from manim import * import numpy as np config.frame_width = 16 config.frame_height = 9 config.pixel_width = 1920 config.pixel_height = 1080 config.background_color = "#0E1116" SAFE_X = config.frame_width / 2 - 0.35 SAFE_Y = config.frame_height / 2 - 0.35 SAFE_Y_BOTTOM = config.frame_height / 2 - 0.7 TEXT_MAX_W = config.frame_width - 1.4 LAPIS = "#7CA6EA" EMER = "#46C58C" GOLD = "#E0AF48" REDD = "#EC7C6C" DIM = "#5A6472" MONO = "Monospace" M = 25214903917 A = 11 MASK = (1 << 48) - 1 def mono(t, size=34, color=WHITE): x = Text(t, font=MONO, font_size=size, color=color) if x.width > TEXT_MAX_W: x.scale_to_fit_width(TEXT_MAX_W) return x def cap(t, size=30, color=DIM): return mono(t, size, color) def title(t): x = Text(t, font=MONO, font_size=38, color=WHITE) x.to_edge(UP, buff=0.45) return x def java_next(seed, bits): seed = (seed * M + A) & MASK return seed, seed >> (48 - bits) def next_int10(seed): seed = (seed ^ M) & MASK while True: seed, r = java_next(seed, 31) val = r % 10 if r - val + 9 <= 0x7FFFFFFF: return val def is_slime(world_seed, x, z): s = (world_seed + x * x * 4987142 + x * 5947611 + z * z * 4392871 + z * 389711) & ((1 << 64) - 1) s = s ^ 987234911 return next_int10(s & MASK) == 0 class MinecraftDice(Scene): def clear_all(self, t=0.6): if self.mobjects: self.play(*[FadeOut(m) for m in self.mobjects], run_time=t) def construct(self): self.opening() self.scene1() self.scene2() self.scene3() self.scene4() self.scene5() self.scene6() self.scene7() self.scene8() self.scene9() def opening(self): t = Text("Running Minecraft's dice backwards", font=MONO, font_size=46, color=LAPIS) t.scale_to_fit_width(min(TEXT_MAX_W, 11)) sub = cap("one multiply, one add, one wrap", 28, DIM) sub.next_to(t, DOWN, buff=0.5) self.play(Write(t), run_time=1.6) self.play(FadeIn(sub, shift=UP * 0.2)) self.wait(1.2) self.clear_all() def scene1(self): ttl = title("the rule") self.play(FadeIn(ttl, shift=DOWN * 0.2)) box = RoundedRectangle(width=7.6, height=1.5, corner_radius=0.1, stroke_color=LAPIS, stroke_width=3, fill_color="#141A22", fill_opacity=1) box.move_to(UP * 1.55) box val = mono("25214902975", 46, WHITE).move_to(box) val lab = cap("state ( 48 bits )", 26, LAPIS).next_to(box, UP, buff=0.3) self.play(Create(box), FadeIn(lab)) self.play(Write(val)) self.wait(0.6) rule = VGroup( mono("state", 40, LAPIS), mono("<-", 40, DIM), mono("( state", 40, LAPIS), mono("* 25214903917", 40, WHITE), mono("+ 11", 40, WHITE), mono(")", 40, LAPIS), mono("mod 2^48", 40, GOLD), ).arrange(RIGHT, buff=0.22) if rule.width > TEXT_MAX_W: rule.scale_to_fit_width(TEXT_MAX_W) rule.move_to(DOWN * 0.55) self.play(FadeIn(rule[0]), FadeIn(rule[1]), FadeIn(rule[2])) self.wait(0.3) for i in (3, 4, 5, 6): self.play(FadeIn(rule[i], shift=RIGHT * 0.25), run_time=0.55) self.wait(0.4) names = VGroup( mono("25214903917 = the multiplier", 28, WHITE), mono("11 = the addend", 28, WHITE), mono("2^48 = the wrap", 28, GOLD), ).arrange(DOWN, buff=0.3, aligned_edge=LEFT) names.next_to(rule, DOWN, buff=0.7).set_x(0) self.play(LaggedStart(*[FadeIn(n, shift=UP * 0.15) for n in names], lag_ratio=0.35)) self.wait(1.6) self.clear_all() def scene2(self): ttl = title("the wrap") self.play(FadeIn(ttl, shift=DOWN * 0.2)) prod = 25214902975 * M + A eq = mono("25214902975 * 25214903917 + 11", 32, LAPIS) res = mono(str(prod), 38, WHITE) top = VGroup(eq, res).arrange(DOWN, buff=0.35) top.next_to(ttl, DOWN, buff=0.45).set_x(0) self.play(Write(eq)) self.play(FadeIn(res, shift=UP * 0.2)) self.wait(0.6) bits = bin(prod)[2:] cells = VGroup() for i, b in enumerate(bits): c = Square(side_length=0.34, stroke_width=1.2, fill_opacity=1) col = REDD if i < 22 else EMER c.set_fill(col, opacity=0.9 if b == "1" else 0.16) c.set_stroke(col, 1.2, opacity=0.6) cells.add(c) cells.arrange_in_grid(rows=2, cols=35, buff=0.05) cells.scale_to_fit_width(min(TEXT_MAX_W, 12.6)) cells.next_to(top, DOWN, buff=0.6).set_x(0) self.play(LaggedStart(*[FadeIn(c, scale=0.7) for c in cells], lag_ratio=0.012), run_time=1.8) lg = VGroup(mono("top 22 bits", 24, REDD), mono("bottom 48 bits", 24, EMER)) lg.arrange(RIGHT, buff=1.4).next_to(cells, DOWN, buff=0.45).set_x(0) self.play(FadeIn(lg)) self.wait(1.0) red = VGroup(*cells[:22]) self.play(FadeOut(lg)) self.remove(lg) self.play(red.animate.set_stroke(REDD, 2.5, opacity=1), run_time=0.5) self.play(LaggedStart(*[FadeOut(c, shift=DOWN * 1.0, scale=0.6) for c in red], lag_ratio=0.03), run_time=1.8) self.remove(red) self.play(FadeOut(res), FadeOut(eq)) self.remove(res, eq) remain = mono("181996700050782", 44, EMER) remain.next_to(cells, DOWN, buff=0.6).set_x(0) self.play(FadeIn(remain, shift=UP * 0.2)) c2 = cap("22 bits thrown on the floor", 30, REDD) c2.next_to(remain, DOWN, buff=0.5).set_x(0) if c2.get_bottom()[1] < -SAFE_Y_BOTTOM: c2.shift(UP * (-SAFE_Y_BOTTOM - c2.get_bottom()[1])) self.play(FadeIn(c2)) self.wait(1.6) self.clear_all() def scene3(self): ttl = title("determinism") self.play(FadeIn(ttl, shift=DOWN * 0.2)) start = mono("start 25214902975", 30, LAPIS) start.next_to(ttl, DOWN, buff=0.4) self.play(FadeIn(start)) states = [181996700050782, 73124365360401, 267783390291528, 66833641684147] outs = [2777049256, 1115789266, 4086050266, 1019800440] head = VGroup(mono("state", 24, DIM), mono("output = state >> 16", 24, DIM)) head[0].move_to(LEFT * 3.6) head[1].move_to(RIGHT * 3.6) rows = VGroup() for s, o in zip(states, outs): a = mono(str(s), 30, WHITE).move_to(LEFT * 3.6) arw = Arrow(LEFT * 0.4, RIGHT * 0.4, buff=0, color=LAPIS, stroke_width=3, max_tip_length_to_length_ratio=0.4) b = mono(str(o), 32, EMER).move_to(RIGHT * 3.6) rows.add(VGroup(a, arw, b)) table = VGroup(head, *rows).arrange(DOWN, buff=0.42) table.next_to(start, DOWN, buff=0.5).set_x(0) self.play(FadeIn(head)) for r in rows: self.play(FadeIn(r[0], shift=RIGHT * 0.2), run_time=0.32) self.play(GrowArrow(r[1]), FadeIn(r[2], shift=RIGHT * 0.2), run_time=0.38) self.wait(0.8) self.play(*[r.animate.set_opacity(0.12) for r in rows], run_time=0.6) self.play(Indicate(start, color=GOLD, scale_factor=1.08), run_time=1.0) for r in rows: self.play(r.animate.set_opacity(1), run_time=0.28) self.wait(0.4) c = cap("same start, same numbers, forever", 30, LAPIS) c.to_edge(DOWN, buff=0.75) self.play(FadeIn(c)) self.wait(1.4) self.clear_all() def scene4(self): ttl = title("why it can be undone") self.play(FadeIn(ttl, shift=DOWN * 0.2)) N = 7 left = VGroup(*[Square(0.22, stroke_width=0, fill_color=LAPIS, fill_opacity=0.9) for _ in range(N)]).arrange(DOWN, buff=0.2) right = VGroup(*[Square(0.22, stroke_width=0, fill_color=EMER, fill_opacity=0.9) for _ in range(N)]).arrange(DOWN, buff=0.2) left.move_to(LEFT * 5.0 + UP * 1.15) right.move_to(RIGHT * 5.0 + UP * 1.15) lt = cap("states before", 22, LAPIS).next_to(left, UP, buff=0.28) rt = cap("states after", 22, EMER).next_to(right, UP, buff=0.28) self.play(FadeIn(left), FadeIn(right), FadeIn(lt), FadeIn(rt)) perm = [3, 0, 5, 1, 6, 2, 4] arrows = VGroup(*[Line(left[i].get_right() + RIGHT * 0.12, right[perm[i]].get_left() + LEFT * 0.12, stroke_width=2, color="#3E6EA8") for i in range(N)]) self.play(LaggedStart(*[Create(a) for a in arrows], lag_ratio=0.18), run_time=2.2) self.wait(0.8) reason = VGroup( mono("25214903917 is odd", 26, WHITE), mono("odd => invertible mod 2^48", 26, GOLD), mono("the step is a bijection", 26, EMER), ).arrange(DOWN, buff=0.3) reason.next_to(VGroup(left, right, arrows), DOWN, buff=0.75).set_x(0) for r in reason: self.play(FadeIn(r, shift=UP * 0.15), run_time=0.55) self.wait(0.6) c = cap("nothing is ever lost from the state", 30, EMER) c.to_edge(DOWN, buff=0.75) self.play(FadeIn(c)) self.wait(1.6) self.clear_all() def scene5(self): ttl = title("the inverse") self.play(FadeIn(ttl, shift=DOWN * 0.2)) prop = mono("25214903917 * 246154705703781 mod 2^48 = 1", 32, GOLD) prop.next_to(ttl, DOWN, buff=0.55).set_x(0) self.play(Write(prop), run_time=1.6) self.wait(0.7) back = mono("previous = ( current - 11 ) * 246154705703781 mod 2^48", 28, EMER) back.next_to(prop, DOWN, buff=0.5).set_x(0) self.play(FadeIn(back, shift=UP * 0.15)) self.wait(0.8) steps = VGroup( mono("181996700050782", 34, WHITE), mono("- 11 -> 181996700050771", 28, DIM), mono("* 246154705703781", 28, DIM), mono("mod 2^48 -> 25214902975", 34, EMER), ).arrange(DOWN, buff=0.32) steps.next_to(back, DOWN, buff=0.55).set_x(0) for s in steps: self.play(FadeIn(s, shift=UP * 0.15), run_time=0.55) self.play(Indicate(steps[3], color=EMER, scale_factor=1.06), run_time=1.0) arw = Arrow(RIGHT * 2.2, LEFT * 2.2, buff=0, color=EMER, stroke_width=6) arw.next_to(steps, DOWN, buff=0.4).set_x(0) if arw.get_bottom()[1] < -SAFE_Y_BOTTOM: arw.shift(UP * (-SAFE_Y_BOTTOM - arw.get_bottom()[1])) self.play(GrowArrow(arw)) self.wait(1.8) self.clear_all() def scene6(self): ttl = title("the catch") self.play(FadeIn(ttl, shift=DOWN * 0.2)) W = 11.5 top = Rectangle(width=W * 32 / 48, height=1.0, stroke_width=2, stroke_color=LAPIS, fill_color=LAPIS, fill_opacity=0.35) bot = Rectangle(width=W * 16 / 48, height=1.0, stroke_width=2, stroke_color=GOLD, fill_color=GOLD, fill_opacity=0.35) bar = VGroup(top, bot).arrange(RIGHT, buff=0) bar.next_to(ttl, DOWN, buff=0.8).set_x(0) blab = cap("48-bit state", 22, DIM).next_to(bar, UP, buff=0.25) top bot n1 = mono("32 bits", 22, WHITE).move_to(top) n1 n2 = mono("16 bits", 22, WHITE).move_to(bot) n2 l1 = mono("you see this", 26, LAPIS).next_to(top, DOWN, buff=0.28) l2 = mono("hidden", 26, GOLD).next_to(bot, DOWN, buff=0.28) self.play(FadeIn(blab), Create(top), Create(bot)) self.play(FadeIn(n1), FadeIn(n2)) self.play(FadeIn(l1, shift=UP * 0.15), FadeIn(l2, shift=UP * 0.15)) self.wait(1.4) c1 = cap("16 bits are withheld on every single call", 28, GOLD) c1.next_to(l1, DOWN, buff=0.6).set_x(0) self.play(FadeIn(c1)) self.wait(1.8) dots = VGroup() rng = np.random.default_rng(7) for _ in range(260): a = rng.uniform(0, TAU) r = np.sqrt(rng.uniform(0, 1)) dots.add(Dot(radius=0.028, color=GOLD, fill_opacity=0.75).move_to( np.array([r * np.cos(a) * 5.2, r * np.sin(a) * 0.85, 0]))) dots.next_to(c1, DOWN, buff=0.45).set_x(0) self.play(LaggedStart(*[FadeIn(d, scale=0.4) for d in dots], lag_ratio=0.004), run_time=2.0) cnt = cap("2^16 = 65536 states fit one output", 28, GOLD) cnt.next_to(dots, DOWN, buff=0.35).set_x(0) if cnt.get_bottom()[1] < -SAFE_Y_BOTTOM: cnt.shift(UP * (-SAFE_Y_BOTTOM - cnt.get_bottom()[1])) self.play(FadeIn(cnt)) self.wait(2.6) self.clear_all() def scene7(self): ttl = title("each observation cuts") self.play(FadeIn(ttl, shift=DOWN * 0.2)) center = UP * 0.55 rng = np.random.default_rng(3) pts = [] while len(pts) < 800: p = rng.uniform(-1, 1, 2) * np.array([3.4, 1.9]) if (p[0] / 3.4) ** 2 + (p[1] / 1.9) ** 2 <= 1: pts.append(p) pts = np.array(pts) cloud = VGroup(*[Dot(point=center + np.array([p[0], p[1], 0]), radius=0.032, color=GOLD, fill_opacity=0.8) for p in pts]) lab = cap("candidates after 1 observation", 26, GOLD) lab.next_to(cloud, DOWN, buff=0.6).set_x(0) self.play(LaggedStart(*[FadeIn(d, scale=0.4) for d in cloud], lag_ratio=0.002), run_time=1.8) self.play(FadeIn(lab)) self.wait(1.6) cx = np.array([0.5, -0.1]) radii = [1.9, 1.0, 0.5] texts = ["2 observations", "3 observations", "4 observations"] for k, rad in enumerate(radii): ring = Ellipse(width=rad * 3.0, height=rad * 1.9, color=EMER, stroke_width=3) ring.move_to(center + np.array([cx[0], cx[1], 0])) self.play(Create(ring), run_time=0.8) anims = [] for d, p in zip(cloud, pts): if d.fill_opacity < 0.1: continue inside = ((p[0] - cx[0]) / (rad * 1.5)) ** 2 + ((p[1] - cx[1]) / (rad * 0.95)) ** 2 <= 1 anims.append(d.animate.set_color(EMER) if inside else d.animate.set_opacity(0.0)) self.play(LaggedStart(*anims, lag_ratio=0.0006), run_time=1.4) newlab = cap("candidates after " + texts[k], 26, EMER).move_to(lab) self.play(FadeOut(ring), Transform(lab, newlab), run_time=0.6) self.wait(0.8) c = cap("each new observation cuts, it does not add", 30, EMER) c.to_edge(DOWN, buff=0.75) self.play(FadeIn(c)) self.wait(2.4) self.clear_all() def scene8(self): ttl = title("slime chunks") self.play(FadeIn(ttl, shift=DOWN * 0.2)) rule = VGroup( mono("seed + chunk x,z -> java.util.Random", 26, LAPIS), mono("nextInt(10) == 0 -> slime chunk", 26, EMER), ).arrange(DOWN, buff=0.26) rule.next_to(ttl, DOWN, buff=0.4).set_x(0) self.play(FadeIn(rule[0])) self.play(FadeIn(rule[1])) n = 16 cell = 0.28 grid = VGroup() for i in range(n): for j in range(n): sq = Square(cell, stroke_width=1, stroke_color="#1E2833", fill_color="#161C24", fill_opacity=1) sq.move_to(np.array([(j - (n - 1) / 2) * cell, ((n - 1) / 2 - i) * cell, 0])) grid.add(sq) grid.next_to(rule, DOWN, buff=0.45).set_x(0) self.play(FadeIn(grid), run_time=0.8) seed = 25214902975 pat1 = [is_slime(seed, j, i) for i in range(n) for j in range(n)] pat2 = [is_slime(seed ^ 1, j, i) for i in range(n) for j in range(n)] on1 = [grid[k].animate.set_fill(EMER, 0.95).set_stroke(EMER, 1) for k in range(n * n) if pat1[k]] self.play(LaggedStart(*on1, lag_ratio=0.02), run_time=1.8) lab = cap("one seed", 26, EMER).next_to(grid, DOWN, buff=0.35).set_x(0) self.play(FadeIn(lab)) self.wait(1.4) flip = cap("flip one bit of the seed", 26, GOLD).move_to(lab) self.play(Transform(lab, flip)) anims = [] for k in range(n * n): if pat2[k] and not pat1[k]: anims.append(grid[k].animate.set_fill(EMER, 0.95).set_stroke(EMER, 1)) elif pat1[k] and not pat2[k]: anims.append(grid[k].animate.set_fill("#161C24", 1).set_stroke("#1E2833", 1)) self.play(LaggedStart(*anims, lag_ratio=0.008), run_time=2.0) self.wait(0.8) c = cap("the map is a fingerprint of the seed", 28, LAPIS) c.to_edge(DOWN, buff=0.75) self.play(FadeOut(lab), FadeIn(c)) self.wait(1.8) self.clear_all() def scene9(self): left = mono("the seed", 32, WHITE) right = mono("what a player sees", 32, WHITE) left.move_to(LEFT * 4.6 + UP * 0.7) right.move_to(RIGHT * 4.6 + UP * 0.7) self.play(FadeIn(left), FadeIn(right)) fa = Arrow(left.get_right() + RIGHT * 0.3 + UP * 0.5, right.get_left() + LEFT * 0.3 + UP * 0.5, buff=0, color=LAPIS, stroke_width=6) ba = Arrow(right.get_left() + LEFT * 0.3 + DOWN * 0.6, left.get_right() + RIGHT * 0.3 + DOWN * 0.6, buff=0, color=EMER, stroke_width=6) fl = cap("what Minecraft does", 26, LAPIS).next_to(fa, UP, buff=0.25) bl = cap("what seed recovery does", 26, EMER).next_to(ba, DOWN, buff=0.25) self.play(GrowArrow(fa), FadeIn(fl)) self.wait(0.7) self.play(GrowArrow(ba), FadeIn(bl)) self.wait(0.9) final = VGroup( cap("public, deterministic, reversible", 30, WHITE), cap("that is the whole trick", 30, EMER), ).arrange(DOWN, buff=0.3) final.to_edge(DOWN, buff=0.75) self.play(FadeIn(final[0], shift=UP * 0.15)) self.play(FadeIn(final[1], shift=UP * 0.15)) self.wait(2.2) self.play(*[FadeOut(m) for m in self.mobjects], run_time=1.2)

Modular Arithmetic