Rules

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The game

• A game is a series of rounds against saved armies. It ends at 8 wins (game won), 3 losses (game lost) or at the end of round 12 (game closed, record kept); a daily challenge also stops at midnight and a duel at its deadline, with their record at that moment; a room plays every round the host picked, with no stop at 8 wins or 3 losses; the tutorial has 3 rounds. A draw counts as neither a win nor a loss.
• The margin of a combat: your HP left minus the opponent's at the end of the combat; that of a game, the sum of its combats. It breaks ties in the daily leaderboard, the room and the duel.
• A round: the Focus (the round's gold), the shop, the Face-off, the combat, the result.
• You can leave whenever you like: the whole state is on the server, and reopening the game picks up where you were. The clocks of a question or of the Face-off keep running with the page closed.
• A game with no action for 7 days expires with its record at that moment.

The Focus

• 6 calculations, one question per page; the answer is one of the 16 buttons, 1 to 16.
• Clock kept by the server: 6 s per question at stages 1 to 3, 8 s at stages 4 to 6. An answer after the deadline counts as wrong; a page left unanswered moves on to the next question.
• Round income: 3 gold, plus 1 per right answer, plus 1 if all are right, so 3 to 10 gold. Unspent gold from the previous round is added (5 at most).
• In the first round of every game, every round of the tutorial and every round of a room, the Focus waits for your click on “Start the Focus”; in the other rounds, it starts with “Continue”.
• A perfect Focus opens the bright offer: one more offer, from the highest open tier, which stays until the end of the shop.

The shop

• 5 offers per draw; for each offer, a tier is drawn according to the round, then a digit of that tier. Tiers open at rounds 1, 3, 5 and 7.
• Prices: tier 1 = 2 gold, tier 2 = 3, tier 3 = 4, tier 4 = 5. Reroll: 1 gold, unlimited; it draws 5 new offers in place of the current draw, bought or not; the bright offer stays.
• A digit you buy goes on the cell you pick (by default the first free front cell, otherwise the first free back cell; supports, Plus, Double and Power, go to the back first). Slate full: sell first.
• Selling: gives back half the price, rounded down, whatever the level; the levels are lost. Moving a digit is free; onto a taken cell, the two swap.
• At the end of the shop, the gold left goes to the reserve, 5 at most; the rest is lost.

Odds of each tier by round

Rounds	Tier 1	Tier 2	Tier 3	Tier 4
1, 2	100 %
3, 4	60 %	40 %
5, 6	30 %	40 %	30 %
7 to 12	15 %	30 %	30 %	25 %

Training

• Once per round, during the shop: pick one of your level 1 or 2 digits and answer a question at the training stage, in 10 s.
• Right: the digit goes up a level (N3 at most) and gains its growth. Wrong, late or unanswered: nothing changes, and the round's Training is used up.

The Face-off

• At the end of the shop, your slate is saved, then the opposing army is drawn and shown in full: digits, levels, ATK and HP, effects.
• You can swap your 4 columns around (each column keeps its front and back) and, in each column, swap the front and the back: 384 arrangements. No buying, no selling, no Training.
• In ranked, the daily challenge and duels, you have 60 s; in a room, the time the host picked; in casual, no clock. An order received too late is refused: the end-of-shop order is kept.
• The army you are shown is another player's at the end of their shop, before their own Face-off: everyone places blind first, then adjusts against a given opponent.

Combat

Your slate, in the order chosen at the Face-off, takes on the opposing army. The combat is fully determined: no draw, both sides follow the same rules, column against column.

Opening

1. Additions and subtractions: Plus, Minus, Abacus (no ATK below 0).
2. Average: Average (only one per side, the highest level, then the leftmost).
3. Twins: each Twin.
4. Power: Power.
5. Sum: each Sigma.
6. Shields: Square.

Exchanges

1. Hits: each digit that hits works out its target and its hit from the state at the start of the exchange.
2. Resolution: all hits land together; for each hit taken: Modulo (hit ignored), Zero (hit cut by its level, never below 1), shield (absorbs up to its value), then HP.
3. Falls: digits at 0 HP or less are removed; the Carry then gives its ATK.
4. Moving up: each empty front cell takes the back digit of its column.
5. End: a side with no digit left loses; both sides emptied at the same exchange: the side with more HP at the start of that exchange wins, a tie is a draw.

Working out a hit

1. Base: the digit's ATK (Fibo: the term of its sequence).
2. Plus the current bonuses: Even, Odd, Prime (read on its HP at the start of the exchange), Delta.
3. Times the factors: Double.
4. Half: the greater of the result and half the target's HP.
5. Splitter: two hits of half, on its target and on the digit behind it (with no digit behind, the full hit); Pi on exchanges that are multiples of 3: the same hit on every digit of the targeted row.

Who targets whom

• The slate has two rows of 4 cells: column c joins front cell c and back cell c + 4.
• Who hits: the digits of the front row, and the reach digits of the back row. A digit only hits if its hit is above 0.
• A digit in column c targets the opposing front digit of column c; if there is none, the nearest opposing front digit in the order c − 1, c + 1, c − 2, c + 2, c − 3, c + 3 (left first). If the opposing front row is empty, the same rule applies to the back row.
• Example: from column 3, with opposing front 3 empty, the order is 2, 4, then 1.
• When a front digit falls, the back digit of its column moves up and keeps its ATK, HP, level and shield.

End

After exchange 12, if both sides are still standing, the side with the most total HP (shields excluded) wins; if tied, the most total ATK; otherwise it is a draw.

Opponents

You always face a saved army (a ghost), never a live player: that of another player at the same round, with the closest record possible (same record, then same wins, then same round), with a rating close to yours (100 points apart, then up to 500), even in casual; failing that, a house ghost generated by Neurostim and shown as such. The Face-off gives the record of a player's army. In a duel, the opponents of each round are drawn when the duel is accepted, the same for both players; in a room, they are the armies of the other players in the room, in turn.

Modes

• Ranked (members): your rating moves with every combat; 60 s Face-off; one ranked game active at a time.
• Casual (members): the same game, without a rating; the Face-off has no clock (the Focus and Training questions keep theirs).
• Daily challenge (members): the same offers for everyone at the same round and the same reroll, each player with their own Focus; one counted game per day (a second one is casual), stopped at midnight (Paris time) with its record; 60 s Face-off. Daily leaderboard: wins, then fewer losses, then margin, then the earliest finish. The podium is a distinction, with no credits. The counted game is the first one started that day, even if forfeited; forfeited, or stopped before its first combat, it leaves the daily leaderboard; the second game of the day is a casual game with its own offers; a Face-off open at midnight is fought in the end-of-shop order.
• Room (2 to 8 players, guests without an account let in by the host's link): rounds in sync with the host's clock; the same offers for everyone at the same round and the same reroll, each player with their own Focus; everyone faces another player's army from the room, and everyone plays every round; each round opens for all at once and the Focus only starts on “Start the Focus”. The host picks 6, 8, 10 and 12 rounds (8 by default), the shop time and the Face-off time, and whether guests may join; the shop clock starts when the round opens, Focus included (a Focus not started in time counts as 0 right answers); a player who joins mid-room counts the missed rounds as losses; a room game cannot be forfeited. Room standings: wins, then fewer losses, then margin, no rating.
• Duel (two members, Duels page): the same game for both (same offers, same questions, same opponents, drawn when the duel is accepted), 12 rounds at most, to play within 24 h, each whenever they like; the best record wins (wins, then fewer losses, then margin, then right answers in the Focus); a full tie is a draw, stakes returned. At the deadline, an unfinished game stops and counts with its record; a player who never started their game loses by forfeit (if neither started, the duel is cancelled). Forfeiting your game loses the duel, whatever the record.
• Tutorial (members): 3 guided rounds against tutorial ghosts; at the Face-off, “Predict” works out the outcome of the chosen order, which no other mode does.
• Trial game (visitors): a game against the house ghosts; nothing is kept in the leaderboards or served as a ghost.
• Your end-of-shop slates (ranked, casual and duel; daily challenge, for the games of the same day) serve as ghosts for other players, under your login, anonymously or not at all, as you choose in your preferences. In a room, your slate only serves the room's players, while the room lasts: under your login if you chose so, anonymous otherwise. At the end of a duel, the other player reads your combat reports only if you chose your login.

Rating

• Elo rating of the ranked mode, starting at 1,000. After each ranked combat, it moves with the result (a win counts 1, a draw 0.5, a loss 0) and the frozen rating of the army you faced: K = 24 for your first 30 ranked combats of the season, 16 after that. The army's owner neither gains nor loses anything: they did not play that combat.
• You play live against a frozen army and you have the last word at the Face-off: that army's rating counts for 190 points less in the calculation, a value measured so that a player of the same level gains nothing for free. Against an army with the same rating, a win is therefore expected: at K = 24, a win earns 6, a draw costs 6 and a loss 18.
• A season lasts a calendar month (Paris time). On the 1^st of each month, the rating moves back towards 1,000: new rating = 1,000 + (rating − 1,000) ÷ 2, rounded, once, even after several months without playing.
• Season leaderboard, among the members with 8 ranked games finished in the month (won, lost or closed after the last round, counted in the month they finish; an expired or forfeited game does not count, but its combats did move the rating): rating, then the most ranked combats; as a last resort, the oldest account. Never a shared place. Credit prizes at the end of the month, by place: 150 for 1st, 100 for 2nd, 60 for 3rd, 25 from 4th to 10th; the leaderboards page shows the prize of each place.
• A guest player, without an account, appears in the leaderboard like a member, marked as a guest; credit prizes only go to members: the prize of their place goes to the next member. Creating an account keeps the rating and the games.
• A Focus or a Training answered faster than a human takes the whole game out of the leaderboards, the prizes and the feed: none of its combats moves the rating, earlier ones included (their changes are taken back).