So let's get into the real meat of the game, which is all about crushing your enemies, seeing them driven before you and hearing the lamentations of their women.
First off, it's essential to understand the counter layout for units in World at War, so let's take a look.
The set of numbers in the upper left and lower left corners of the counter represents the Armor Piercing (AP) attack rating and the High Explosive (HE) attack rating. The AP attack numbers are used when attacking hard targets, which are basically armored targets like tanks. The HE attack rating is used when attacking soft targets, such as infantry. You can tell whether a target is hard or soft by the picture on the counter. If it's an actual picture of a unit then it's a hard target. If it's a NATO symbol (such as the X for infantry units), then it's a soft targets. In the example above, the Abrams tank is a hard target.
Anyway, when the Abrams attacks a hard target and uses its AP attack, there are three numbers to consider, the 10/4/4. The "10" represents the range of the Abrams unit in hexes. The Abrams can actually shoot up to twice this range but will incur some penalties, but I'll discuss that later.
The next number, the big red "4" represents the firepower of the Abrams. It rolls this number of six sided dice in an attack. The final number, the small white "4" represents the "to hit" number. Each die roll of 4, 5, or 6 will result in a hit on the enemy unit.
Let's have an example, shall we?
The Abrams in the picture above is going to shoot at the T-72, which is six hexes out. The T-72 counter has a picture of a tank on it so we know it's a hard target. Therefore, we are going to use our AP rating to fire at it with the Abrams.
The T-72 is well within the 10 hex range of the Abrams so there are no penalties to hit. We roll four six sided dice and hope for results of 4 or better.
We get...6/3/6/3 on the dice results. We have scored two hits on the T-72. Terrific!
Long Range Fire
As I stated above, the "10" hex range can be exceeded by some units. If the number for the range is NOT underlined, the unit can fire at extended range, up to twice the printed range. In this case, the Abrams can fire up to 20 hexes. However, the penalty for firing at such long range is a +1 for the "to hit" number. This means the Abrams would be rolling 4 dice (same firepower number) but only results of a "5" or "6" would count as a hit.
In the example above, you can see that the range of "10" for the Abrams is not underlined, so therefore it is capable of extended range (it can fire up to 20 hexes with a small penalty). The Soviet BRDM-AT, however, has its AP range of 14 underlined so it actually cannot fire beyond this range ever.
Let's have an example with extended range combat:
In the above example, the Abrams is firing at the T-72, which is 11 hexes out. This is beyond the "10" hex range of the Abrams but we can still take a shot. We roll 4 dice for the firepower and hope for a result of "5" or "6". We get...3/1/5/1
The Abrams has scored a single hit on the T-72. Good stuff.
Close Range Fire
Just as there are penalties for firing at extended range, there are bonuses for firing at an enemy in close range. Again, these bonuses and penalties are only available for units whose range is NOT underlined. So the Soviet BRDM-AT will not get a close range bonus.
When a unit attacks another unit that is half of its printed range or less, the "to hit" roll decreases by 1. This means that the Abrams will get the "to hit" bonus at firing at units that are 5 hexes or closer to it.
In the example above, the T-72 is only three hexes away from the Abrams so it's in close range. The Abrams will roll 4 firepower dice and will hit on a result of 3, 4, 5, and 6. So we roll and get: 1/1/3/5. The Abrams scores two hits on the T-72.
Note that these range bonuses and penalties also apply for HE attacks.
In the example above, the Abrams is firing at the Soviet infantry two hexes away. Since it's a soft target, we're using the HE attack numbers for the Abrams, which are "5" range, "3" firepower, and "5" to hit. Since the range to the Soviet infantry is half the printed range for the Abrams "5" HE range, it gets a bonus to its "to hit" numbers. So now the Abrams will roll 3 dice firepower and will hit on a 4 or 5. The Abrams rolls and gets... 4/3/5, resulting in two hits on the Soviet infantry.
Since we're dealing with relatively modern combat vehicles, World at War provides some units with the capability of firing while on the move. Units that are moving-fire capable are indicated by an underlined firepower rating on the counter. These units can move up to half their movement rate (rounded down) and then fire. Note that they cannot fire and then move or move, fire and then move. It's only move and then fire - full stop.
The penalties related to movement fire can be found on the trusty play aid included with your World at War game but, as a general rule of thumb, most units that conduct moving fire will suffer both a -1 to their firepower and a +1 to their "to hit" rating. Note that the Abrams is a really high tech unit so it only suffers the +1 for the "to hit" rating.
Let's get an example going:
In the example above, the T-72 has moved from J12 to J11 and will fire at the Bradley sitting in G9. The T-72 would normally be rolling four 6-sided dice with a "to hit" of "5" or greater. However, since the T-72 is moving, we roll only three dice with a "to hit" number of "6".
The T-72 rolls: 5/5/2. No hits are scored. After this, the T-72 is marked Ops Complete. It cannot continue to move.
Note that bonuses related to range can still kick in here and modify the numbers. Here's another example:
In the case above, the T-72 is also conducting moving fire. However, it has moved within close range of the Bradley (the T-72 has a "7" range for AP attacks, which means that close range counts as 3 hexes or less). The T-72 still suffers a -1 to firepower and a +1 to hit for conducting moving fire, but it also gets a close range bonus of -1 to hit, which negates the "to hit" moving fire penalty. As a result, the T-72 is rolling three dice and will hit on a "5" or "6". The T-72 rolls: 2/5/1. It scores one hit on the Bradley. Again, the T-72 is marked Ops Complete and can move no further.
Defensive Rolls: Hard Targets
So far, we've only been talking about attacking and hitting. It's important to note that when an enemy unit is hit, it gets a chance to negate those hits. How units can negate hits is determined partly by whether it's a hard target or a soft target, so let's start with hard targets.
All hard targets have a defensive armor rating in the upper right hand corner of their counter.
The Abrams defensive armor rating is 3/5. So when the Abrams gets hit, it will roll 3 six sided dice and each roll that is 5 or 6 will negate one hit. The number of dice the Abrams gets to roll can be modified by certain conditions such as being in certain terrain but I'll talk about that later.
Here's an example:
The T-72 above fires at the Abrams, scoring two hits on it. The Abrams now gets a chance to negate those hits by rolling three 6-sided dice and hoping for a result of 5 or 6. The Abrams rolls a 4/5/2. One of the two hits is negated.
Defensive Rolls: Soft Targets
As stated above, hard targets and soft targets make defensive rolls differently. Taking a look at an infantry counter, for example, you'll notice that it has no defensive armor rating:
Soft targets like infantry get a defensive die roll based on the terrain which they are in. For open terrain, they get nothing. They are automatically hit and cannot roll defensive die to negate hits (so keep your infantry out of the open). Infantry in woods hexes get a single defensive die. This means they roll one 6-sided die and they negate a hit on a "5" or "6" result. Infantry in cities get two defensive dice. So they roll two 6-sided dice and negate a hit on a "5" or "6" result.
Here's an example:
In the above example, the Abrams can attack infantry in L12 (clear hex), L13 (woods), or the infantry in the city (H14). Let's say the Abrams fires at the infantry in L12 and scores two hits. The infantry is in a clear hex so it gets no defensive die roll. It simply takes the two hits and tries to walk it off.
If the Abrams fires at the infantry in the woods hex (L13) and scores two hits, however, the infantry can roll one 6-sided die and negates one of the hits on a roll of "5" or "6". No matter what, though, it's definitely going to take one hit.
Finally, if the Abrams fires at the infantry in the city hex (H14) and scores two hits, the infantry gets to roll two 6-sided dice and each result of "5" or "6" will negate one hit. So if the infantry in the city rolls and gets a 5 and a 6, it takes no hits.
This covers basic combat - attacking and defensive rolls. Of course, there are several more aspects to ranged combat such as terrain bonuses and concealment but I believe this is enough to give beginners a grasp of the basics and as a supplement to the rulebook. If you see any mistakes here, please let me know in the comments.