Story · Minpentai

Fig. 1LiveGliders bouncing between pairs of sigils, on the rule our game uses, in the colors of the book’s device view. Each lane is longer than the one to its left, so each round trip takes a few more steps, and the row drifts out of phase like a pendulum wave. Every 640 steps the field runs backward until every glider is back in line at t = 0 (§4).

§1The novel

Snowmoon is a 32-chapter novel by Vitalik Buterin, published in full on September 27, 2026, under the GPL v3. It opens in the month of Snowmoon in the year 3724 and moves between two countries. In Veridia, Gladias is picked by cryptographic sortition to vote on how much he likes a building he is walking past. In Dzego, two students, Zei and Fin, hurry to a class whose location is decrypted only minutes before it starts. Security is tight because, as Zei puts it, the Arctic Empire is getting more aggressive. By chapter 15 it has invaded Redshire.

Through the Dzego chapters runs a game. Zei plays Minpentai, a game on a grid of squares that evolves by one fixed rule, almost like physics. The book shows enough of it to make you want to play, but never prints the full rules, so readers could watch it but not play it. This site fills the gaps and runs the result on Ethereum. Where we filled a gap, this page says so.

§2A game called Minpentai

In chapter 4, Zei hands Fin his hand device. On it is an animation from his last game (Fig. 2). Fin asks what every newcomer asks: Okay, so what am I seeing here?

Fig. 2The animation on Zei’s hand device: black cells on squared paper, a step counter in the corner. The book’s own illustration.Snowmoon, ch. 4 · GPL v3 · embedded unchanged except for renamed internal ids

The answer comes over the next chapters, through Zei’s matches. A match is played in a hall inside a mountain, before more than a thousand people, by players in cabled glasses whom drones have just scanned for cheating devices. The field at the beginning appeared completely black. Each player has a symbol, a pattern of dots that lets its owner see everything within thirty squares of any copy of it. So everyone starts by drawing copies of their symbol in their own corner, then builds walls and glider factories around them.

Then a loud voice counts down: MU GU GEI FA, LE MU GEI FA, PA GU GEI FA. After a second countdown that runs through the single digits comes TAU FA. Gliders started flying. Gliders are small shapes that keep their form as they move. They bounce off walls, and when they hit rock formations they can make more gliders. Larger spaceships carry a player’s symbol across the field and, crashing into rocks in a controlled way, leave copies of it behind. Every so often comes an intervention turn, when all players were able to put down more squares near any copy of their symbols. Players keep score in symbols, and a game usually ends when someone resigns (Fig. 3).

Quarterfinal · ch. 4

Zei Bai Pan Dza

The final · ch. 7 · Zei : Bai

17 : 16early

17 : 22Bai’s trap

14 : 11the factory

12 : 6Bai resigns

Zei in the final Bai in the final

Fig. 3The summary view the audience follows on their devices: an indigo field, each player’s symbols as small pastel shapes. The quarterfinal has four players; the final is shown at four moments, as small multiples. The scores are the book’s (Seventeen symbols to twenty-two, Fourteen to eleven, Twelve to six); the first we counted from the picture. Shapes and colors belong to each match, not to each player.Snowmoon, ch. 4 and ch. 7 · GPL v3 · embedded unchanged except for removed inline margins

§3“Rotate one eighty if three”

Zei names the rule his last game used: We were playing with the ‘rotate one eighty if three’ rule. The book says no more. The phrase matches a rule that already existed outside the book, Critters, described by Tommaso Toffoli and Norman Margolus in 1987. Our game runs it, and so does every live figure on this page.

Cut the grid into 2×2 blocks and count the live cells in each block. With two, leave the block alone. With three, flip every cell and give the block a half turn. Otherwise, flip every cell. On the next step, shift the cuts by one square and do it again. In symbols, with $|b|$ the number of live cells in a block $b$, $\bar b$ the block with every cell flipped, and $\rho$ a half turn:

\[ f(b)=\begin{cases} b, & |b| = 2\\[2pt] \rho(\bar b), & |b| = 3\\[2pt] \bar b, & |b|\in\{0,1,4\} \end{cases} \tag{1} \]

Click any cell in the top row to make your own block.

Fig. 4Top: Equation (1) on one block for each count of live cells; the bottom block is the result. Below: a glider over four steps. The thin lines show how the next step cuts the grid; the cuts shift by one square each step. After four steps the glider has moved two squares up (dashed: where it started). Odd steps are drawn inverted; see the note.

From that one line come things that move and things that stand still. Four cells in the right shape travel two squares every four steps: a glider. A solid 2×2 block on the right squares never moves at all; in our game such blocks are the sigils and the posts. And a row of posts eight squares apart does what Zei bragged about in chapter 2:

“Finally figured out a wall structure that reflected their gliders right back at them.”
Zei · chapter 2

glider cells posts (2×2 blocks) recent path

Fig. 5LiveTwo walls of posts, eight squares apart, and two gliders. Glider A hits a post head-on, tangles with it for a few steps and leaves the way it came; the post is restored cell for cell. Glider B is aimed at a gap and is turned back all the same. A comes home every 132 steps and B every 84, so the whole field repeats every $\operatorname{lcm}(132, 84) = 924$ steps.

§4“Anything that can be created can be destroyed”

Fin asks Zei whether he is chasing a perfect wall. Zei’s answer is the deepest idea in the game:

“Because the laws of physics are time‑reversible, nothing can be invincible. Anything that can be created can be destroyed.”
Zei · chapter 4

Time-reversible means every step can be undone exactly. Equation (1) never sends two different blocks to the same result: blocks with two live cells keep two, zero and four swap, one and three swap, and flips and half turns lose nothing. So the rule has an inverse, which is the same rule with the three replaced by a one:

\[ f^{-1}(b)=\begin{cases} b, & |b| = 2\\[2pt] \rho(\bar b), & |b| = 1\\[2pt] \bar b, & |b|\in\{0,3,4\} \end{cases} \tag{2} \]

Apply Equation (2) with the cuts in reverse order, and the field runs backward (Fig. 6).

sigil posts rams and debris where the fort stood at step 0

Fig. 6LiveThree rams, each a pair of gliders in tandem, hit a sigil guarded by four posts. Forward 80 steps with Equation (1): the sigil and three of the posts break into debris, and a glider flies off. Backward 80 steps with Equation (2): the debris reassembles and the rams fly back out. The curve counts the cells that differ from step 0. The backward pass (dashed) retraces the forward one exactly and ends at 0.

Read backward, Fig. 6 shows debris assembling itself into a fort. That is how Zei cracks the hex board in chapter 12: to find gliders that would create his symbol, he decides to put that end state into the sandbox, and play it backwards. Read forward, it says what Zei tells Fin: whatever stands can be knocked down. There has to be a weakness somewhere, he says, and it’s just a matter of how well you hide it.

§5The tournament

The games that carry the plot are Zei’s matches in the Minpentai tournament of Pafogai Du. In the quarterfinal (ch. 4), teams are supposed to be random, and Zei is paired with Bai, who has beaten him three times out of three, against Pan and Dza. Zei and Bai barely speak. Late in the game Zei finally says, Bai, next intervention turn is your chance to shine. His spaceship crashes into the wall of Pan’s base, and for one single turn the wreck forms Bai’s symbol. They win.

Afterward Mu, who has the priests’ ear, explains. The priests decide the rules shut away in a dungeon. This time the doors stayed sealed for five days, twice as long as usual, and the change was that the team selection was not done through cryptographic randomness. The real test was whether two people who dislike each other would cooperate when it counted. Every match in the book has a rule like this (Table 1).

Ch.	Match	The priests’ new rule
4	Quarterfinal	Teams are not picked by cryptographic randomness.
7	Final	Each player gets a personal AI, and each side can read the other AI’s thought transcript.
12	Nationals, qualifier	Played on a hex grid: the serene battle over a field of snowflakes.
14	Nationals, semifinal	Intervention resources cut by 80%; five shrines apply any changes made inside them as an XOR to the regions near their corners, every thousand turns.

Table 1A new rule every match. Every game there’s always some kind of new rule, and the most important tactic of all was knowing how to adapt (ch. 4).Hex board: Snowmoon, ch. 12 · GPL v3 · embedded unchanged except for removed inline margins

In the final (ch. 7), Zei faces Bai alone, and the new rule gives each player a powerful AI, with a catch: the players and their AIs will have full read access to the thought transcripts of each other’s AI. Bai sets a trap and leads twenty-two symbols to seventeen. Zei answers with something no transcript can show. He has his AI build a factory that can run sixteen strategies, chosen by the type of glider and the time it hits. Then, by hand, he blocks the AI’s own trigger glider with a single brick and sends two gliders of his own. The factory fires a strategy no one planned for: fourteen to eleven, then twelve to six, and Bai resigns.

In the open: the factory and all sixteen strategies

····

Fig. 7Our sketch of the idea, not the book’s. Bai’s AI could read the whole factory; it could not read the choice. Choosing one of sixteen strategies takes $\log_2 16 = 4$ bits: here, two from the type of the trigger glider (rows) and two from when it arrives (columns).

Mu draws the lesson: In the real world, it’s not just thinking well that matters, it’s also hiding your thoughts. Later Den, the councilor who has Zei studying cryptography, points out that Zei won his semifinal the same way: by pre-publishing most of the data and using only a few bits of manipulation at the end (ch. 15). The national final is never played. The Arctic Empire invades Redshire. The other finalist, Deluin, is caught there, and the finals are canceled.

§6“Minpentai isn’t just a sport”

Why would a city fill a hall inside a mountain for what Zei thinks of as a high-school programming game? In chapter 10, Mu tells him.

“It’s because Minpentai isn’t just a sport.”

“It’s military training.”
Mu · chapter 10

The tournament exists to raise the next generation of defenders: people who cooperate with allies they do not get along with, who can deal with suddenly changing rules, who use AI and know its limits. Every rule in Table 1 is a lesson. The disguise is the point: Minpentai was only safe because it pretended to be just a game.

From Veridia it looks the same. Senator Verdow says that against the Arctics he would almost trust Dzego’s Minpentai players and their trainers over the actual Dzego army (ch. 17). Gladias reports that Dzego has been making the bet this whole time that the war that will matter will be a drone war, one more similar to Minpentai than to anything like Helisport (ch. 21). The game was a rehearsal for a war against the Arctic Empire.

Fig. 8The door to the Minpentai hall as chapters 4 and 12 describe it: stone bricks curving into an arch, and at the top two horizontal arms, meeting in the middle with clashing fists. The book has no picture of it; this is our drawing, with the field glimpsed through the door.

§7From the book to the chain

Here is what we kept, what we changed and what we added. The left column is the book; the right is our game.

In the book		In our game
RuleRotate one eighty if three.	→	Critters, the rule in Figs. 4 to 6, on a 256 × 256 field run by an Ethereum contract. It is the rule card for season 1.
SymbolsEach copy gives vision within thirty squares; players keep score in symbols.	→	Sigils: a 2×2 block on one of 512 plots. Each sigil is one share of the season.
Intervention turnsPlayers put down squares near their symbols.	→	Bells every 4 hours. Between bells the field runs 128 generations; at each bell, everyone’s moves land at once.
The priestsA secret new rule for every match, decided in a sealed dungeon.	→	A rule card for each 14-day season.
Walls and glidersWalls reflect gliders; spaceships carry symbols.	→	Posts (2×2 blocks) are walls. A single glider bounces off a sigil; a ram, two gliders in tandem, cracks it.
The Arctic EmpireA drone war against the Arctic Empire.	→	Arctic rams enter from the edge of the board at each bell, zero to four of them.
The countdownMU GU GEI FA … TAU FA	→	The same words count down to every bell. TAU FA also names our token, $TAUFA.
DamageSymbols are simply destroyed.	→	oursA rammed sigil cracks and stops earning. Repair it before the next bell, or it falls.
EndgameSomeone resigns.	→	oursSiege weekend: the last two days, with no new sigils, no repairs and no shields.
MoneyNone. The book’s Minpentai has no buy-in.	→	oursSigils cost ETH. The floor is 0.003 ETH; each sale multiplies the price by 1.003, and the premium above the floor halves every 6 hours. Each purchase is split: 45% tribute to the standing sigils (not the buyer’s own), 42% to the war chest, 8% to buy and burn $TAUFA, and 5% to operations. Stockereum creator fees (2% of $TAUFA volume) all go to standing sigils. At season end, 90% of the war chest is paid by claim weight (a living sigil counts 1, plus half of each ruin it captured; a ruin keeps 0.5), and 10% carries over.

Table 2The book on the left, our game on the right. Rows marked ours have no counterpart in the book.

Later sigils pay earlier ones. If new sigils stop, tribute stops. You can lose what you pay.

Where each sigil’s ETH goes

82.8% returns to this season’s players

17.2% does not

45%tribute to standing sigils 37.8%war chest, paid at season end 4.2%war chest, carried over 8%buys and burns $TAUFA 5%operations

\[ 0.45 + 0.9 \times 0.42 = 0.828 \tag{3} \]

Players who lost money in a simulated season, by daily trading fees

0.05 ETH/day33–58%

0.2 ETH/day2–9%

0.5 ETH/day0–1%

0%25%50%75%100%

At 0.05 ETH a day, the median player among the last tenth to arrive lost about 30–40% of what they paid.

Fig. 9Top: the split of every sigil purchase. Tribute and the 90% of the war chest paid at season end go back to players; Equation (3) adds them up. Bottom: the share of players who lost money across our simulated seasons (100 to 1,000 players each). The players are simulated and their behavior is assumed, so read these numbers as a sketch, not a forecast.

So, as a group, players get back 82.8% of what they pay for sigils, plus all the trading fees, minus what they spend on cells. Whether the group comes out ahead depends on trading volume, not on the game itself. With few fees, most players lose some of what they paid, and the later a player arrives, the more they lose.

Play it

The characters of Snowmoon learned Minpentai by playing it. The demo runs a whole season with simulated players and fake ETH, in about ten minutes at normal speed.

Play a season→ Read the math→

Notes and credits

Quotes, and the illustrations in Figs. 2 and 3 and Table 1, are from Snowmoon by Vitalik Buterin, GPL v3, vitalik.eth.limo/snowmoon. The book’s SVGs are embedded with their internal ids renamed and inline margins removed. Every other figure is ours.
Critters: T. Toffoli and N. Margolus, Cellular Automata Machines, MIT Press, 1987. The live figures use the same step function as our game, on small fields that wrap around.
Money figures come from our integrated simulation of the board and the economy with tuned parameters. The players are simulated and their behavior is assumed.