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The 20 digits

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Each digit has an attack (ATK) and hit points (HP) at level 1; each level adds its growth. N stands for the digit's level (N1 to N3). ⌊x⌋ rounds down, ⌈x⌉ rounds up. Support digits only act from the back, on the front digit of their column; reach digits hit from the back.

Tier 1 · 2 gold · open from round 1

  • Plus tier 1 support

    N1 N2 N3
    1/3 2/4 3/5

    Support. Opening: the front digit of its column gains 2N ATK (2, 4, 6).

    Example: Plus N1 behind an Odd N1 (ATK 3): the Odd opens at 3 + 2 = 5 ATK.

  • Minus tier 1

    N1 N2 N3
    2/3 3/4 4/5

    Opening: the opposing front digit of its column loses N + 1 ATK (2, 3, 4), never below 0. Works from the front or the back.

    Example: Minus N2 facing an Odd N1 (ATK 3): the Odd opens at 3 − 3 = 0 ATK.

  • Double tier 1 support

    N1 N2 N3
    1/3 2/4 3/5

    Support. The front digit of its column hits × 2 for the first N exchanges (1, 2, 3).

    Example: Double N2 behind a front digit with 3 ATK: hits of 6, 6 and 3.

  • Zero tier 1

    N1 N2 N3
    0/6 0/9 0/12

    Every hit it takes is cut by N (1, 2, 3), never below 1.

    Example: Zero N2 (9 HP) takes 2 then 5: 2 − 2 → 1 (floor 1), 5 − 2 → 3, HP 9 → 8 → 5.

  • Even tier 1

    N1 N2 N3
    2/4 3/6 4/8

    Its hit gains N + 1 (2, 3, 4) when its HP is even.

    Example: Even N1 (2/4): at 4 HP its hit is 2 + 2 = 4; at 3 HP, 2.

  • Odd tier 1

    N1 N2 N3
    3/3 4/5 5/7

    Its hit gains N + 1 (2, 3, 4) when its HP is odd.

    Example: Odd N2 (4/5): at 5 HP its hit is 4 + 3 = 7; at 4 HP, 4.

Tier 2 · 3 gold · open from round 3

  • Prime tier 2

    N1 N2 N3
    3/5 4/7 5/9

    Its hit gains 2N + 1 (3, 5, 7) when its HP is a prime number.

    Example: Prime N3 (5/9): at 9 HP (not prime) its hit is 5; at 7 HP, 5 + 7 = 12.

  • Half tier 2

    N1 N2 N3
    2/6 3/8 4/10

    Its hit is the greater of: its hit, ⌊target's remaining HP / 2⌋.

    Example: Half N1 (ATK 2) against a target with 13 HP: max(2, ⌊13 / 2⌋ = 6) = 6.

  • Square tier 2

    N1 N2 N3
    2/5 3/6 4/7

    Opening: a shield of (N + 1)² (4, 9, 16), which does not regenerate.

    Example: Square N2: a shield of (2 + 1)² = 9, which absorbs the first hits taken.

  • Twin tier 2

    N1 N2 N3
    2/4 3/5 4/6

    Opening: +N ATK and +N HP for each other allied digit with the same ATK as it.

    Example: Twin N1 (2/4) with two other allies at 2 ATK: +2 ATK and +2 HP, now 4/6.

  • Abacus tier 2

    N1 N2 N3
    1/5 2/7 3/9

    Opening: +1 ATK per right answer in its owner's Focus that round (0 to 6).

    Example: Abacus N1 (ATK 1) after a Focus of 4 out of 6: 1 + 4 = 5 ATK.

Tier 3 · 4 gold · open from round 5

  • Splitter tier 3

    N1 N2 N3
    6/6 8/7 10/8

    Hits its target and the back digit of its target's column, each for ⌊hit / 2⌋; with no second digit, the full hit lands on the target.

    Example: Splitter N1 (ATK 6): 3 on its target and 3 on the digit behind it; with no digit behind, 6 on the target.

  • Modulo tier 3

    N1 N2 N3
    3/8 4/11 5/14

    Ignores any hit it takes whose value is a multiple of 3.

    Example: Modulo N1 (8 HP) takes 12: ignored; then 13: HP 8 → −5, it falls.

  • Sigma tier 3 reach

    N1 N2 N3
    1/6 1/8 1/10

    Reach. Opening: its ATK becomes ⌊sum of the ATK of the allied front row, itself excluded, ÷ (5 − N)⌋ (÷ 4, ÷ 3, ÷ 2).

    Example: Sigma N1 with a front row at 5, 0, 4 and 2 ATK: ⌊11 / 4⌋ = 2 ATK.

  • Average tier 3

    N1 N2 N3
    2/6 3/8 4/10

    Opening: each digit of the allied front row takes ⌈average ATK of that row⌉ + (N − 1) as its ATK (0, 1, 2).

    Example: Average N1 with a front row at 2, 2, 3 and 0 ATK: ⌈7 / 4⌉ = 2, each front digit goes to 2 ATK.

  • Carry tier 3

    N1 N2 N3
    4/6 5/8 6/10

    When it falls, the back digit of its column (failing that, the nearest living allied front digit) gains its ATK.

    Example: Carry N2 (ATK 5) in front of an Odd N2 (ATK 4): when it falls, the Odd goes to 9 ATK and moves up to the front.

Tier 4 · 5 gold · open from round 7

  • Power tier 4 support

    N1 N2 N3
    2/6 3/8 4/10

    Support. Opening: the ATK of the front digit of its column becomes its square, capped at (N + 2)² (9, 16, 25); never less than its ATK.

    Example: Power N1 behind a front digit with 6 ATK: 36, capped at 9; at N2, 16.

  • Pi tier 4 reach

    N1 N2 N3
    3/9 4/12 5/15

    Reach. On exchanges 3, 6, 9 and 12, hits every digit of the opponent's foremost row; on the others, hits normally.

    Example: Pi N1 (ATK 3): on exchange 3, hits every digit of the opposing front row for 3; before that, a single target.

  • Delta tier 4

    N1 N2 N3
    3/10 4/13 5/16

    Its hit gains ⌊HP lost / 2⌋ (HP lost = HP after the opening minus current HP).

    Example: Delta N1 (3/10) down to 3 HP: 3 + ⌊7 / 2⌋ = 6.

  • Fibo tier 4

    N1 N2 N3
    3/8 4/10 5/12

    Base hit: its ATK on exchanges 1 and 2, then the sum of the two previous base hits (1, 1, 2, 3, 5, 8…).

    Example: Fibo N2 (ATK 4): base hits 4, 4, 8, 12, 20, 32…

Order of the opening

Before the first exchange, the opening effects apply in six steps. Each step reads the state left by the one before; within a step, all effects are worked out on the same state, then applied together.

  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.

The school rule: additions come before products and powers.

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.