One of the key differences between OD&D and 3.5+ is the way the game handles Armor Class (AC) and Attack Rolls. Early Editions of Dungeons and Dragons (LBB, Supplements, Holmes, BX, BECMI, and Cyclopedia) make use of a simple chart based system for resolving attack rolls. You find the attackers row on the relevant chart, slide over to the defenders armor class, and you’ve got your target number (ToHit) for the attack roll. Later editions of D&D (3, 3.5, 4, Pathfinder) utilize Ascending AC where the AC is replaced by the first row of ToHit numbers on the Attack Table such that a first level PC’s ToHit number is identical to the targets AC. The attack roll has to be ≥ the ToHit number to succeed, Swords and Wizardry offers both systems.
Ascending AC
In the simplest imaginable example its clear that Ascending AC is the winner, but the problem with this AC=ToHit model is that as a combatants level progresses so too does the ToHit number, so in Descending AC the ToHit number is found on the chart, whereas in Ascending AC a little math is used to derive the ToHit number, and this math is then hidden so as to seem simpler. So Mathematically it looks like:
D20+mods ≥ AC is a success
But more accurately
D20+AttackBonus+Mods ≥ AC is a success
Which can be written as
D20+Mods ≥ ACAttackBonus
Thaco
So ACAttackBonus is the same as the ToHit number from the Attack Table. You can get the same exact result if you take the score from the AC0 column on the correct row for your Character Based on class and level (the Thac0 or ToHit AC 0) and subtract the Descending AC from that, or mathematically:
D20+Mods ≥ Thac0AC
So they are basically the same when the Thac0 based formula is written as below:
D20+Mods+AttackBonus ≥ AC D20+Mods+AC ≥ Thac0
Additional Subtraction with Descending AC
A commonly cited problem with Descending AC (as described above) is that any modifications which affect your AC are described as a + when beneficial and – when harmful, but are subtracted from the Base AC. This means that a +1 shield or a +1 dexterity bonus subtracts 1 from the armor class, but since subtracting one from the armor class is the same as adding one to the ToHit number this really isn’t that confusing, but for new players it may be easier to leave AC alone and keep the Defense bonuses seperate, as below:
D20+Mods ≥ ToHit+DefenseBonus
or using Thac0
D20+Mods+AC ≥ Thac0+DefenseBonus
Since you can prework ACDefenseBonus (which again is ToHit+DefenseBonus) and write it down as Modified AC to simplify things the above is slightly less efficient, but manageable if you genuinely dislike the idea of subtracting a +1 bonus.
AC in Play
Ascending AC

Player gives their D20 roll plus any bonuses they know about

Game master compares that to AC Plus penalties PC doesn’t know about
Descending AC

Player gives their D20 roll plus their bonuses and level/class if GM doesn’t remember

Game Master looks at chart, gets target number and adds Penalties PC doesn’t know about
or

Player gives D20 roll plus any bonuses, also gives Thac0 (by subtracting roll from Thac0 Player knows what AC they should hit).

Game master adds bonuses to Thac0 or the D20 Roll as necessary and subtracts Roll from Thac0 to see if if they can hit the opponents AC.
Why I prefer Descending AC with a Defense Bonus to Ascending AC with an Attack Bonus
Although the Swords and Wizardry rules do not set a Maximum or Minimum AC for either Ascending or Descending AC, the rules do generally keep Ascending AC between AC 10 and AC 28, with all AC’s above AC 19 requiring supernatural bonuses, and Descending AC is kept between AC 9 and AC 9 with 0 being the boarder between the mundane armor and supernatural gifts.
I like the numbers for Descending AC with a Defense Bonus for entirely aesthetic reasons. AC 0 is a logical barrier, creatures with natural AC’s below 0 are something special, and being unable to hit someone with a roll that should have struck an AC 0 or higher opponent tells you that something is not quite right about this person, either they are something out of the ordinary, or they have access to something out of the ordinary.
Armor Classes with Descending AC
2 Plate with shield 3 Plate 4 Chain with shield 5 Chain 6 Leather with shield 7 Leather 8 Just a shield 9 Regular clothing or bare skin
Universal AC on the Character Sheet
To make converting between all of the above easier you can put the following on your Character Sheet:
(Ascending AC ___ + Defense Bonus ___) + (Descending AC ___  (Defense Bonus ___)) = 19 19  Thac0 ___ = Attack Bonus ___
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