exclusions in the last column are not a separate table. They are derived from the circle and diamond columns by the rule above, and doing that derivation reproduces the badge's own exclusion key exactly, with no discrepancies.
Build one
The keypad below is the badge's interaction, rebuilt from the same data as the figures: select a key to place that tile, and the keys that would then want the same circle or diamond grey out.
∅
Select keys to build a combination. With JavaScript off this panel stays empty and the worked examples below are drawn as figures.
Worked examples, drawn at build time · A, A·D, A·D·G, A·D·G·L
The count
Build a graph on sixteen vertices, one per arrangement, with an edge wherever two arrangements share a circle or a diamond: 22 edges. A legal combination is an independent set in that graph, and sixteen vertices is small enough to settle exhaustively.
| Tiles | Combinations | |
|---|---|---|
| 1 | 16 | the base arrangements, one tile each |
| 2 | 98 | pairs — the tier where legality and wholeness most disagree |
| 3 | 288 | |
| 4 | 417 | the widest tier |
| 5 | 274 | |
| 6 | 64 | the ceiling: no legal combination holds seven |
| all | 1,157 |
Six is the ceiling. No independent set of seven exists, which is what closed the library's original shelf scheme at six. Book I carries the derivation; this page carries the table.
Two corrections on the record
The interactive model carries an audit note against the source spreadsheet, and both corrections are in the data this page is built from:
ENP was wrongly marked excluded. It is valid, and the enumeration here returns it.
GJLNP was mislabelled as a duplicate of the unrelated CJLNP. They are different combinations.
The model also computes its total at load rather than quoting it, which is what makes it an independent check. The 16-point badge model has the rest of that page's reference.
Note
Aaron's slot.
Open questions
- The badge silhouette — the outline of all sixteen filled at once — exists as an exact-arc derivation (the badge shapes), but the copy embedded in the interactive model is a traced path mapped in from a drawing at a different radius. The exact version and the embedded version are two objects.
- Nothing on this page draws the sixteen in their ring order. The arrangement midpoints form a closed star ring with six chords across it, and that structure is recorded in the geometry constants but not pictured anywhere.
Checks
| What | How | Source |
|---|---|---|
| The sixteen tiles | Hub, circle node, rotation and diamond for each of A–P | nuna-combination-shapes-reference.md |
| Table against the geometry | All sixteen hub letters and rotations match the library's own arrangement table; all sixteen circle nodes match its hub coordinates | _src/core.py |
| Exclusions are derived, not quoted | Deriving “shares a circle or a diamond” from the table reproduces the exclusion key exactly | nuna-combination-shapes-reference.md + _src/core.py |
| Edges | 22, recomputed | _src/core.py |
| Totals | 16 · 98 · 288 · 417 · 274 · 64 = 1,157, recomputed by exhaustive independent-set search | _src/core.py |
| Ceiling | No independent set of size seven exists | _src/core.py |
| Eight circles, eight diamonds | Counted from the table: 8 distinct hubs, 8 distinct diamond nodes | nuna-combination-shapes-reference.md |
| ENP and GJLNP | “ENP was wrongly marked excluded (it's valid), and GJLNP was mislabeled as a duplicate of the unrelated combo CJLNP” | nuna-badge-interactive (1).html |
| Live enumeration in the model | Computed at load from the key, not copied from the spreadsheet | nuna-badge-interactive (1).html |
| Prior assertion | 1,157, spreadsheet verified | Nuna-Starting-Kit (1).md |