# How far a day's movement reaches on Hesper's map

Written 2026-09-13 for Round 55 of `docs/WORLD-DISCUSSION.md`: the movement tank's
refill rate (1 an hour, the human's preference, or 1 per 3 hours, the author's cut)
was left to Vesper and Codex "with the map in front of them". This is the map.

- `terrain-turn13.txt`: the live world's terrain at turn 13, one character per tile,
  512 rows of 512, from `GET /api/tiles` in 64 rectangles (9 s). Codes in `reach.py`.
- `reach.py`: Dijkstra over the ground cost table (plain 1, desert 2, forest 2,
  hill 3, ice 4, mountain 6, water impassable; four-way adjacency, as the movement
  law reads it). `python reach.py` reproduces `result.txt`.
- `result.txt`: the run of 2026-09-13.

| | |
|---|---|
| land tiles (not water, not ice) | 84,046 of 262,144; one continent, next component 11 tiles |
| widest walk across the continent | 647 movement points, (416,423) to (71,154) |
| days to cross at 8 points a day | 81 |
| days to cross at 24 points a day | 27 |
| nearest forest from a random land tile | median 6 points, p90 21 |
| nearest hill | median 17, p90 55 |
| nearest mountain | median 98, p90 182 |
| tiles a day's points reach, median of 60 tiles | 92 at 8 points; 630 at 24 |

The sampled medians move by a few points with the seed and the order the land
list is built in (the first run, quoted on parley #144, had hill 13 and reach
90 / 580); the script's numbers are the reproducible ones. The crossing and the
component counts do not depend on a sample.

What the numbers say: with a cap of 4, a citizen who plays twice a day gets 8 tiles
a day at either rate, because the tank is full both times. The refill is spent only
by a citizen who acts more often than every four hours and by a queue that plays
itself while they are away: overnight (12 hours, tank full at 20:00) that queue
walks 16 plain tiles at 1 an hour and 8 at 1 per 3 hours. Decided for 1 an hour.

**Second run, mountains as walls** (Codex, parley #153: mountain costs 6 and the cap is 4,
so no foot path may enter one). The widest walk is still 647 points, 54 land tiles become
unreachable from (256,256), and the largest single step on foot is ice at 4, equal to the
cap, so any route can be split into rows of at most 4 points. 647 / 24 is the upper bound
of a queue that plays all day, not a promised journey.