*Welcome to *Guidance*, Private Sanctuary’s source for tips and techniques for the Pathfinder Roleplaying Game, written by Everyman Gamer Alexander Augunas. Today, we’re going to be talking about the mathematics behind the Challenge Rating system.*

Before I was the Everyman Gamer, I created a little Google Drive document called the GM’s Guide to Challenging Encounters, and its actually how I got my start on offering tips and tricks to players and GMs alike. Back in the day, I got tons of messages from GMs asking me how it works, so I went and wrote this article to break down the mathematical mysteries of the CR system to 1) better compartmentalize the guide’s thoughts and ideas and 2) to provide a wider array of content and informative articles to the Pathfinder Roleplaying Game Community. I’ve noticed that there are a surprisingly few number of blogs that give detailed system analysis like this. And if there’s a niche to be filled, you can bet your stars that I’m going to try and fill it!

# Pathfinder’s Ever-Beating Heart

When you sit down at your gaming table, do you ever think to yourself, “I’m ready for a hardcore session of mathematics?” Perhaps you should. Far beyond calculating the amount of damage you deal with your +5 keen bastardizing longsword, the entire Pathfinder Roleplaying Game system is built upon the shoulders of mathematics. In nowhere is this more evident than in the beast that is the CR system.

CR, or Challenge Rating, is a numerical value given to all challenges in the Pathfinder Roleplaying Game. The CR mechanic is traditionally associated with monsters, but it has expanded to include traps, haunts, spirits, and more. In order to grasp the CR mechanic, there are several concrete facts that all GMs must understand.

- A challenge’s CR and the amount of XP that it rewards to players when slain are related. Compare all CR 7 monsters in the game. They all reward the same amount of XP when conquered. If you find an exception, it’s a misprint; there are no exceptions.
- A challenge’s XP reward increases exponentially with its CR. The formula is CR + 2 = 2(XP), or “Increasing an encounter’s CR by 2 doubles its XP reward.” Likewise, increasing an encounter’s CR by 4 quadruples its XP reward. This formula does not hold true with any other number; only 2s and 4s. You can verify this for yourself in Chapter 12: GameMastering in the Core Rulebook.
- A challenge’s CR assumes that the PCs are a 4-person party. I’ll be explaining this fact shortly, but for now, you’ll have to settle for my word.

# CR and APL — What’s the Connection?

When we’re dealing with encounter design, the other abbreviation you’ve likely seen before is APL, or Average Party Level. Simply put, this is the combined level of all your players averaged together. Now, the Core Rulebook flat-out tells you that APL assumes four to five members in a party. Here’s what it doesn’t tell you; that fifth party member, he should be increasing the CR too, and the amount he should be increasing the encounter’s CR by grows larger and larger with each passing level. Let’s look at some mathematics!

Before we can analyze the connection between APL and CR, we need to convert them into numerical values that can actually be compared. In order to do this, we need to discover the CR equivalence for a PC. For ease of use, we’ll discuss 1st level PCs because 1st level is a little murky. According to Chapter 12 of the Core Rulebook, a 0 HD character has a CR equal to its class levels –1; for a 1st level character, this translates to CR ½. However, you might have noticed that NPCs are typically outfitted with less wealth than PCs; Chapter 12 also notes that a character with PC wealth increases its CR by +1. Effectively, a properly equipped PC has a CR equal to his character level, so our 1st level PC is CR 1 and has an XP reward of 400 xp. When comparing apples to apples, a fight against a 1st level PC is equivalent to any CR 1 monster you’ll find in any Bestiary. Combined together, a full team of 1st Level PCs is an APL 1 party worth 1,600 XP. If you remember our Rule of 4 (CR + 4 = 4(XP)), it probably won’t surprise you that an APL Party is actually a CR 5 encounter according to mathematics; according to the Core Rulebook, an appropriate encounter of average difficulty (APL + 0) for a 5th level party. Would a team of 1st level PCs prove challenging to a team of 5th level PCs? In all, it’s likely to be as difficult as a team of 1st level PCs going up against a CR 1 darkmantle. The verdict? Mathematically speaking, pulling the veil off of the APL system and converting it to a CR ranking shows that even an encounter of average difficulty is heavily slanted towards the PCs.

# The Linear Guild Combat

So, what happens then if you pit a party of PCs against their equals? Imagine we take a few dozen scrolls of simulacrum and perfectly duplicate each member of your PCs’ party and make the two groups fight each other. I call combats like this Linear Guild encounters; its namesake comes from the Order of the Stick webcomic, in which the Linear Guild is an antagonistic party to the PC party with an evil opposites theme.

Well, as 1st level characters with NPC wealth all members of the brawl are worth 400 XP. This means that each side is worth 1,600 XP (400 XP * 4 participants), which is the equivalent of a CR 5 encounter. That said, the PC party’s APL does not change; they are still an APL 1 party. This means your players are an APL 1 party fighting a CR 5 encounter; encounters of this level of difficulty aren’t even outlined in the Core Rulebook; the highest modifier noted is CR +3; titled epic encounters. But as you can plainly see, in a Linear Guild encounter (APL +4) both sides are perfectly even. For every modifier less than CR +4 that your encounter is, your players receive roughly 25% more XP in their favor; But don’t take my word for it; check the numbers:

- APL 1 (1,600 XP) / CR 5 (1,600 XP) – Equal
- APL 1 (1,600 XP) / CR 4 (1,200 XP) – Challenge has 75% of party’s XP
- APL 1 (1,600 XP) / CR 3 (800 XP) – Challenge has 50% of party’s XP
- APL 1 (1,600 XP) / CR 2 (600 XP) – Challenge has 37.5% of party’s XP
- APL 1 (1,600 XP) / CR 1 (400 XP) – Challenge has 25% of party’s XP

Doesn’t match up perfectly, but you can see the obvious progression. In this case, that extra XP translates into resources; additional hit points, on-use abilities, wealth, actions taken per round, and so on. In a nutshell, mathematics shows us that encounters cannot be truly even unless the party’s APL is equal to the challenge’s CR + 4.

# STOP.

Put away your Core Rulebook and Bestiaries. I’m not finished yet.

Yes. I just said encounters cannot be even unless their CR equals the party’s APL + 4. What I did not say is that they cannot be challenging unless their CR equals the party’s AP 4, so that doesn’t mean you should just thumb through your Bestiaries, find a monster whose numbers match up, and plop it onto the battlefield. There are far too many factors to consider.

The method I listed for challenging PCs works when pitting your PCs against many smaller creatures that combine their CR in the same way a party of PCs does. Tribes of goblins, a pride of lions, the Linear Guild. This system works for them. You should not use these mathematical facts to justify throwing a balor against your APL 16 PCs without serious consideration for the consequences. Here is another important fact.

Math does not consider Special Abilities and vice versa.

Yes. Some special abilities scale with Hit Dice, and therefore scale with the CR system and math. But not all abilities do, and you need to consider what your players have access to when they face the encounter.

For example, mathematically speaking, we’ve proven that a CR 5 encounter is perfectly even to an APL 1 party. That said, would you throw a green hag (CR 5) at a group of 1st level adventurers? She has an AC of 19, so even your party fighter isn’t likely to hit her without a natural 15 or better. She, on the other hand, has a +13 to her attack rolls at a level where full plate is too expensive to purchase and her attacks require a DC 16 Fortitude save to prevent her from sapping 2 points of Strength from her victims. A Con 16 Fighter (bonus of +5) has a 55% chance of failing his save against her. This is ignoring the fact that with a Stealth bonus of +13, she’s almost always going to get the jump on even your best 1st level rogue or monk. As you can see, “balanced” means “equal number of resources at each side’s disposal. Packing all of those resources into one container is likely to utterly destroy your PCs.

That’s it for this installment of the GM’s Guide. Next week we’ll be looking at using XP Rewards instead of CRs to design encounters, and this is where we’ll talk more about how adding a fifth member to your PC party really affects your group’s APL. Until then, what are some strategies you use when planning encounters? Do you prefer a myriad of easy encounters that your PCs can breeze through, or do you prefer a small number of extremely difficult encounters? Have you ever used an extremely high CR monster against your PCs (+4 or greater)? What was the result? Did you feel I covered this topic sufficiently, or do you have any lingering questions remaining? Leave your answers in the comments below!

*Alexander “Alex” Augunas has been playing roleplaying games since 2007, which isn’t nearly as long as 90% of his colleagues. Alexander is an active freelancer for the Pathfinder Roleplaying Game and is best known as the author of the Pact Magic Unbound series by Radiance House. Alex is the owner of Everyman Gaming, LLC and is often stylized as the Everyman Gamer in honor of Guidance’s original home. Alex’s favorite color is blue, his favorite Pathfinder Race/Class combination is kitsune rogue, and he loathes long walks on the beach because he sunburns easily.*

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