717 connected shapes
Claim
Of the 1,157 combinations, exactly 717 fuse into a single connected shape.
Construction
Every tile in a combination docks to the centre, so counting the centre in would make all 1,157 connected and the question empty. The test therefore leaves the centre out and asks whether the docked tiles hold together among themselves.
Contact comes in two kinds. An edge join is one tile's diamond sitting one step from another's circle, the two boundaries sharing a full arc. A point join is two circles on diagonal nodes, exactly tangent, or two diamonds sharing a single corner.
Proof
For each combination, build the contact graph on its docked tiles — the centre excluded — and count the combinations whose graph is connected. The technical reference gives the tier-by-tier result: 16, 40, 100, 223, 274 and 64. Leave the centre in and the count collapses to a trivial 1,157; that single qualification is what makes 717 reproducible.
16 + 40 + 100 + 223 + 274 + 64 = 717
The remaining 440 are legal arrangements whose tiles leave the figure in separate pieces. Note what happens at the top: at five and six tiles every legal arrangement is connected, 274 of 274 and 64 of 64, so above four tiles the connectivity test stops doing any work.
Why it matters downstream
This number matters more to the physical work than to the mathematics. A shape joined only at a point cannot be cast, cut or handled as one piece, so 717 is the count of combinations that could become an object — and even that is generous, since it counts point joins as contact.
Note
Aaron's slot.
Open questions
- Whether point joins should count as contact at all. The decision is discretionary and it is the likely source of the family-count dispute recorded on the combination-shapes page.
- Why the threshold falls between four and five tiles rather than somewhere else. Observed in the data, not derived.
Checks
| What | How | Source |
|---|---|---|
| Tier sums | 16 + 40 + 100 + 223 + 274 + 64 = 717, recomputed from the published table | nuna-combination-shapes-reference.md |
| Subset of Claim I | 717 ≤ 1,157 necessarily; both appear in the same reference | nuna-combination-shapes-reference.md |
| Contact definition | Edge join and point join, stated in the construction section of the reference | nuna-combination-shapes-reference.md |
| Centre excluded | “The center tile isn’t counted in connectivity”; the rule reproduces 717 only with the centre left out | nuna-combination-shapes-reference.md |