This paper discusses the mechanics and mathematics of Experience Point awards given to characters after they overcome challenges. After defining and explaining the terminology, I present two methods.

The first method is the chart from the Dungeon Master's Guide. I identify at least seven minor inconsistencies in it. After correcting the inconsistencies, I derive three formulae to explain the table.

I then derive a single consistent formula for calculating experience based on ECL and CR and present a new chart using this formula.

Introduction

Characters earn experience by succeeding in "encounters" that present them with "challenges". A challenge can be a monster, a trap, a tricky diplomatic negotiation, or anything else that the characters must get past in order to achieve their goals. Every such challenge overcome by a party earns each member of the party a certain number of "experience points", or XP. How much XP is earned by a particular character is determined by comparing the difficulty of the challenge with the capability of that character.

In order to make that determination, you need metrics to judge both aspects: how capable is a particular character, and how difficult is the challenge?

The metric for a character is simply the "character level" (CL), or "effective character level" (ECL), if the character is a member of a "powerful race". The corresponding metric for a challenge is the "challenge rating" (CR).

As I discuss later, those two numbers determine how much XP a character earns for helping to overcome a challenge.

Neither characters nor challenges tend to run solo: characters form into adventuring parties, and groups of monsters often also run together in packs.

Since overcoming a group of monsters is harder than overcoming just one, an encounter with multiple monsters will have a higher difficulty metric. The metric for an encounter as whole is called the "Encounter Level" (EL). If there is a single challenge in the encounter, the EL equals the CR of the challenge. There are rules and charts for scaling up the EL for encounters incorporating multiple CRs.

The average ECL of the characters in a party is called the "party level". The typical party has four characters with a well-balanced variety of classes. Published adventures choose the difficulty of their encounters assuming a party of that size. Smaller or larger parties are possible but will find the challenges more or less difficult than intended. I've not seen any rules or charts for scaling "party level" if there are fewer or more than four characters in the party.

When assessing the difficulty of an encounter, the DM compares "party level" to "encounter level".

The typical D&D encounter pits a group of four characters of average level X against something with EL X. An encounter of that sort is defined to be "challenging" and is expected to use up about 20% of the party's resources.

From the V3.5 Dungeon Master's Guide, page 49:

Encounter Difficulty

Encounter	Description
Easy	EL lower than party level
Challenging	EL equal to party level
Very difficult	EL 1-4 higher than party level
Overpowering	EL 5+ higher than party level

XP method from the Dungeon Master's Guide

After succeeding in an encounter, each member of the party gains XP for each challenge within the encounter based on her ECL, the individual challenge's CR, and the number of characters in her party.

XP(ECL, CR) ÷ N

Let's see if we can derive the formula for XP(ECL, CR). The V3.5 Dungeon Master's Guide, has a table for this on page 38:

Experience Point Awards (Single Monster)

ECL	CR 1	CR 2	CR 3	CR 4	CR 5	CR 6	CR 7	CR 8	CR 9	CR 10	CR 11	CR 12	CR 13	CR 14	CR 15	CR 16	CR 17	CR 18	CR 19	CR 20
1	300	600	900	1,350	1,800	2,700	3,600	5,400	7,200	10,800	**	**	**	**	**	**	**	**	**	**
2	300	600	900	1,350	1,800	2,700	3,600	5,400	7,200	10,800	**	**	**	**	**	**	**	**	**	**
3	300	600	900	1,350	1,800	2,700	3,600	5,400	7,200	10,800	**	**	**	**	**	**	**	**	**	**
4	300	600	800	1,200	1,600	2,400	3,200	4,800	6,400	9,600	12,800	**	**	**	**	**	**	**	**	**
5	300	500	750	1,000	1,500	2,250	3,000	4,500	6,000	9,000	12,000	18,000	**	**	**	**	**	**	**	**
6	300	450	600	900	1,200	1,800	2,700	3,600	5,400	7,200	10,800	14,400	21,600	**	**	**	**	**	**	**
7	263	350	525	700	1,050	1,400	2,100	3,150	4,200	6,300	8,400	12,600	16,800	25,200	**	**	**	**	**	**
8	200	300	400	600	800	1,200	1,600	2,400	3,600	4,800	7,200	9,600	14,400	19,200	28,800	**	**	**	**	**
9	*	225	338	450	675	900	1,350	1,800	2,700	4,050	5,400	8,100	10,800	16,200	21,600	32,400	**	**	**	**
10	*	*	250	375	500	750	1,000	1,500	2,000	3,000	4,500	6,000	9,000	12,000	18,000	24,000	36,000	**	**	**
11	*	*	*	275	413	550	825	1,100	1,650	2,200	3,300	4,950	6,600	9,900	13,200	19,800	26,400	39,600	**	**
12	*	*	*	*	300	450	600	900	1,200	1,800	2,400	3,600	5,400	7,200	10,800	14,400	21,600	28,800	43,200	**
13	*	*	*	*	*	325	488	650	975	1,300	1,960	2,600	3,900	5,850	7,800	11,700	15,600	23,400	31,200	46,800
14	*	*	*	*	*	*	350	525	700	1,050	1,400	2,100	2,800	4,200	6,300	8,400	12,600	16,800	25,200	33,600
15	*	*	*	*	*	*	*	375	563	750	1,125	1,500	2,250	3,000	4,500	6,750	9,000	13,500	18,000	27,000
16	*	*	*	*	*	*	*	*	400	600	800	1,200	1,600	2,400	3,200	4,800	7,200	9,600	14,400	19,200
17	*	*	*	*	*	*	*	*	*	425	638	850	1,275	1,700	2,550	3,400	5,100	7,650	10,200	15,300
18	*	*	*	*	*	*	*	*	*	*	450	675	900	1,350	1,800	2,700	3,600	5,400	8,100	10,800
19	*	*	*	*	*	*	*	*	*	*	*	475	713	950	1,425	1,900	2,850	3,800	5,700	8,550
20	*	*	*	*	*	*	*	*	*	*	*	*	500	750	1,000	1,500	2,000	3,000	4,000	6,000

* No "standard" XP is defined for monsters whose CR is eight or more levels below the ECl of the character. A horde of such monsters may very well be a difficult challenge, but a single monster in the horde provides negligible difficulty.

** No "standard" XP is defined for monsters whose CR is eight or more levels above the ECL of the character, as such an encounter should be too difficult for the character to overcome.

There are some irregularities in this table: aside from levels 1 - 3 being identical, there are a few odd numbers. We'd like to come up with a formula that can be applied to higher ECLs, as well as higher CRs for the listed ECLs.

ECL appears to be a factor in this table. Let's divide each number by the ECL it refers to and see if that clarifies things:

XP(ECL, CR) ÷ ECL

ECL	CR 1	CR 2	CR 3	CR 4	CR 5	CR 6	CR 7	CR 8	CR 9	CR 10	CR 11	CR 12	CR 13	CR 14	CR 15	CR 16	CR 17	CR 18	CR 19	CR 20
1	300	600	900	1,350	1,800	2,700	3,600	5,400	7,200	10,800	**	**	**	**	**	**	**	**	**	**
2	150	300	450	675	900	1,350	1,800	2,700	3,600	5,400	**	**	**	**	**	**	**	**	**	**
3	100	200	300	450	600	900	1,200	1,800	2,400	3,600	**	**	**	**	**	**	**	**	**	**
4	75	150	200	300	400	600	800	1,200	1,600	2,400	3,200	**	**	**	**	**	**	**	**	**
5	60	100	150	200	300	450	600	900	1,200	1,800	2,400	3,600	**	**	**	**	**	**	**	**
6	50	75	100	150	200	300	450	600	900	1,200	1,800	2,400	3,600	**	**	**	**	**	**	**
7	37.5	50	75	100	150	200	300	450	600	900	1,200	1,800	2,400	3,600	**	**	**	**	**	**
8	25	37.5	50	75	100	150	200	300	450	600	900	1,200	1,800	2,400	3,600	**	**	**	**	**
9	*	25	37.5	50	75	100	150	200	300	450	600	900	1,200	1,800	2,400	3,600	**	**	**	**
10	*	*	25	37.5	50	75	100	150	200	300	450	600	900	1,200	1,800	2,400	3,600	**	**	**
11	*	*	*	25	37.5	50	75	100	150	200	300	450	600	900	1,200	1,800	2,400	3,600	**	**
12	*	*	*	*	25	37.5	50	75	100	150	200	300	450	600	900	1,200	1,800	2,400	3,600	**
13	*	*	*	*	*	25	37.5	50	75	100	150	200	300	450	600	900	1,200	1,800	2,400	3,600
14	*	*	*	*	*	*	25	37.5	50	75	100	150	200	300	450	600	900	1,200	2,400	3,600
15	*	*	*	*	*	*	*	25	37.5	50	75	100	150	200	300	450	600	9500	1,200	1,800
16	*	*	*	*	*	*	*	*	25	37.5	50	75	100	150	200	300	450	600	900	1,200
17	*	*	*	*	*	*	*	*	*	25	37.5	50	75	100	150	200	300	450	600	900
18	*	*	*	*	*	*	*	*	*	*	25	37.5	50	75	100	150	200	300	450	600
19	*	*	*	*	*	*	*	*	*	*	*	25	37.5	50	75	100	150	200	300	450
20	*	*	*	*	*	*	*	*	*	*	*	*	25	37.5	50	75	100	150	200	300

One formula we can derive is simple: the XP earned for defeating a challenge whose CR equals the ECL: the bold numbers in the table.

XP_ECL=CR = 300 × ECL

For simplicity, we'll reason further with ECL 10. Lets normalize by dividing by 3000.

XP(10, CR) ÷ 3000

ECL	CR 1	CR 2	CR 3	CR 4	CR 5	CR 6	CR 7	CR 8	CR 9	CR 10	CR 11	CR 12	CR 13	CR 14	CR 15	CR 16	CR 17	CR 18	CR 19	CR 20
10	*	*	1/12	1/8	1/6	1/4	1/3	1/2	2/3	1	3/2	2	3	4	6	8	12	**	**	**

It's symmetrical: for a CR x levels above a particular ECL, you multiply by a factor. For a CR x levels below the ECL, you divide by the same factor.

Additionally, we can see that overcoming a challenge of CR (X + 2) is worth double the XP of defeating CR X with one exception: going from CR (ECL - 1) to CR (ECL + 1) goes from 2/3 the XP to 3/2 the XP.

Unfortunately, this all means that there's not one formula that describes the data in this chart. Remembering that

XP_ECL=CR = 300 × ECL

We can derive the following three formulae:

(CR - ECL) is even:

XP(ECL, CR) = XP_ECL=CR × 2 ^{(CR - ECL) ÷ 2}

(CR - ECL) is odd and greater than zero:

XP(ECL, CR) = 3/2 × XP_ECL=CR × 2^{(CR - ECL - 1) ÷ 2}

(CR - ECL) is odd and less than zero:

XP(ECL, CR) = 2/3 × XP_ECL=CR × 2^{(CR - ECL + 1) ÷ 2}

Three formulae or not, the numbers end up being nice and even, generally, in the big chart, since the fractions chosen divide nicely into 300.

Now, with a single row describing the XP for ECL 1 - 3, two (at least) of those ECLs will not fit the formulae well. A few other specific values are not what one would expect, either:

Bad values in XP table

XP(ECL, CR)	is	should be
XP(3,1)	100	150
XP(4,1)	75	100
XP(4,5)	400	450
XP(4,7)	800	900
XP(4,9)	1,600	1,800
XP(4,11)	3,200	3,600
XP(5,1)	60	75

Let's assume that those are typos, and ignore ECL 1 and 2 for now. Then, it's easy to extend the table to ECLs and CRs higher than the table in the Dungeon Master's Guide. Further, it's obvious how to calculate an XP for monsters whose CR is farther removed from the character's ECL than the table normally accounts for.

Use this table for encounters within the "expected" range:

XP(ECL, CR)

CR - ECL	Earned XP
-7	25 × ECL
-6	37.5 × ECL
-5	50 × ECL
-4	75 × ECL
-3	100 × ECL
-2	150 × ECL
-1	200 × ECL
0	300 × ECL
+1	450 × ECL
+2	600 × ECL
+3	900 × ECL
+4	1,200 × ECL
+5	1,800 × ECL
+6	2,400 × ECL
+7	3,600 × ECL

Using that table, if we allow ECL 1, ECL 2, and ECL 3 to all use the formulae, rather than sharing one line in the table, the first five levels of the XP table now look like this:

Experience Point Awards (Single Monster)

ECL	CR 1	CR 2	CR 3	CR 4	CR 5	CR 6	CR 7	CR 8	CR 9	CR 10	CR 11	CR 12	CR 13	CR 14	CR 15	CR 16	CR 17	CR 18	CR 19	CR 20
1	300	450	600	900	1,200	1,800	2,400	3,600	**	**	**	**	**	**	**	**	**	**	**	**
2	400	600	900	1,200	1,800	2,400	3,600	4,800	7,200	**	**	**	**	**	**	**	**	**	**	**
3	450	600	900	1,350	1,800	2,700	3,600	5,400	7,200	10,800	**	**	**	**	**	**	**	**	**	**
4	400	600	800	1,200	1,600	2,400	3,200	4,800	6,400	9,600	12,800	**	**	**	**	**	**	**	**	**
5	375	500	750	1,000	1,500	2,250	3,000	4,500	6,000	9,000	12,000	18,000	**	**	**	**	**	**	**	**

Single Formula XP

It should be possible to rationalize the XP chart such that a single formula can be used. From the chart:

XP(10, CR) ÷ 3000

ECL	CR 1	CR 2	CR 3	CR 4	CR 5	CR 6	CR 7	CR 8	CR 9	CR 10	CR 11	CR 12	CR 13	CR 14	CR 15	CR 16	CR 17	CR 18	CR 19	CR 20
10	*	*	1/12	1/8	1/6	1/4	1/3	1/2	2/3	1	3/2	2	3	4	6	8	12	**	**	**

Observe that the XP doubles when the CR goes up by two, but that the XP goes up by a factor of either 3/2 or 4/3 when the CR goes up by one. If the XP always goes up by a factor of √2 when the CR goes up by one, XP will still double every two CRs, but we can express the XP formula with a single formula.

The chart showing the factors now looks like:

XP(10, CR) ÷ 3000

ECL	CR 1	CR 2	CR 3	CR 4	CR 5	CR 6	CR 7	CR 8	CR 9	CR 10	CR 11	CR 12	CR 13	CR 14	CR 15	CR 16	CR 17	CR 18	CR 19	CR 20
10	*	*	1/(8√2)	1/8	1/(4√2)	1/4	1/(2√2)	1/2	1/√2	1	√2	2	2√2	4	4√2	8	8√2	**	**	**

The single formula expressing this rule is this:

XP(ECL, CR) = 300 × ECL × 2 ^{(CR - ECL) ÷ 2}

Here's an earned XP chart for ECL 10 comparing this new rule with the old rule from the Dungeon Master's Guide:

XP(10, CR)

Rule	CR 1	CR 2	CR 3	CR 4	CR 5	CR 6	CR 7	CR 8	CR 9	CR 10	CR 11	CR 12	CR 13	CR 14	CR 15	CR 16	CR 17	CR 18	CR 19	CR 20
New	*	*	265	375	530	750	1,061	1,500	2,121	3,000	4,243	6,000	8,485	12,000	16,971	24,000	33,941	**	**	**
Old	*	*	250	375	500	750	1,000	1,500	2,000	3,000	4,500	6,000	9,000	12,000	18,000	24,000	36,000	**	**	**
Δ	*	*	+15	0	+30	0	+61	0	+121	0	-257	0	-515	0	-1,029	0	-2,059	**	**	**

We can quantify the differences as follows.

1. There is no change when (CR - ECL) is even.
2. When the difference is odd and CR is greater than ECL, the original formula used a factor of 3/2 and the new formula uses √2. Therefore, the ratio between the new and the old formulae in that case is:

   √2 ÷ (3 ÷ 2)

   (2 ÷ 3) × √2

   (2 × √2) ÷ 3

   .9428

3. When the difference is odd and CR is less than ECL, the original formula used a factor of 2/3 and the new formula uses 1/√2. Therefore, the ratio between the new and the old formulae in that case is:

   (1 ÷ √2) ÷ (2 ÷ 3)

   3 ÷ (2 × √2)

   1.0607

Just as 3/2 is the reciprocal of 2/3, √2 is the reciprocal of 1/√2 and the two combined ratios are also reciprocals of each other.

The whole XP chart looks like this:

Experience Point Awards (Single Monster)

ECL	CR 1	CR 2	CR 3	CR 4	CR 5	CR 6	CR 7	CR 8	CR 9	CR 10	CR 11	CR 12	CR 13	CR 14	CR 15	CR 16	CR 17	CR 18	CR 19	CR 20
1	300	424	600	849	1,200	1,697	2,400	3,394	**	**	**	**	**	**	**	**	**	**	**	**
2	424	600	849	1,200	1,697	2,400	3,394	4,800	6,788	**	**	**	**	**	**	**	**	**	**	**
3	450	636	900	1,273	1,800	2,546	3,600	5,091	7,200	10,182	**	**	**	**	**	**	**	**	**	**
4	424	600	849	1,200	1,697	2,400	3,394	4,800	6,788	9,600	13,576	**	**	**	**	**	**	**	**	**
5	375	530	750	1,061	1,500	2,121	3,000	4,243	6,000	8,485	12,000	16,971	**	**	**	**	**	**	**	**
6	318	450	636	900	1,273	1,800	2,546	3,600	5,091	7,200	10,182	14,400	20,365	**	**	**	**	**	**	**
7	262	371	525	742	1,050	1,485	2,100	2,970	4,200	5,940	8,400	11,879	16,800	23,759	**	**	**	**	**	**
8	212	300	424	600	849	1,200	1,697	2,400	3,394	4,800	6,788	9,600	13,576	19,200	27,153	**	**	**	**	**
9	*	239	338	477	675	955	1,350	1,909	2,700	3,818	5,400	7,637	10,800	15,274	21,600	30,547	**	**	**	**
10	*	*	265	375	530	750	1,061	1,500	2,121	3,000	4,243	6,000	8,485	12,000	16,971	24,000	33,941	**	**	**
11	*	*	*	292	412	583	825	1,167	1,650	2,333	3,300	4,667	6,600	9,334	13,200	18,668	26,400	37,335	**	**
12	*	*	*	*	318	450	636	900	1,273	1,800	2,446	3,600	5,091	7,200	10,182	14,400	20,365	28,800	40,729	**
13	*	*	*	*	*	345	488	689	975	1,379	1,960	2,758	3,900	5,515	7,800	11,031	15,600	22,062	31,200	44,123
14	*	*	*	*	*	*	371	525	742	1,050	1,485	2,100	2,970	4,200	5,940	8,400	11,879	16,800	23,759	33,600
15	*	*	*	*	*	*	*	398	562	795	1,125	1,591	2,250	3,182	4,500	6,364	9,000	12,728	18,000	25,456
16	*	*	*	*	*	*	*	*	424	600	849	1,200	1,697	2,400	3,394	4,800	6,788	9,600	13,576	19,200
17	*	*	*	*	*	*	*	*	*	451	638	902	1,275	1,803	2,550	3,606	5,100	7,212	10,200	14,425
18	*	*	*	*	*	*	*	*	*	*	477	675	955	1,350	1,909	2,700	3,818	5,400	7,637	10,800
19	*	*	*	*	*	*	*	*	*	*	*	504	712	1008	1,425	2,015	2,850	4,031	5,700	8,061
20	*	*	*	*	*	*	*	*	*	*	*	*	530	750	1,061	1,500	2,121	3,000	4,243	6,000

And the &Delta chart between the Old and New methods is this:

Experience Point Awards (Single Monster): &Delta (New - Old)

ECL	CR 1	CR 2	CR 3	CR 4	CR 5	CR 6	CR 7	CR 8	CR 9	CR 10	CR 11	CR 12	CR 13	CR 14	CR 15	CR 16	CR 17	CR 18	CR 19	CR 20
1	0	-26	0	-51	0	-103	0	-206	**	**	**	**	**	**	**	**	**	**	**	**
2	+24	0	-51	0	-103	0	-206	0	-412	**	**	**	**	**	**	**	**	**	**	**
3	0	+36	0	-77	0	-154	0	-309	0	-618	**	**	**	**	**	**	**	**	**	**
4	+24	0	+49	0	-97	0	-194	0	-388	0	-776	**	**	**	**	**	**	**	**	**
5	0	+30	0	+61	0	-129	0	-257	0	-515	0	-1,029	**	**	**	**	**	**	**	**
6	+18	0	+36	0	+73	0	-154	0	-309	0	-618	0	-1,235	**	**	**	**	**	**	**
7	0	+21	0	+42	0	+85	0	-180	0	-360	0	-721	0	-1,441	**	**	**	**	**	**
8	+12	0	+24	0	+49	0	+97	0	-206	0	-412	0	-824	0	-1,647	**	**	**	**	**
9	*	+14	0	+27	0	+55	0	+109	0	-232	0	-463	0	-926	0	-1,853	**	**	**	**
10	*	*	+15	0	+30	0	+61	0	+121	0	-257	0	-515	0	-1,029	0	-2,059	**	**	**
11	*	*	*	+17	0	+33	0	+67	0	+133	0	-283	0	-566	0	-1,132	0	-2,265	**	**
12	*	*	*	*	+18	0	+36	0	+73	0	+146	0	-309	0	-618	0	-1,235	0	-2,471	**
13	*	*	*	*	*	+20	0	+39	0	+79	0	+158	0	-335	0	-669	0	-1,338	0	-2,677
14	*	*	*	*	*	*	+21	0	+42	0	+85	0	+170	0	-360	0	-721	0	-1,441	0
15	*	*	*	*	*	*	*	+23	0	+45	0	+91	0	+182	0	-386	0	-772	0	-1,544
16	*	*	*	*	*	*	*	*	+24	0	+49	0	+97	0	+194	0	-412	0	-824	0
17	*	*	*	*	*	*	*	*	*	+26	0	+52	0	+103	0	+206	0	-438	0	-875
18	*	*	*	*	*	*	*	*	*	*	+27	0	+55	0	+109	0	+218	0	-463	0
19	*	*	*	*	*	*	*	*	*	*	*	+29	0	+58	0	+115	0	+231	0	-489
20	*	*	*	*	*	*	*	*	*	*	*	*	+30	0	+61	0	+121	0	+243	0