aspirinV2: tap to start with a random solvable deal

The first tap (or SHUFFLE) loads the solved board and deals it with a
Fisher-Yates pass driven by the free-running logistic-map RNG, so tap
timing picks the board. A parity latch adds one swap of cells 1 and 2
when needed, so every deal is solvable.

Tile moves and deal swaps share one cell-swap mechanism, cutting RAM
from 56.4 kB (failed to start on the Orion) to 34.6 kB (runs). Adds
deal checks to verify_logic.py --v2 and the hint label to the viewer.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

verify_logic.py

@@ -345,7 +345,7 @@ def main():
     print("re-shuffle OK")
 
 
-def astar(s0):
+def astar(s0, weight=1.5):
     """Weighted A* on a 16-tuple (0 = blank); returns the list of cell indices to press."""
     import heapq
     goal = tuple(list(range(1, 16)) + [0])
@@ -370,11 +370,19 @@ def astar(s0):
             ns = tuple(l)
             if seen.get(ns, 1e9) > gc + 1:
                 seen[ns] = gc + 1
-                heapq.heappush(pq, (gc + 1 + 1.5 * h(ns), gc + 1, ns, p + [n]))
+                heapq.heappush(pq, (gc + 1 + weight * h(ns), gc + 1, ns, p + [n]))
         if len(seen) > 2_000_000:
             return None
 
 
+def solvable(s):
+    """4x4: permutation parity (blank as 16) == parity of the blank's distance from home."""
+    perm = [v if v else 16 for v in s]
+    inv = sum(1 for a in range(16) for b in range(a + 1, 16) if perm[a] > perm[b])
+    z = s.index(0)
+    return inv % 2 == (3 - z // 4 + 3 - z % 4) % 2
+
+
 def main_v2():
     """aspirinV2.sax: fixed cells, numbers shown via each cell's `in`, empty cell recoloured."""
     tops, paths = load("aspirinV2.sax")
@@ -416,11 +424,18 @@ def main_v2():
         cell[k].v["touch"] = False
         step(2)
 
+    hint = paths["/ox/Game/Hint"]
     step(3)
     assert state() == goal, "boot did not load the solved board: %s" % (state(),)
-    step(120)
+    step(random.randint(20, 80))           # idle: no shuffle until the first tap
     check()
-    print("after shuffle:\n" + show())
+    assert state() == goal and hint.g("in", False) and not solved() and moves() == 0
+    press(0)                               # first tap (cell 1, not next to the gap) deals
+    assert not hint.g("in", False)
+    step(30)
+    check()
+    print("after deal:\n" + show())
+    assert solvable(state()), "dealt an unsolvable board"
     assert state() != goal and not solved() and moves() == 0
 
     z = state().index(0)
@@ -442,7 +457,7 @@ def main_v2():
     assert state() == before and moves() == 2, "hold did not move exactly once"
     check()
 
-    path = astar(state())
+    path = astar(state(), weight=4)        # random deals are deep; any solution will do
     assert path is not None, "solver gave up"
     print("solving in", len(path), "presses")
     for k in path:
@@ -456,10 +471,28 @@ def main_v2():
     paths["/ox/Game/Shuffle"].v["touch"] = True
     step(2)
     paths["/ox/Game/Shuffle"].v["touch"] = False
-    step(120)
+    step(30)
     check()
-    assert state() != goal and not solved() and moves() == 0
-    print("re-shuffle OK")
+    assert state() != goal and not solved() and moves() == 0 and solvable(state())
+    print("re-deal OK")
+
+    # many deals: always solvable, all different, every number reaches every cell
+    seen, where = set(), [set() for _ in range(16)]
+    for _ in range(300):
+        step(random.randint(1, 40))
+        paths["/ox/Game/Shuffle"].v["touch"] = True
+        step(1)
+        paths["/ox/Game/Shuffle"].v["touch"] = False
+        step(20)
+        s = state()
+        check()
+        assert solvable(s), s
+        seen.add(s)
+        assert s[15] == 0, "deal moved the blank out of cell 16"
+        for k, v in enumerate(s):
+            where[k].add(v)
+    assert all(len(w) == 15 for w in where[:15]), [len(w) for w in where]
+    print(f"300 deals: {len(seen)} distinct, all solvable, every number seen in cells 1-15")
 
 
 if __name__ == "__main__":