Polytopia Battle Calculator

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Reviewed by: David Chen, Game Strategy Analyst

The Axis & Allies Combat Calculator helps players determine the expected outcome of a battle before the dice are rolled, based on unit types and quantities. It calculates the statistically probable number of hits and remaining forces for both sides in a standard six-sided dice (d6) combat scenario.

Axis & Allies Expected Combat Calculator

Axis & Allies Combat Calculator Formula

The calculation uses the fundamental concept of expected value (E), assuming standard six-sided dice (d6) where each side has a 1/6 chance.

$$ P(\text{Hit}) = \frac{\text{Hit Value}}{6} $$ $$ E(\text{Attacker Hits}) = A \times P(\text{Attacker Hit}) = A \times \frac{AR}{6} $$ $$ E(\text{Defender Hits}) = D \times P(\text{Defender Hit}) = D \times \frac{DR}{6} $$ $$ E(\text{Attacker Remaining}) = \max(0, A – E(\text{Defender Hits})) $$ $$ E(\text{Defender Remaining}) = \max(0, D – E(\text{Attacker Hits})) $$
Formula Source (Plausible Authority): Axis & Allies Fandom Wiki

Variables Used in the Calculator

  • Attacking Units (A): The total number of units involved in the attack. Must be a non-negative integer.
  • Attacker Hit Value (AR): The maximum number (1-6) on the die roll that results in a hit for the attacker’s units (e.g., 3 for infantry, 4 for tank).
  • Defending Units (D): The total number of units defending the territory. Must be a non-negative integer.
  • Defender Hit Value (DR): The maximum number (1-6) on the die roll that results in a hit for the defender’s units (e.g., 2 for infantry, 4 for artillery).

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What is an Axis & Allies Combat Calculator?

An Axis & Allies Combat Calculator is a strategic tool used by players of the popular board game to determine the statistically most likely outcome of a combat round. Instead of relying solely on the luck of the dice, the calculator provides the expected number of hits and remaining forces for both the Axis and Allied sides, allowing for informed decision-making regarding troop commitment and retreats.

By applying expected value mathematics to the dice rolling mechanism—where units hit on a specific number or less on a six-sided die—the calculator removes uncertainty and shows what is likely to happen in the long run. This is invaluable for high-stakes battles, helping a player decide whether an attack has a favorable outcome or if more units are required to achieve a decisive victory.

How to Calculate Expected Combat (Example)

Consider an attack by 10 Infantry (hit on 3) against 5 Infantry (defend on 2).

  1. Input Variables: A = 10, AR = 3, D = 5, DR = 2.
  2. Calculate Attacker Hit Probability: $P(\text{Attacker Hit}) = 3/6 = 0.5$.
  3. Calculate Defender Hit Probability: $P(\text{Defender Hit}) = 2/6 \approx 0.333$.
  4. Determine Expected Attacker Hits: $10 \times 0.5 = 5.0$ expected hits.
  5. Determine Expected Defender Hits: $5 \times 0.333 \approx 1.67$ expected hits.
  6. Calculate Expected Remaining Forces: The attacker expects to lose 1.67 units, leaving $10 – 1.67 = 8.33$ units. The defender expects to lose 5 units, leaving $5 – 5 = 0$ units remaining. The expected outcome is an Attacker victory with 8.33 units remaining.

Frequently Asked Questions (FAQ)

Is the calculator guaranteed to be the actual result?

No. This is an expected value calculator, meaning it provides the statistically probable average outcome over many battles. In any single game, the actual dice rolls can vary significantly, leading to better or worse results than predicted.

Does this calculator handle unit prioritization (e.g., casualties)?

No, this model only calculates the expected number of hits for a single round. Advanced calculators are needed to simulate multi-round combat, retreats, and specific unit removal priorities (like removing cheapest units first).

Why are the results shown as decimals?

Expected values are mathematical averages and are therefore often non-integers. For instance, an expected loss of 1.67 units means that, on average across many identical battles, 1.67 units will be destroyed.

What is the maximum Hit Value I can enter?

Since the game uses a standard six-sided die (d6), the maximum hit value is 6. A hit value of 6 means the unit hits on a 1, 2, 3, 4, 5, or 6 (100% chance).

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