diff --git a/game/scripts/avatar.gd b/game/scripts/avatar.gd
new file mode 100644
index 0000000..cde223c
--- /dev/null
+++ b/game/scripts/avatar.gd
@@ -0,0 +1,31 @@
+class_name Avatar
+extends Node2D
+## The local player's figure. It moves outside the lockstep simulation; only
+## its pickaxe hits become commands.
+
+## Walking speed: five tiles per second.
+const SPEED := 5.0 * GridMotion.TILE_SIZE
+const RADIUS := 5.0
+## Longest frame that moves the figure in one go. Longer frames are cut, so
+## that the figure cannot pass through a wall after a stall.
+const MAX_STEP_SECONDS := 0.1
+
+## Callable(x: int, y: int) -> bool that says whether a tile can be walked on.
+var is_walkable: Callable
+
+
+func _process(delta: float) -> void:
+	walk(Input.get_vector("move_left", "move_right", "move_up", "move_down"), delta)
+
+
+## Moves the figure in `direction` for `delta` seconds, sliding along walls.
+func walk(direction: Vector2, delta: float) -> void:
+	if direction == Vector2.ZERO or not is_walkable.is_valid():
+		return
+	var motion := direction * SPEED * minf(delta, MAX_STEP_SECONDS)
+	position = GridMotion.move(position, motion, RADIUS, is_walkable)
+
+
+func _draw() -> void:
+	draw_circle(Vector2.ZERO, RADIUS + 1.0, Color(0.15, 0.1, 0.05))
+	draw_circle(Vector2.ZERO, RADIUS, Color(0.95, 0.85, 0.6))
diff --git a/game/scripts/avatar.gd.uid b/game/scripts/avatar.gd.uid
new file mode 100644
index 0000000..f54956f
--- /dev/null
+++ b/game/scripts/avatar.gd.uid
@@ -0,0 +1 @@
+uid://bpltng2n5f61g
diff --git a/game/scripts/grid_motion.gd b/game/scripts/grid_motion.gd
new file mode 100644
index 0000000..75d9fc2
--- /dev/null
+++ b/game/scripts/grid_motion.gd
@@ -0,0 +1,44 @@
+class_name GridMotion
+extends RefCounted
+## Movement of a body over the tile grid. Figures move outside the lockstep
+## simulation, so this code may use floating point.
+
+## Edge length of a tile, in pixels.
+const TILE_SIZE := 16.0
+
+
+## The tile that contains `point`.
+static func tile_of(point: Vector2) -> Vector2i:
+	return Vector2i(floori(point.x / TILE_SIZE), floori(point.y / TILE_SIZE))
+
+
+## The centre of `tile`, in pixels.
+static func tile_center(tile: Vector2i) -> Vector2:
+	return (Vector2(tile) + Vector2(0.5, 0.5)) * TILE_SIZE
+
+
+## True if a square body of half width `radius` around `center` touches
+## only walkable tiles. `is_walkable` is a Callable(x: int, y: int) -> bool.
+static func fits(center: Vector2, radius: float, is_walkable: Callable) -> bool:
+	var first := tile_of(center - Vector2(radius, radius))
+	# The far edge lies just inside its tile, so a body that ends exactly on
+	# a tile border does not count the next tile.
+	var last := tile_of(center + Vector2(radius, radius) - Vector2(0.001, 0.001))
+	for y in range(first.y, last.y + 1):
+		for x in range(first.x, last.x + 1):
+			if not is_walkable.call(x, y):
+				return false
+	return true
+
+
+## Moves a body by `motion`, one axis after the other. An axis that would
+## end in a wall is skipped, so a body pushed into a wall slides along it.
+static func move(center: Vector2, motion: Vector2, radius: float, is_walkable: Callable) -> Vector2:
+	var result := center
+	var along_x := result + Vector2(motion.x, 0.0)
+	if fits(along_x, radius, is_walkable):
+		result = along_x
+	var along_y := result + Vector2(0.0, motion.y)
+	if fits(along_y, radius, is_walkable):
+		result = along_y
+	return result
diff --git a/game/scripts/grid_motion.gd.uid b/game/scripts/grid_motion.gd.uid
new file mode 100644
index 0000000..148586e
--- /dev/null
+++ b/game/scripts/grid_motion.gd.uid
@@ -0,0 +1 @@
+uid://dn0m8pxkigqig
diff --git a/game/scripts/tick_clock.gd b/game/scripts/tick_clock.gd
new file mode 100644
index 0000000..cd89d18
--- /dev/null
+++ b/game/scripts/tick_clock.gd
@@ -0,0 +1,29 @@
+class_name TickClock
+extends RefCounted
+## Turns frame time into a whole number of simulation ticks.
+
+## Most ticks one frame may run. After a long stall the simulation catches
+## up only this far and drops the rest, instead of freezing to catch up.
+const MAX_TICKS_PER_FRAME := 5
+
+var seconds_per_tick: float
+var paused := false
+
+var _accumulated := 0.0
+
+
+func _init(ticks_per_second: int) -> void:
+	seconds_per_tick = 1.0 / ticks_per_second
+
+
+## Returns how many ticks to run for a frame that took `delta` seconds.
+func advance(delta: float) -> int:
+	if paused:
+		return 0
+	_accumulated += delta
+	var ticks := int(_accumulated / seconds_per_tick)
+	if ticks > MAX_TICKS_PER_FRAME:
+		_accumulated = 0.0
+		return MAX_TICKS_PER_FRAME
+	_accumulated -= ticks * seconds_per_tick
+	return ticks
diff --git a/game/scripts/tick_clock.gd.uid b/game/scripts/tick_clock.gd.uid
new file mode 100644
index 0000000..717fbf3
--- /dev/null
+++ b/game/scripts/tick_clock.gd.uid
@@ -0,0 +1 @@
+uid://bmrv5nfb0wu6r
diff --git a/game/tests/test_avatar.gd b/game/tests/test_avatar.gd
new file mode 100644
index 0000000..87dbfa8
--- /dev/null
+++ b/game/tests/test_avatar.gd
@@ -0,0 +1,33 @@
+extends TestCase
+
+
+func _open_field(x: int, y: int) -> bool:
+	return x >= 0 and y >= 0 and x < 100 and y < 100
+
+
+func test_the_figure_walks_five_tiles_per_second() -> void:
+	var avatar := Avatar.new()
+	avatar.is_walkable = _open_field
+	avatar.position = GridMotion.tile_center(Vector2i(50, 50))
+	var start := avatar.position
+	for frame in 10:
+		avatar.walk(Vector2.RIGHT, 0.1)
+	check(is_equal_approx(avatar.position.x - start.x, 5.0 * GridMotion.TILE_SIZE), "five tiles in one second, moved %s" % (avatar.position.x - start.x))
+	avatar.free()
+
+
+func test_a_long_frame_moves_the_figure_no_further_than_a_tenth_of_a_second() -> void:
+	var avatar := Avatar.new()
+	avatar.is_walkable = _open_field
+	avatar.position = GridMotion.tile_center(Vector2i(50, 50))
+	var start := avatar.position
+	avatar.walk(Vector2.DOWN, 3.0)
+	check(is_equal_approx(avatar.position.y - start.y, Avatar.SPEED * Avatar.MAX_STEP_SECONDS), "cut to a tenth of a second")
+	avatar.free()
+
+
+func test_without_a_map_the_figure_stays() -> void:
+	var avatar := Avatar.new()
+	avatar.walk(Vector2.LEFT, 0.1)
+	check_eq(avatar.position, Vector2.ZERO, "no walkability callable")
+	avatar.free()
diff --git a/game/tests/test_avatar.gd.uid b/game/tests/test_avatar.gd.uid
new file mode 100644
index 0000000..41aa696
--- /dev/null
+++ b/game/tests/test_avatar.gd.uid
@@ -0,0 +1 @@
+uid://b7gc0kkgk5xg0
diff --git a/game/tests/test_grid_motion.gd b/game/tests/test_grid_motion.gd
new file mode 100644
index 0000000..569ef0c
--- /dev/null
+++ b/game/tests/test_grid_motion.gd
@@ -0,0 +1,46 @@
+extends TestCase
+## A 10 x 10 test grid in which tile (3, 1) and every tile outside are walls.
+
+const SIZE := GridMotion.TILE_SIZE
+const RADIUS := 5.0
+
+
+func _walkable(x: int, y: int) -> bool:
+	if x < 0 or y < 0 or x >= 10 or y >= 10:
+		return false
+	return Vector2i(x, y) != Vector2i(3, 1)
+
+
+func test_tile_of_and_tile_center_agree() -> void:
+	check_eq(GridMotion.tile_of(Vector2(0.0, 0.0)), Vector2i(0, 0), "origin")
+	check_eq(GridMotion.tile_of(Vector2(15.9, 16.0)), Vector2i(0, 1), "tile borders")
+	check_eq(GridMotion.tile_of(Vector2(-0.1, 3.0)), Vector2i(-1, 0), "left of the map")
+	check_eq(GridMotion.tile_center(Vector2i(2, 3)), Vector2(40.0, 56.0), "centre")
+	check_eq(GridMotion.tile_of(GridMotion.tile_center(Vector2i(7, 4))), Vector2i(7, 4), "round trip")
+
+
+func test_free_movement_covers_the_full_distance() -> void:
+	var start := GridMotion.tile_center(Vector2i(5, 5))
+	var moved := GridMotion.move(start, Vector2(3.0, -2.0), RADIUS, _walkable)
+	check_eq(moved, start + Vector2(3.0, -2.0), "unblocked")
+
+
+func test_a_wall_stops_the_body_before_it() -> void:
+	var position := GridMotion.tile_center(Vector2i(1, 1))
+	for step in 40:
+		position = GridMotion.move(position, Vector2(1.0, 0.0), RADIUS, _walkable)
+	check(position.x + RADIUS <= 3.0 * SIZE, "the body stays left of the wall at x = 48, ends at %s" % position.x)
+	check(position.x + RADIUS > 3.0 * SIZE - 1.0, "and gets close to it, ends at %s" % position.x)
+
+
+func test_a_body_slides_along_a_wall() -> void:
+	var start := GridMotion.tile_center(Vector2i(2, 1))
+	var moved := GridMotion.move(start, Vector2(4.0, 2.0), RADIUS, _walkable)
+	check_eq(moved.x, start.x, "the blocked axis stays")
+	check_eq(moved.y, start.y + 2.0, "the free axis moves")
+
+
+func test_the_edge_of_the_map_is_a_wall() -> void:
+	var start := Vector2(RADIUS, RADIUS)
+	var moved := GridMotion.move(start, Vector2(-1.0, -1.0), RADIUS, _walkable)
+	check_eq(moved, start, "no step off the map")
diff --git a/game/tests/test_grid_motion.gd.uid b/game/tests/test_grid_motion.gd.uid
new file mode 100644
index 0000000..9ac22a8
--- /dev/null
+++ b/game/tests/test_grid_motion.gd.uid
@@ -0,0 +1 @@
+uid://bjgyllbxp5ty8
diff --git a/game/tests/test_tick_clock.gd b/game/tests/test_tick_clock.gd
new file mode 100644
index 0000000..5c7f463
--- /dev/null
+++ b/game/tests/test_tick_clock.gd
@@ -0,0 +1,31 @@
+extends TestCase
+
+
+func test_one_tick_lasts_a_twentieth_of_a_second() -> void:
+	var clock := TickClock.new(20)
+	check_eq(clock.advance(0.05), 1, "a full tick")
+	check_eq(clock.advance(0.025), 0, "half a tick")
+	check_eq(clock.advance(0.025), 1, "the second half completes the tick")
+
+
+func test_a_second_of_short_frames_runs_about_twenty_ticks() -> void:
+	var clock := TickClock.new(20)
+	var ticks := 0
+	for frame in 60:
+		ticks += clock.advance(1.0 / 60.0)
+	# Rounding in floating point may leave the last tick for the next frame.
+	check(ticks == 19 or ticks == 20, "19 or 20 ticks, got %d" % ticks)
+
+
+func test_a_long_stall_runs_at_most_five_ticks_and_forgets_the_rest() -> void:
+	var clock := TickClock.new(20)
+	check_eq(clock.advance(10.0), TickClock.MAX_TICKS_PER_FRAME, "capped")
+	check_eq(clock.advance(0.0), 0, "the rest of the stall is dropped")
+
+
+func test_a_paused_clock_runs_no_ticks() -> void:
+	var clock := TickClock.new(20)
+	clock.paused = true
+	check_eq(clock.advance(1.0), 0, "paused")
+	clock.paused = false
+	check_eq(clock.advance(0.05), 1, "running again")
diff --git a/game/tests/test_tick_clock.gd.uid b/game/tests/test_tick_clock.gd.uid
new file mode 100644
index 0000000..d61e09a
--- /dev/null
+++ b/game/tests/test_tick_clock.gd.uid
@@ -0,0 +1 @@
+uid://bhwedt1ik28sq