@general-6-stars I’m really glad you tested this out, because it gives me a chance to explain more carefully what I intended!
Dice should only be compared within each color, not between colors.
So for the land battle, attacker has 3 fighters, which could roll 1 yellow die – but the defender doesn’t have any planes, and yellow dice can only hit planes, so there’s no point in rolling the yellow die. Instead, it would be better to assign the planes to boost the attacking infantry.
Attacker has 3 tanks, so rolls 1 red die, and boosts it by +1 because of the mech. infantry. You had the attacker rolling a 6, so the attacker is showing a 6 + 1 = 7 on his red die. The defender also rolled a 6, but has nothing to boost it with. The attacker’s 7 beats the defender’s 6, so the attacker gets to inflict one casualty on the defender, and the attacker gets to choose who dies. In this case I would suggest choosing the artillery, since the artillery has not fired yet and the artillery is more immediately useful than the second defending tank.
Next up is white dice. As you correctly figured out, the attacker rolls 2 white dice and boosts each of them by +1 for the supporting artillery. The attacker can also boost each die +1 for the supporting fighters. With both artillery support and air support, the white dice each get +2. You rolled a 2 and a 4, so that becomes a 4 and a 6.
The defender also rolls 2 white dice, only one of which gets boosted (because the other artillery got killed by the attacking tank). The boost is only +1, because the defender only has artillery support, not air support. Defender rolls a 2 and a 4. The defender can choose which die to boost, but it doesn’t really matter; you get the same results either way. Let’s say they boost the 4, so defenders’ rolls are a 2 and a 5.
The attacker’s 6 beats the defender’s 5, and the attacker’s 4 beats the defender’s 2. The defender has to take two casualties. They can choose which casualties to take. I’d suggest taking 1 infantry and 1 tank, because taking 2 infantry would cost the defender a white die in the next round of combat, since the defender would then only have 3 infantry left.
So, total casualties from the first round are zero for the attacker, and 1 inf, 1 art, 1 tnk for the defender. The tanks and planes didn’t roll very many dice directly, but because the attacker had better support from stuff like tanks and planes, they were able to boost their dice to high numbers, which means that the attacker got much better results in the end.