Try everything, then check it exactly
The machine tries every legal move, in order of length, until it reaches the goal. The winner is then rebuilt in exact arithmetic, where it either works or does not.
| Problem | Euclid | His moves | Fewest | Status | Cost | Checked |
|---|---|---|---|---|---|---|
| equilateral triangle | I.1 | 5 | 2 | fewest possible | 2C1 + 2C2 + 2C3 | yes |
| perpendicular bisector | I.10 | 29 | 3 | fewest possible | 2S1 + 1S2 + 2C1 + 2C2 + 2C3 | yes |
| midpoint | I.10 | 29 | 4 | fewest possible | 4S1 + 2S2 + 2C1 + 2C2 + 2C3 | yes |
| perpendicular at a point | I.11 | 21 | 5 | fewest possible | 2S1 + 1S2 + 4C1 + 4C2 + 4C3 | yes |
| double a segment | I.3 | 13 | 2 | fewest possible | 2S1 + 1S2 + 1C1 + 1C2 + 1C3 | yes |
| square on a segment | I.46 | 37 | 5 | fewest possible | 2S1 + 1S2 + 4C1 + 4C2 + 4C3 | yes |
| equilateral triangle (compass-only) | I.1 | 5 | 2 | fewest possible | 2C1 + 2C2 + 2C3 | yes |
| double a segment (compass-only) | I.3 | 13 | 3 | fewest possible | 3C1 + 3C2 + 3C3 | yes |
| midpoint (compass-only) | I.10 | 29 | 6 | fewest possible | 6C1 + 6C2 + 6C3 | yes |
The two counts measure different things, and the gap is mostly that. His moves counts every line and circle drawn once the earlier propositions he leans on are expanded — his I.10 reaches through three other results, so fifteen things get drawn. Fewest stops as soon as the goal exists and draws nothing to tidy up.
More to the point, Euclid was building constructions that could be proved from what came before. Reusing an earlier result costs moves but buys certainty. Counting moves was a nineteenth-century preoccupation, not his.
A construction is a program, and these are different machines to run it on: compass and straightedge, compass alone, straightedge alone with one circle given, or a compass stuck at one setting. The classical theorems say the compass alone loses nothing, and the search agrees — at a price in moves.
Three different things, and they used all to be called fewest possible. Fewest possible means every shorter figure was enumerated in the exact kernel and none reached the goal. No floating point enters that claim, and neither do the bounds below: the exact enumeration applies no scope limit, no point limit and no deduplication at all. It is a theorem.
Shortest found. The search covered every shorter figure it could represent, but in floating point. A point pair merged at a tolerance of 10⁻⁷, a tangency lost at 10⁻⁹, or two different figures sharing a rounded fingerprint would each hide a construction and turn a longer answer into a false minimum. Certifying these depths exactly is out of reach.
Upper bound. The search hit its node budget. Only the construction shown is meaningful.
The float search is what makes the problem tractable at all, and it carries two further limits: points straying far outside the figure are ignored, and a figure of more than twenty points is abandoned. Neither touches a fewest possible row.