# TOAW IV: Game calculations

### Equipment Density

Up to nine units may be grouped in any location, but in many cases, this is a bad idea. Each location has a specific allowed Equipment Density:

${\displaystyle 50+2*Scenario\ physical\ scale^{2}}$

Scale Allowed Density
2.5km/hex 68
5km/hex 100
10km/hex 250
15km/hex 500
20km/hex 850
25km/hex 1300
50km/hex 5050

Any location with more than the allowed number of “active defender” equipment suffers from increased losses in the Event of combat. Any location with more than the allowed number of Vehicles or Horse Teams suffers from traffic jams (increased movement costs to enter).

### Unit Quality Calculation

${\displaystyle Quality={\frac {(2*proficiency+readiness)}{3}}}$

### Unit Combat Strength Calculation

${\displaystyle Strength=equipment\ strength*{\frac {(2*proficiency+readiness+supply)}{4}}}$

## Notes On Combat Resolution

Artillery fire is based on cumulative mass fire. Anti-Tank and Anti-Personnel fire are handled at the level of individual weapons firing at individual targets. Each item has a maximum of from one to three shots on any given round.

### Maximum Rate of Fire, per Round of Combat:

Equipment Characteristics                            Attacking Defending
Motorized, Long Range, Helicopter 3 shots 3 shots
Static, Fixed, or Infantry 1 shot 2 shots
Everything Else                                                  2 shots 3 shots

Individual weapons try to engage appropriate targets. There is some randomization, but the toughest targets will tend to draw fire from the most lethal enemy systems. Note that this has numerous side effects.

Among them: Large towed Anti-Tank weapons are now much less effective on the attack than before. AFV’s are relatively more lethal than before. Losses to very large forces attacking very small forces will be smaller than under the original TOAW system. The actual number of shots fired by Engaged units is dependent upon relative overall Engaged Strength in each individual combat but will not exceed the theoretical maximum sum expected for all weapons based on the table above. All shots fired by Anti-Personnel weapons are assumed to be lethal. In most cases, this will give results very similar to those expected in the original version of TOAW.

Anti-Armor fire is quite different. In order to be lethal, any given Anti-Armor shot must first hit the target (see Targeting, below). Each hit must then be able to defeat the armor protection of the target equipment. Each armored target has one Strength vs. HEAT weapons and another vs. Kinetic weapons (also includes Top-Attack weapons), although the two values are identical for almost all early period armored vehicles. See the “Equipment List.doc” file in the “Docs” folder for effective armor values.

#### Anti-Armor Chance to Hit, per shot

Equipment Characteristics Restricted Vision Normal Open Vision
Targeting++++ 85% 100% 100%
Targeting+++ 85% 85% 72%
Targeting++ 85% 75% 56%
Targeting+ 75% 50% 25%
Everything Else 50% 33% 11%

Note that Restricted Vision locations benefit simpler weapon systems while reducing the capability of more advanced systems. On the other hand, Open Vision rewards more advanced systems (nominal chances to hit are squared). Open Vision locations have no precipitation and no terrain other than Open, Arid, Roads, Rivers, Rocky, Escarpments, Canals, or Sandy. Restricted Vision locations have Heavy Precipitation, Heavy Cultivated, Urban, Urban Ruin, or Forest terrain. Very few early-period weapons have enhanced targeting capabilities.

#### Anti-Armor Chance to Defeat Armor, per hit

Penetration quotient

${\displaystyle pq=100*{\frac {Anti\ Armor}{Defensive\ Armor}}}$

Chance to Kill
100+ 100%
99...25 ${\displaystyle ({\frac {pq^{2}}{100}})\%}$
24- 0%

This means there is no chance to kill armor unless the firing weapon has at least 25% of the nominal penetration necessary to defeat the target’s armor. Because of the square involved, the chance to kill declines rapidly with decreasing pq (penetration quotient). Defensive armor values are divided by three during penetration checks when attacked by aircraft.

Examples: Chances to Kill
pq Chance
99 98%
90 81%
80 64%
70 49%
50 25%
40 16%
30 9%
25 6%

Example calculation of a situation where T-34/76 (late) Anti-Armor 7 (kinetic), fires on PzKpfw V Panther armor 13 under normal conditions:

Chance to hit = 33%

pq = 100 x 7 / 13 = 54.

Chance to kill if hit = ((54 ^ 2) / 100) % = 29%

Overall chance to kill = 33% x 29% = 10%.

## Artillery vs. Entrenchments

Artillery can lower the effectiveness of prepared defensive positions during combat. The effect is intended to model the earth-churning tendencies of heavy Artillery and is tied to the weight of individual shells. Heavier pieces are much more effective than lighter pieces. MRL’s (Multiple Rocket Launchers) generally do not receive this advantage. While the Anti-Personnel strengths of heavy Artillery may seem weak (due to very low rates of fire), weapons of 150mm or larger can be very effective against entrenched enemies. Bombardments won't reduce entrenchment, but Bombardments may knock a defender out of Defend, Entrenched or Fortified status.

### Armored Troop Transports

The contribution of Armored Troop Transports (APC, MICV, etc.) to unit strength is based upon unit Loss Tolerance.

At higher Loss Tolerances, more of these vehicles are assumed to be directly involved in combat. They contribute directly to unit Strengths. Infantry assigned to the unit receives less protection from enemy Artillery fire and tends to be lost when Troop Transports are hit by Anti-Armor fire. At lower Loss Tolerances, fewer of these Vehicles are on the front lines. They do not contribute directly to unit Strengths, but they do provide more protection to Infantry during Artillery fire. Infantry is generally assumed to be dismounted if the Transport is hit by Anti-Armor fire and will not be affected by Transport losses.

When an APC class item of equipment is destroyed in combat, there is a chance (based on loss tolerance and the proportion of Infantry and Transports in the unit) that an Infantry squad of some kind belonging to the same unit will also be destroyed.

### Nuclear Weapon Yields

Atomic-capable systems are individually rated for a Nuclear Attack Strength. The following values are defined for various equipment. These values are added to a unit’s conventional Attack Strength. At sub-kiloton levels, the addition may not be particularly significant.

Yield Equivalent
0.1kT 25
0.2kT 50
0.5kT 125
1kT 250
2kT 500
5kT 1250
10kT 2500
20kT 5000
25kT 6250
40kT 10000
50kT 12500
60kT 15000
100kT 25000
200kT 50000
300kT 75000
500kT 125000
1mT 250000
4mT 1000000

## Seeing the Elephant

When a unit’s actual Proficiency is determined upon conversion from Untried to Veteran status, its Proficiency can vary by ± 33% the original value. Veteran values will tend to fall near the expected Proficiency.

## Naval Combat Procedure

### Step 1: Determine the Target Ship’s Durability, Armor, and Agility Factors

Use the equipment’s (not unit icon) naval flag as follows:

For Carrier Naval:

• Durability = 0.2251 x DF
• Armor = 0.203395 x DF
• Agility = 93 x (1-damage%)
• For Heavy Naval:
• Durability = 0.159445 x DF
• Armor = 0.334704 x DF
• Agility = 75 x (1-damage%))
• For Medium Naval:
• Durability = 0.114241 x DF
• Armor = 0.248594 x DF
• Agility = 160 x (1-damage%))
• For Light or Riverine Naval:
• Durability = 0.082939 x DF
• Armor = 0.051987x DF
• Agility = 210 x (1-damage%))

For Embarked Units:

• Durability = 25
• Armor = 0
• Agility = 18

However, these values are superseded by any explicit values the designer may have coded for the ship class in the scenario’s Equipment.nqp file.

If the ship is in port, divide Agility by 2.

Note that if the attacker is a ship, then its durability will have to be determined too, for the second hit check, below.

### Step 2: Determine the Number of Shots/Attacking Planes

Note that this is where the Naval Attrition Divider is applied – scaling the number of shots.

Number of attacking planes = Assigned x (2 x Proficiency + Readiness + Supply) / 4 / (Naval Attrition Divider/10)

Fractions from this are evaluated as the percent chance of one more plane. Each resulting plane attacks individually and is evaluated for hit/miss, penetration/damage individually. This is done for each type of plane in the unit.

Number of shots per attacking coastal battery = (102.922 x Gun’s Anti-Naval Value x Assigned x ((2 x Proficiency + Readiness + Supply) / 4)/ Shell Weight) / (Naval Attrition Divider/10)

Fractions from this are evaluated as the percent chance of one more shot. Each resulting individual shot is evaluated for hit/miss, penetration/damage individually. This is done for each type of gun in the unit.

Number of shots from each attacking ship TO&E line = ((#Ships-Damage%) x 102.922 x Ship’s Anti- Naval Value x ((2 x Proficiency + Readiness + Supply) / 4) / Shell Weight) / (Naval Attrition Divider/10)

Fractions from this are evaluated as the percent chance of one more shot. Each individual shot resulting is evaluated for hit/miss, penetration/ damage individually. Note that the formula evaluates as 10 salvos.

### Step 3: Determine if a Hit Occurs

For each Individual Attacking Airplane or each Shot from each Gun of each Ship/ Coastal Battery, determine if a Hit Occurs.

There are two checks:

First hit check passes if:

Attacking Unit Proficiency x Attacker Shock Level > Random (Target Ship Agility Rating) x Defender Shock Level

Note that this is where shock, for both sides, is applied.

Second hit check for airplanes passes if:

Attacking plane’s Anti-Naval Value x Visibility > Random (Target Ship Agility Rating) x Defender

Shock Level

There is an exception for torpedo bombers (any plane with the Torpedo flag), where the Anti- Naval Value is divided by 4 first.

Visibility is a value based upon the combination of weather x lighting: clear/hazy/overcast x day/ mixed/night. Clear is 1, hazy is .66, and overcast is .33. Day is 1, mixed is .66, and night is .33. Note that the square root of this product is used for surface naval combat. Weather value is the average of the target hex and attacker hex values.

Second hit check for guns (including surface ships) passes if:

100 x SQRT(Visibility) / ((1 + (0.002 + .13 / Attacking Ship’s Durability) x Range x Range) > Random (100)

Here is where the Attacking Ship’s Durability will be needed. For Coastal Guns, a value of 100,000 is used for durability. Note that the actual parameter used is the Accuracy – but this is identical to the Durability though, unless the designer has specified a different value in the .nqp file. See the curve below for how this equation plots out.

Note that Range is the number of hexes between the attacker and defender – not including their hexes – times the hex scale in kms. So adjacent combat would be point-blank range (=0). However, in that adjacent case, there is a random check as follows:

Range = random (minimum (max range of all units involved, hexscale/2)

This is only determined once per naval unit per combat round.

Both checks must pass for a hit to occur.

### Step 4: Determine the Damage any Hit may have caused

If a hit occurred, it causes 1 damage point if:

(Attacking Weapon’s Shell Weight > Random (Defending Ship’s Durability / 10)) AND (Target ship’s damage < 20)

(This is intended to be automatic unless the shell weight is trivially small. But note that there is a limit to the damage that can be inflicted by such superficial hits: 5 max for Heavy Naval, 10 max for Medium Naval, 15 max for Carrier Naval, and 20 max for all others.)

Then, it must be determined if the hit penetrated the ship’s armor and what additional damage it may have caused:

Penetration occurs if:

Random (Attacking Weapon’s Shell Weight /2) > Random (Defending Ship’s Armor Value)

If this check passes:

Additional damage points = Random (Attacking Weapon’s Shell Weight / 10) x Random (50) / Defending Ship’s Durability Value

There is an exception for torpedo Bombers, whose Shell Weight is quadrupled for the above checks.

(So, there is a /4 effect for the hit check, followed by a x4 effect for the damage – harder to hit with a torpedo, but if you do…the damage is underwater, making it much more severe).

There is then a chance that the additional damage is a critical hit:

5% x (Attacker Shell Weight / (Defender Durability * 6.25)) * (Defender Force Critical Hit Scalar / 10)

Note that there is a new Force parameter for scaling the chance of critical hits as the designer desires.

Also note that if the shell weight is the same as the weighted durability and the scalar is left at the default value of 10, then the equation reduces to a 5% chance.

If a critical hit occurs, additional damage is incurred due to the target ship’s magazine detonating. This additional damage is:

The result of a normal distribution centered at 50 with a standard deviation of about 13.5.

### Step 5: Finally, Determine what Defending Equipment has been Destroyed

A ship that accumulates 100 or more damage points sinks (goes to the dead pile). If the ship has accumulated less than 100 damage points, that value is stored in the ship’s TO&E line and affects several ship abilities. If that line has more than one ship in it, only the “lead” ship has accumulated that damage and is affected by it.

If an embarked unit suffers damage points, a randomly selected weight of equipment equal to those points is destroyed as follows. Each TO&E item in the embarked unit loses:

Assigned x damage accumulated / Unit Weight Fractions of this are evaluated as the chance of one more kill.

If an aircraft carrier suffers more than 66 damage points it ceases to function as an aircraft carrier. This will cause an air unit to be eliminated if there are then more air units in the hex than aircraft carrier bases. The air unit to be eliminated is selected randomly. Otherwise, the planes on the carrier are not targeted or included in target calculations. (But they do engage in AS combat).

The following curve shot shows how the gunnery range equation plots out for various Durability values: The top curve is for a coastal gun. The next is for a 130-Durability BB. The next is for a 50-Durability CA. Then next is for a 30-Durability CL. And the last is for a 10-Durability DD. Vertical lines are every 20%; Horizontal range is out to 50 kms.

For example, at 30km, a coastal gun has a 36% hit chance, a BB has a 27% chance, a CA has a 19% chance, a CL has a 15% chance, and a DD has a 7% chance.

Note that these values assume a stationary target, a perfect crew, and perfect visibility. The agility of the target vs. the proficiency of the crew is tested in the first hit check, and the visibility factor scales these values.

Note that the square root of the visibility factor is used for gunnery. This is to account for the fact that gunnery enjoys better optics, radar, star shells, searchlights, etc. that planes normally don’t have.

## Naval Repair Procedure

During the inter-turn calculation period, all damaged ships get the following checks for damage repair

IF the ship’s unit is in a supplied anchorage hex AND it did not move in the previous turn THEN

# of damage points repaired = 1500 / ((Durability + Armor + AP + AAA) x # of turns per week).

Fractions of this are evaluated as the chance of one more point repaired.

ELSE

# of damage points repaired = 150 / ((Durability + Armor + AP + AAA) x # of turns per week).

Fractions of this are evaluated as the chance of one more point repaired AP and AAA values are the ship class’s intrinsic values. They are not the damage-adjusted values.

Note that repair is 10x faster in port than at sea.

These figures will typically allow about the following weekly rates of repair in port:

Carrier – 5 points.

Battleship – 1 point.

Heavy Cruiser – 3 points.

Light Cruiser – 5 points.

Destroyer – 14 points.

Note that a true port repair simulation would require modeling port and shipyard capacities. But adding such features will have to wait. The simplistic version above will have to do for now.

## Sea Interdiction Procedure

Chance of surface interdiction per in-range surface interdictor

Target Value of Moving Unit or Stack / Target Value of all detected in-range Units.

Note that if the moving stack is the only one detected, then the chance of interdiction is 100%. Also, if there are 100 friendly ships and 100 enemy ships in spotting range, the chance of interdiction will be 1% per ship, averaging 1 interdiction per moving enemy ship – it pairs off both side’s ships (on average). Chance of air interdiction per in-range air interdictor:

Target Value of moving Unit or Stack * (Naval Attrition Divider / 10) / Target Value of all detected in-range Units / (interdictor’s spotting range / 2). This will tend to limit air interdiction to no more than once or twice as the fleet passes through its scouting range. The point is to limit planes to realistic mission rates. There is an exception for carrier movement, though. If the moving carrier hasn’t been attacked in the turn, it is automatically interdicted in the first detected in-range hex it enters. (Who gets the first shot really matters in carrier-vs.-carrier combat.)

Both formulae pose a tactic for the phasing player: bring the entire fleet to be moved through the interdictor’s range into its detection before moving through. The more in-range targets detected the less interdiction there will be per individual move.

## Naval Spotting Procedure

First, understand the necessity for this feature: Naval units, unlike ground units, can’t hide – due to their great size – and operate in a terrain that has no cover.

They must be handled separately from ground units for recon purposes. Within the limit of the horizon, subject to visibility, they should be detected.

### Surface Recon

These models the observation of the seas from the coastline or from the conning towers of ships.

The daytime observation range is 25km and the nighttime observation range is 10km. Whenever a land unit moves into a coastal hex, it reveals Deep- Water hexes out to the appropriate range – just as if it had entered a peak hex. And note that, like for peak hexes, if the observer is under a cloud, it can’t observe. As a naval unit moves, it reveals Deep-Water hexes around it out to the appropriate range – just as if it had entered a peak hex (same comment about clouds).

The range of hexes revealed at the various hex scales is, therefore, as follows:

A range of 1 means only adjacent hexes are revealed. Note that the range to the target doesn’t include

the firing or target hexes.

That’s why there may seem to be an extra hex included in some of the above. For example, eleven 2.5km hexes out from an observing naval unit would be a target naval unit that was ten 2.5km hexes (25km) from that observing naval unit.

Note that radar is modeled as well: Any ship with the “All Weather” flag set will see out to 50km, day or night. Furthermore, if a force has any aircraft on map with the “All Weather” flag set, then all ships are assumed to have radar and see accordingly.

(This is needed for existing modern scenarios that won’t see further designer attention).

### Air Recon

These models the observation of the seas by air units committed to sea operations. That includes ground-based air units assigned to Sea Interdiction or Air Superiority and Carrier-based air units assigned to Sea Interdiction or Air Superiority.

The range that can be fully searched for a given number of planes =

Minimum (50Km x Square Root (# planes available for search), max range of planes)

Fractions of this are evaluated as the chance of one more hex in the range.

So, for example, if the total ready planes/2 assigned to Sea Interdiction/Air Superiority in a hex total 9, then the max range that can be searched will be 150km. If the planes had an assigned range of 200, then the range of 150 will be substituted for it for search purposes. If the planes had an assigned range of 100, then 100 will, of course, still be used instead. Note that the “max range of planes” is the average max range of all the planes in the hex.

So, for 2.5km hexes, that 150km range would be 60 hexes. For 5km, 30 hexes. For 10km, 15 hexes, for 15km, 10 hexes. For 20km, 8 hexes. For 25km, 6 hexes. For 50km, 3 hexes. Within these ranges, all enemy Deep-Water hexes would be spotted.

Note that the number of planes available for search = Assigned x readiness / 2. So, if the hex had 75% readiness, they would have needed 24 planes in it to have 9 planes available to search. Also note that only all-weather aircraft can search at night.

Note that there is now a crop circle designed to show naval spotting range (since this can be less than the nominal range of the aircraft onboard the carrier). It is distinguished by a dashed line as shown here:

Note that in the above case, the carrier spots out to three hexes (150km). But the planes on the carrier have ranges of 12 (600km). So, the dashed line is critical for players to know how far their carriers can see.

Note that if the range was extended one hex by chance evaluation of any fractional value, that extra hex is not reflected in the circle. So, players will not know if they got the extra hex or not – adding a little risk to carrier recon. This is true for non-phasing and phasing unit’s ranges. The phasing unit’s range varies randomly with each hex entered.

Note that hazy and overcast locations affect the chances of spotting. If the weather is in the phasing unit’s hex, it affects all the unknown hexes in range of the unit. If in the target unit’s hex, it only affects the chances of detecting that specific hex. Hazy locations reduce spotting chances by 15%. Overcast locations reduce it by 30%.