Tank on Tank Love

The Grant rolls on.  What a weird looking beast!

This is an article about the mathematics of tanks shooting at tanks.  Specifically, it is about the odds and tactics pertaining to different ratios of AT to FA.  Across the various eras, the differential between these two numbers dictates most of the effectiveness of armored vehicles against each other. 



Difference
Front Armor (down) vs Anti-Tank (across)

FA/AT	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
1	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
2		1	2	3	4	5	6	7	8	9	10	11	12	13	14
3			1	2	3	4	5	6	7	8	9	10	11	12	13
4				1	2	3	4	5	6	7	8	9	10	11	12
5					1	2	3	4	5	6	7	8	9	10	11
6						1	2	3	4	5	6	7	8	9	10
7							1	2	3	4	5	6	7	8	9
8								1	2	3	4	5	6	7	8
9									1	2	3	4	5	6	7
10										1	2	3	4	5	6
11											1	2	3	4	5
12												1	2	3	4
13													1	2	3
14														1	2

This chart shows the difference between AT and Front Armor for various values.  Okay, so it's just subtracting.  But notice the wedge shape below which armor provides absolute protection.

Save at Short range
Front Armor (down) vs Anti-Tank (across)

	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
1	G/2+	3+	4+	5+	6+	G6+	-	-	-	-	-	-	-	-	-
2		G/2+	3+	4+	5+	6+	G6+	-	-	-	-	-	-	-	-
3			G/2+	3+	4+	5+	6+	G6+	-	-	-	-	-	-	-
4				G/2+	3+	4+	5+	6+	G6+	-	-	-	-	-	-
5					G/2+	3+	4+	5+	6+	G6+	-	-	-	-	-
6						G/2+	3+	4+	5+	6+	G6+	-	-	-	-
7							G/2+	3+	4+	5+	6+	G6+	-	-	-
8								G/2+	3+	4+	5+	6+	G6+	-	-
9									G/2+	3+	4+	5+	6+	G6+	-
10										G/2+	3+	4+	5+	6+	G6+
11											G/2+	3+	4+	5+	6+
12												G/2+	3+	4+	5+
13													G/2+	3+	4+
14														G/2+	3+

I'm going to follow informal convention and use the term "Glanced" to mean an armor save that equaled the AT rating.  I will use the term "Penetrated" to refer to an armor save that fails outright.

This chart shows the score a give Armor value needs to Save against different AT values. Here, the useful values make a band down the chart.  Above this band, Penetration is automatic.  Below that band, the target will automatically pass its save. On the lower border of the band, you can see that the target can only be Bailed, and even then if it rolls a 1 (marked G/2+).  At the top end, it will at least suffer a Glance (marked G6+), even if it so lucky as to roll a 6.  For a number like 3+, that is the number needed to escape Penetration.  It can still be Glanced at 1 less.

Each differential between FA and AT has its inherent tactics. I will give common examples from different eras.  Some of these are surprising at first glance, particularly in EW. 

Eight (or more)
Common examples:

• Jagdpanther vs. just about anything
• Panther v Sherman 75
• Sherman 75 vs German half-track

At this difference, even at long range, no possible roll can save the target.  He can't roll a seven. With such a superiority, your main concern is whether the target merits such overwhelming attention.

Seven 
Common examples:

• Tiger IE vs Sherman 75
• Firefly vs. Panzer IVH

At a difference of 7, the target can successfully save a Glance at long range.  As such, the difference is largely negligible, and is more annoying (or miraculous) than anything else.  Use tactics as above.

Six
Common examples:

• T34/85 vs. Panzer IVH
• Firefly vs. StuG

At a difference of 6, the target can ninja a Glance even at short range.  At long range, it has a more respectable 33% chance of survival.  Again, with this match-up, your main consideration should be fire and threat priority for your big guns.

Five
Common examples:

• Panzer IVH vs Sherman 75
• T34/85 vs StuG

This is the turning point in our math.  At this difference, when you engage your target at range, it has a 50% chance of bouncing a hit (including the possibility of a Glance).  Any long range firefight is therefore likely to take a while.   

Four
Common examples:

• Sherman 75 vs. Panzer IVH 
• Firefly vs.Tiger IE
• Somua vs. Panzer IIE (late)
• Panzer IVD vs Hotchkiss

At difference of 4, at close range, the target has a 50% chance of bouncing a hit (including the possibility of a glance.) At long range, the target will save (including Glances) 66% of the time. If you engage tanks with this chance of hitting, you are going to need greater numbers or skill or some other advantage if you don't want the fight to take multiple turns.

Three
Common examples:

• Sherman 75 vs. StuG 
• Panzer IVH vs. T34/85
• T34 vs Panzer IV
• Panzer III (long 75) vs. Sherman
• Panzer III (long 75) vs. T34
• Somua vs. Panzer IVD
• Panzer IVD vs Somua

At a difference of 3, you will struggle to hurt the target.  At a close range, they will bounce hits (including Glances) 66% of the time.  At long range, the target must roll a 1 before you can score a Penetrating Hit.  That's not good odds for you.  Pursue it only if you must.

Two
Common examples:

• T34/85 vs. Panther
• Panzer IVD vs. Char 1B
• Panzer II vs. Hotchkiss

At a difference of 2, you can only score Glancing hits at long range.  At close range, the target must roll a 1 in order for you to Penetrate it.  These odds are so bad, that you might as well not even bother.  Pursue it only if you must.

One
Common examples:

• Rifles or MGs against Halftracks
• Sherman 75s against Tiger IE
• Panzer IVHs against IS-2s
• Panzer III (long 75) vs Churchill III 

At a difference of 1, you can only score a Glancing hit at close range, and if the target flubs it with a 1.  I really wouldn't even bother with this shot, unless it's your only choice.


Tactical Considerations
There are many factors to consider when you apply this math.

If you have the bigger gun, remember, no matter how superior it is, you still have to hit the target.  If the enemy is skilled and concealed, he may die slowly, indeed, and your gun may be better used elsewhere.  Likewise, every weapon has a Firepower rating.  Usually a 3+.  This means, that even if you Penetrate a target, usually it will still only Bail 33% of the time.  Against Fearless tanks or tanks with Protected Ammo, Bails may not mean much at all.

The degree of superiority matters a lot, too.  Ideally, your tanks should vaporize the enemy before they can fight back.  The less of an advantage you have, the more tanks (or the more time) it will take to destroy a target. 

Conversely, if your tanks face the crappy end of the math, bear in mind -- they still have to hit you.  Try to stay at range and in concealment.  Or better yet, out of LoS together.  Force the big gun to move, cutting down on its RoF.  Depending on your own armament, you may be able to snipe at him, or just swarm or flank.

You need to look at the chart from both directions.  Your tank may have an awesome gun but terrible armor -- Nashorn, anyone?  Or it may have great armor and a pathetic gun -- like the KV-1.

Finally, FA vs AT considerations completely ignore the possibility of flanking shots on side armor.  Most tanks have a significantly weaker side armor.  The chess of tank battles requires you to think ahead to prevent side shots and to close or open range, depending on where and how you have the advantage.