Monsters have levels too

In D&D there are rules for how many men should there be of each level in a large group of men. Delta’s has observed that the

number and levels for leaders of groups of men in the OD&D campaign (in several different places) is surprisingly close to a simple divide-by-2 at each level

This seems like quite good system for any game, so how about an easy method to work out for a group of men, how many there are at each level?

One week later in Tenkar's Tavern, Eric mentioned "creatures with levels" and Nate McD commented

I have always liked the idea of ditching fixed HD for monsters, and adjusting their HD as appropriate... if there are 10 HD humans, why can't there be 10 HD orcs, goblins, or kobolds?

It's an idea I've considered before, and it's simple enough to give creatures levels - in D&D you add one HD per level, and in Explore you just give +1 attack, +1 parry, +1 Incap. But I don't want to have a group of fifth level orcs, I want a group of orcs with a fifth level leader, and what level you has always seemed a bit arbitrary.

So how about combing the two?

Calculating Levels for a Group of Monsters or Men

Whenever you meet a group of men or monsters, calculate the number at each level as follows: half the total (rounded up) are level one. Take the remainder, half of these (rounded up) are level two, etc.

An Example
A group of 35 bandits:

35 = 18 + 17 = 18 + 9 + 8 = 18 + 9 + 4 + 4 = 18 + 9 + 4 + 2 + 2 = 18 + 9 + 4 + 2 + 1 + 1.

You'd actually calculate this by repeatedly striking out the remainder and replacing it with a sum, and as soon as you get to a power of 2 you know the rest of the sequence thus:

35 = 18 + 17

Then you cross out the 17 to get:

35 = 18 + 17 9 + 8

Then you cross out the 8 and write in the remainder to get:

35 = 18 + 17 9 + 8 4 + 2 + 1 + 1

So 35 orcs, 18 first, 9 second, 4 third, 2 fourth, 1 fifth, 1 sixth.

Second Example

A group of 97 orcs:

97 =  49 + 48 24 + 24 12 + 12 6 + 6 3 + 3 2 + 1

So 49 first, 24 second, 12 third, 6 fourth, 3 fifth, 2 sixth, 1 seventh.

Observations

You always ends up with only one individual of the highest level, a single leader.

A leader of level n always has a group smaller than 2ⁿ. (rather than this being the average number). For example, you have a fifth level leader for groups sized 16-31.

The total strength of the party ends up being about 50% stronger than if they were all first level, so you might need to reduce the number in the group by one third.

I tried a random system, but it was time consuming and you get a wide variation in the highest level (roll d6 for all people. 1-3 means first level, discard these dice. Roll again to see who is second level. Repeat).

I didn't feel a need to persevere looking for a better random system - you can just jiggle them around a bit to make them more varied.

This seems appropriate for humanoids, but what about other creatures? What about a pack of Wolves? Giant Rats? Giant Hornets? I'm going to try it out!

Matching Points in Spacetime

What's in a name?The “Beneath” in the name of this blog (and my game) is meant to indicate not just the literal “beneath the surface” of a dungeon, but also meaning intrigue and puzzle solving. Similarly the “Beyond” means literally to travel beyond locked doors or chasms, or to other planes, but also meaning beyond the stars, or beyond human powers, or that there is no limit in RPGs. The name is purposely chosen to work for Fantasy, Space Opera, Mystery, Horror, or Super Heroes.

ExploreSciFi 

It’s looking more and more likely that my homebrew Explore will support SciFi as a first class citizen alongside Fantasy. This isn't surprising, since after all the skill system originates in house rules for our Star Frontiers campaign. Hence yesterday's in-game description of how interstellar travel might work in ExploreSciFi (quite influenced by Star Frontiers – but not Knight Hawks) along with a scientific explanation of how this doesn't violate the laws of Relativity. (Truly - it doesn't). Then on the way into work this morning the mathematician in me pondered about the canonical nature of the CMB, why is it the 'unique distinguished exemplar'?

A Recap

To recap, yesterday I was saying that every observer has a concept of what is happening “now” across the universe, and this is applicable to all physical phenomena. This is different for another observer with a different relative velocity; for example the two observers will disagree about whether two events, one occurring next to each observer, are simultaneous or not. The difference is greater the larger the separation between the observers, and the larger the difference between their velocities. This is how come accelerating to faster than the speed of light causes time travel paradoxes.

Secondly I discussed the concept of the “age of the universe”, which measured from any planet in the universe will give almost identical answers, since with only a small velocity change you can match the CMB. A universal time can thus be given to any event in space & time, it is the time since the universe began at the point where the event occurred, measured relative to the CMB. If you jump instantaneously from one point in space to another, appearing at the same universal time as you left, then you can never travel backwards in time, so no paradox.

Why the CMB? 

So, to return to today's question - what is uniquely special about the CMB? Why can I only jump from here to there at the same point in universal time? Why not some other reference frame?

Let's rephrase that: why can I only jump from here to there at the point where the age of the universe is the same?

With this rephrasing it's clear – when space is bent to join A to B, the point in time you must arrive is the moment when the expansion of the universe at B is the same as it was at A when you left. That is, you can only bend space to join “matching points” in space-time.

That not only seems eminently sensible, it looks like it is the only possible answer, and I've no idea why that didn't occur to me before!

Footnote: let’s not forget that relativity is based upon the concept that there is no universal reference frame, and we still have that rule for all physical phenomena except instantaneous interstellar travel. You still cannot accelerate to faster than the speed of light as that would cause time travel paradoxes.

The Law of Instantaneous Interstellar Travel

The Mayflower and Kapetyn-b

In the mid 23^rd century, mankind discovered a new fundamental law of physics: The Law of Instantaneous Interstellar Travel. This led to a flurry of activity, leading to the development of Interstellar Spaceship Travel a mere 27 years later. Every school child knows the story of the first instantaneous interstellar mission, the rescue mission to Kapteyn-b.

150 years to the day after the robot colony had been founded on Kapteyn-b, the first instantaneous interstellar mission carrying the second group of human colonists arrived and re-occupied the ruined settlement. In one of the most celebrated stories of human endeavour, they successfully reactivated the robotic shuttle and Captain Iain M. Banks piloted it into orbit and docked with the inactive Mayflower, overrode the glitching flight computer which had kept it stuck in orbit for 98 years, and successfully performed a manual landing. The original colonists had been asleep for 127 years, nearly 100 years longer than originally planned. "One cool move by me, one giant leap for sentient beings" the Caption famously tweeted.

The Law of Instantaneous Interstellar Travel

An object can instantaneously jump between one point in space and any other point in space, so long as its velocity relative to the CMB (Cosmic Microwave Background) is within 1/adm km/s, where m is the mass of the object, d is the distance between the two points, and a is the Asimov constant. The power required to initiate the jump is given by cd²m GW, where c is the Clarke constant.

Practial Issues for the Interstellar Traveller

Unfortunately, as all modern space tourists know, “Instantaneous Intersteller Travel” is not quite so instantaneous in practice!Firstly you have to accelerate to be almost stationary relative to the CMB, which on Earth is approx. 360 km/s; this means you have to accelerate at 1G in the direction of Leo for approximately 10 hours.Further, since the power required is proportional to the distance squared, interstellar travel is performed as a series of short jumps, with one hour's recalibration between each jump, hence it takes around 1 day to travel each light year.Finally, you have to decelerate at the further end.Hence the total journey time is roughly 1 day + 1 day / light year.Fortunately the direction of approach and departure from any given planet are in opposite directions and are the same for all destinations, hence collisions between accelerating/decelerating spaceships is not a risk.

Isn't FTL Travel Impossible?

Everyone knows that Relativity says that FTL (faster than light) travel would lead to travelling backwards in time, which is impossible, hence faster than light travel isn't possible.

That is true.

However, instantaneous travel might be possible...

Let us consider the paradox:

The FTL Paradox

Imagine that you take off from the earth in a spaceship and fly away from earth at high velocity. Relativity says that your perception of time passing on the earth is altered. Time now appears to run slower on the earth. After two hours of flight, only one hour appears to have passed on earth. Thus if you could travel instantaneously back to the earth, you would arrive one hour after you left. But similarly back on the earth, time on your spaceship appears to be running slower, and only half an hour appears to have passed on the spaceship, so if you now instantaneously travel back to the spaceship, you would arrive only half an hour after it had left the earth. That is, you would have arrived before you left.

The Age of The Universe

This paradox reveals that the concept of “now” is not a constant across the universe – it depends upon your reference frame. If you can travel instantaneously in one reference frame, and then again in a different frame, you could then travel backwards through time.

Thus we have seen that the age of the earth, in fact the age of the Universe, depends upon the observer. It should depend upon which galaxy you’re observing the universe from. So how can scientists talk of measuring the age of the Universe?

Cosmic Microwave Background 

The universe is expanding, hence all of the galaxies in the universe are accelerating away from all of the other galaxies due to the expansion of the universe. What is very surprising is that almost all of the relative velocity of all of the other galaxies in the universe compared to us is due to the expansion of the universe.

The CMB (Cosmic Microwave Background radiation), the “thermal radiation left over from the Big Bang”, provides a universal frame of reference throughout the universe: it is expanding, but relative to it every galaxy in the universe is almost stationary. If we measure the CMB we find that we are moving relative to it at approx. 360 km/s, or only 0.1% of the speed of light. Hence the age of the universe is the same, no matter which galaxy or star system you measure it from.

To quote NASA:

The largest reference frame we know about is that of the cosmic background radiation itself...the fireball light left over from the Big Bang. It samples a scale of the universe over 15 billion light years in radius. If you have no 'peculiar' speed relative to this frame, your galaxy is exactly partaking of the expansion of the universe with no additional local speed added to it. The NASA Cosmic Background Explorer confirmed many previous measurements of the motion of our earth's speed and found that it has a peculiar speed of 360 +/- 20 kilometers/sec in the direction of the constellations Leo and Crater.

CMB as a Univeral Frame of Reference

Now imagine that this cosmic microwave background radiation is the single canonical universal frame of reference for the universe, and that “now” throughout the universe means “now” as perceived by an observer in this particular frame of reference.

In this system, instantaneous interstellar travel between the spaceship and the earth takes you from “now” on the Spaceship to “now” on the earth, and vice-versa, where “now” is measured by the CMB. Since this earth is moving slowly enough against the CMB to ignore relativistic effects, the traveller was two hours on the spaceship before they jump, which to observers on the earth didn't occur until 4 hours after the ship left, so they find themselves back on earth four hours after they left. They jump back to the spaceship and find themselves back at the same moment they left.

There is no paradox*.

Bending The Laws Of Relativity

But haven't we just broken the Laws of Relativity?

The general principle of relativity states

All systems of reference are equivalent with respect to the formulation of the fundamental laws of physics.

The principle has to be bent to say:

All systems of reference are equivalent with respect to the formulation of the fundamental laws of physics except for the Law of Instantaneous Interstellar Travel.

The principle can be bend this much and not break.

In one of the largest ironies in human history, this precise scenario was first discussed in the 21st century by Science Fiction aficionados, but was mistaken for the ravings of nay-saying flat earthers, and dismissed out of hand.

*Note that this only works for instantaneous travel – FTL travel still leads to a paradox!

Lego, Grids, and Smugglers

I've never had the desire to paint miniatures, and unpainted ones seemed very unattractive, so I never really used them until the mid 90s when I bought some old painted miniatures from a friend. When we started playing in late 2009 I dug them out as a way of it being clear to the children what was going on, which worked well - especially as I had a lot of Orc miniatures which doubled for every villainous humanoid tribe in the Caves of Chaos.

I soon ran out of appropriate miniatures, and even when I had one I'd waste a minute finding it, so we switched to using miniatures for PCs, tokens for monsters. This was originally out of necessity (an Orc miniature is NOT an appropriate substitute for a T-Rex!) but was soon embraced whole heartedly. The party was attacked the other night by a horde of giant Hornets, and I felt no shame or regret at not having 20+ giant hornet miniatures!

I went through the obligatory phase of trying out grid based combat and discovered like countless hordes before me how it transformed the game overnight from "What does your character do now?" to "Your turn to move your character." I knew what problem it was trying to solve, as I had experienced the problem, but that was not the solution! The realisation that the grid could be used purely for drawing on a freehand map, and miniatures used purely for communicating who was where, was a revelation that made me feel quite sheepish.

Our homegrown battlemat has gone through various incarnations, all involving transpaseal - those rolls of plastic you get from WHSmiths (or your local equivalent) on squared backing paper. I went through several of these, with varying size grids, until I realised you can see the squares through the plastic already. I now have two large sections attached together which cover the whole table, and a black dot every three squares to mark 10'.

When we played Star Frontiers we had no SciFi miniatures, so we used counters until someone co-opted the Lego.

The party prepares to exit their Explorer

The next campaign was Marvel Super Heroes FASERIP, and we have loads of them in Lego so that was a no-brainer.

So anyway, here's some shots of Mazak, Elanor, Osran, Khazem and Flairin recently storming the Smugglers' Ship. For this session we knew the plan, and we have loads of Pirates*, so we had to use them:

The party approach the ship in a boat, having created a distraction on the other side. The smuggler in the lookout hasn't noticed anything.

The battle is underway, one smuggler down already. Mazak is still in the boat, Khazem has summoned a monster, Osran is standing on the hatch to stop it from opening, Cap'n Hardask has come out of his cabin, and his Monkey is sat on top of the cabin. The d6 show the wounded people - ignore the other dice, they were an idea for tracking bonuses I dropped from the system! You can see why soon after this I went for colour coded dice for tracking wounds (red for PCs, green for monsters). The lookout is climbing down.

Cap'n Hardask is down, and one of the smugglers (the lookout!) is trying to make good his escape in the boat. All the PCs are wounded except Flairin, and Elanor is incapacitated and wounded 10.

The smugglers are all rounded up, Osran is healing Elanor, Khazem is looting the dead, and Flairin is making friends with the Monkey! The lookout is now one of the prisoners, so you can see he has surrendered and climbed back on board :-)

*I should actually clarify that they were Smugglers not Pirates as this was an important plot point when they got convicted of Piracy and the players ended up having to clear their names!

Character Sheet Origami

Are character sheets taking up too much space at your table?

Do they get in the way?

Don't let your game be ruled by A4!

Reclaim space for Miniatures! Drinks! Snacks! Props!

Leverage the power of Origami today!!!

A couple of years ago halfway through a gaming session I had a flash of inspiration - I could do character sheets on a sheet of A4 folded in half. For those of you not familiar with the ancient art of Origami, here is an illustrative example of folding a sheet of paper in half to make a 4 page A5 booklet:

I know this is groundbreaking stuff and may be taking you outside your comfort zone, but I hope you're managing to follow this.

With this system, you can put say put stats and skills on the front, combat information on the back, equipment and notes inside.

Unfortunately as is often the way with great leaps forward in the progress of mankind, I discover I am not the first to make this discovery. Or the second. Not even the tenth.

Here is a fifth edition D&D character sheet. Then there is the eight page Pathfinder / 3.5 character sheet (I think there's a clue that your game's too complicated!). Some people have done A5 old school D&D character sheets, but Delving Deeper has gone one step further with sideways A5 character sheets. That's bold!

In story game world they have gone one step further to the ultimate - tri fold character sheets. "this is almost as important as "character sheet" was when someone thought of it" (possibly they're overselling it).

But my favourite for sheer off-the-wall wacky ideas has to be the character sheet as an envelope for storing things in!

Weapons and Armour

Here's the last in the current series of posts about my wounds-instead-of-hit-points-combat-system. There are a couple of points to cover, so I'll dive straight in...

WEAPONS

Balanced but meaningful weapon choices
In RPGs there are often "better" weapon choices which conflicts with "in character" choices, or alternatively all weapons are the same. I want players to have a variety of weapon options with equivalent good but different game mechanics.

Weapons give you reach, so make it easier to hit someone. You can also keep an attacker at bay or parry their blows. Thirdly, weapons make it easier to kill someone. Hence each weapon gives you bonuses to attack, parry and kill, and since bonuses in my system are equivalent, you simply make each weapon give different bonuses but the same total bonus.

A shield, 2-handed weapon or two weapons?
Characters also have the option of a 1-handed weapon + shield, a 2-handed weapon, or a 1-handed weapon + dagger. I don't want characters with two weapons to make two attacks (for speed and for balance), instead they should get the same total bonus as a character with a two handed weapon.

The bonuses
1-handed weapons give you a total of +6, offhand weapons and shields give you total of +3, and 2-handed weapons give you a total of +9. They always give you at least +1 onto each of attack, parry, and kill. The weapons in brackets are sub-optimal weapons with lower bonuses.

Weapon	Type	PARRY	ATTACK	KILL	Length	Weight
Halberd	2-H	+3	+4	+2	7'	8 lb
Long Sword	2-H	+2	+4	+3	4’6''	6 lb
Flail	2-H	+2	+3	+4	4'6''	6lb
(Quarterstaff)	2-H	+4	+2	+1	7'	8 lb
(Spear)	2-H	+3	+3	+2	7'	6 lb
Battle Axe	1-H	+1	+2	+3	3’6''	4 lb
Sword	1-H	+2	+2	+2	3’	2 lb
Falchion	1-H	+2	+1	+3	3'	2 lb
Morning Star	1-H	+2	+1	+3	3'	4 lb
(Mace)	1-H	+2	+1	+2	3’	4 lb
(Club)	1-H	+2	+1	+1	3'	2 lb
Shield	Offhand	+3	0	0	3'	16lb
Dagger	Offhand	+1	+1	+1	1'4’’	4 oz
Short Sword	Small	+1.5	+2	+1	2'	12 oz

Why these particular bonuses?

Firstly, note that quarterstaff, spear, mace and club are sub-optimal weapons. A quarterstaff  is useful because you can pretend it's not a weapon, whereas a spear can also be thrown. On the other hand, a club is just not as good as a mace, and a mace is not as good as a morning star (a spiked mace).

The only offhand weapon - dagger - has to be 1/1/1.
Shields are simply +3 parry.

Swords are better than daggers in every way, so have to be 2/2/2.
A falchion or a morning star are deadly if you hit someone with them, and you can parry with them, but you cannot stab someone with them, so they are 2/1/3.
A battle axe is also deadly if you hit someone, they are more difficult to parry, but you cannot easily parry with them, so is 1/2/3.

A longsword compared to sword & dagger is easier to attack (better reach), but no better parry than a normal sword, hence is 2/4/3.
A flail is dangerous, but not great at parrying, so is 2/3/4.
A halberd is great at attacking as you can thrust and hack with it, and is good at parrying as you can fend off your opponent, so is 3/4/2.

Finally, we have the single small weapon - short sword. You can wield one of these in either hand. Hence they are +4.5. The half is rounded up. E.g. one short sword gives you 2/2/1, two short swords gives you 3/4/2 (the same as a halberd). Thus two short swords is no deadlier than a longsword, but gives you more attacking opportunities.

What about other weapons?
Minor variations of a weapon all count as the same weapon, so there's no long list of polearms as they all just varieties of halberd (though that's a cool link). Anyone else bugged by the weird insistence on arbitrarily changing the spelling of Poleaxe to "Pollax" in Wikipedia on the basis of that the OED says it derives from the Middle English Pollax or Polax which comes from Poll+Axe.

Pikes, warhammers, rapiers, sabres, cutlasses are all missing because they're from later historical periods.

There are potential other weapons to add such as the Trident (in case they meet Mermen), or improvised weapons (such as Scythe, Pitchfork, Chair, Bottle, Pitchfork, Spade). Whip and Net will need some special rules.

ARMOUR

Since weapons give +9, there are no prizes given for guessing that the maximum bonus armour can give you is +9.

Type	Weight	Armour Bonus
Leather	3lb	+2
Quilted (can be worn under Mail)	3lb	+2
Arming doublet (can be worn under Mail/Plate)	3lb	+2
Mail Hauberk	16lb (+ 3lb)	+2 (+2)
Scale / Segmented / Lamellar / Plate Armour	12lb (+ 3lb)	+3 (+2)
Steel Cap/Leather helm	1lb	+1
Steel Helm	2lb	+2
Full leather gauntlets & greaves	1lb	+1
Steel Gauntlets	1lb	+1
Steel Greaves	1lb	+1

Note that Mail is worn over quilted armour or an arming doublet, but Plate armour has to be tied onto an arming doublet - that's what all those strings hanging off them in the pictures are for!
You only get a bonus for leather gauntlets and greaves together.
I've not distinguished between different high grade armour types as it seems to be mostly about fashion/tradition or status, and note plate armour is not a full suit of armour..

MISC

Extra Damage for high attack rolls

If you beat the target to hit by 10 or more, you get an extra dice on your kill roll. 

If it's 20 or more then you get two extra dice etc.
This is for when someone has a massive attack bonus and the hit is almost automatic and they feel cheated. It's "beat by 10" because that's easy to spot. It also makes the calculation of the attack roll more interesting as you want to beat it by 10.

Ranged Attacks
We'll leave these for another day :-)

Future Features To Be Developed
Grappling
Mounted combat.
Rules for nets & whips.
Disarming?
Dismount?
Wound locations?
Descriptions for severe or incapacitation wounds? Thanks to Lloyd for this mine of ideas and links on the subject.

You could also have optional rules such as different shields, e.g. an offensive shield 2/1/0, or a larger shield 4/-1/0.
You could make heavier armour have a better armour value in return for a reduction on parry and attack. E.g. +11 armour with -1 attack, -1 parry.

These omissions in the rules at the moment are due to the way they get developed - it is on a need-to-know basis. That is, when I need-to-know I make up a rule. If it floats in play then it stays. If it sinks (or stinks) then it's thrown out. I try not to be too sentimental!

Combat Advancement

Back to Combat again to answer the question of  what rate characters progress at. I've got just one more post after this about this combat system, then it's on to pastures new. So here's the answer:

Parry, attack and incapacitated values go up by 1 every level. (Kill doesn't).

But here's the rub - this is true for every class not just Fighters! What differentiates the class is how much better the Fighter starts off at melee and parry. If they start off +2 in melee and parry, then they are always +2 in melee and parry, so they are always 1.5 times as likely to hit, and 2/3 as likely to be hit.

In addition, a Fighter also has several attacks (melee / bow / throw / unarmed), whereas other characters would only have one or two of these.

So how does this rate of advancement work out in practice? (If you get bored, just skip to the conclusion section at the bottom).

What are the stats for an nth level fighter?
Assuming a Fighter has Plate & Shield & a sword, then a typical value for 1st level would be

Sword & Shield: P17, A4, K5. 
11/14/17/20 (23)
Full Armour: 9

So at nth level a Fighter's stats would be

Sword & Shield: P16+n, A3+n, K5. 
11/-/-/- (22+n)
Full Armour: 9

Note that this also applies to level 0.

(The other wound thresholds are just averages rounded up, so at 3rd level the middle one W3 is average of 11&25 = 18. Then W2 is average of 11&18 = 15. W4 is average of 18&25 = 22).

How good at fighting is this monster?
Everything has a Fight Rating, which is

Fight Rating = Parry + Attack + Kill + W3 (i.e. the middle value)

Anything with the same Fight Rating is roughly the same threat. That is, bonuses on parry, attack, kill or armour are roughly equivalent, but bonuses on incapacitation alone are only half as good.

Thus every level you get +2.5 fight rating, plus every so often you'll find some magic item or other to improve you, so we take that +3 fight rating for a monster makes it +1 level. Our 1st level fighter above had 43 fight points, and we'll take 42-44 as level 1:

Fight Points	Level	XP
24	-5	8
27	-4	12
30	-3	16
33	-2	24
36	-1	32
39	0	48
42	1	64
45	2	96
48	3	128
51	4	192
54	5	256
57	6	384
60	7	512
63	8	768
66	9	1024
69	10	1536

I'm not talking about XP today, but you can see the XP gained from killing a monster is doubling every other level.

So how good is that Fighter?
A fighter battles a never-ending supply of 1st level opponents - how many can he slay on average before he is slain himself?

We want to see the fighter improving markedly every level, but not too fast.

I wrote a short Java program to test this, and to calculate the same values for AD&D for comparison purposes.

Level	Explore	AD&D
0	0.4
1	0.7	0.9
2	1.3	2.1
3	2.1	3.9
4	3.4	6.3
5	5.3	9.2
6	8.4	12.7
7	12.6	16.7
8	18.4	21.4
9	25.5	26.9
10	37.3	32.9

(I took the AD&D fighter to be d10 hp/lvl, AC2 (plate & shield), d8 damage, "To hit" smoothed out to be 19-lvl.)

As you can see, it's a fairly similar rate between the two systems.

In Explore the rate of improvement is about double per level, slowing down to about half that rate as you get far far better than your opponent.

In contrast AD&D is slightly more linear, as per the oft quoted criticism "linear fighters, quadratic wizards" which causes an issue at high levels.


How long do fights last?
How many rounds does the average fight-to-the-death between two fighters of the same level take?

We want to see the length of the fight increase a bit at high levels, but not too much.

Here's the output from the program:

Level	Explore	AD&D
0	4.1
1	4.4	6.7
2	4.9	9.4
3	5.2	11.6
4	5.6	13.3
5	5.9	14.4
6	6.4	15.4
7	6.8	16.2
8	7.2	16.9
9	7.4	17.4
10	8.1	18.0

Note that in Explore the duration increases linearly to about double, whereas in AD&D it starts higher, there's an initial fast increase before levelling off, and it ends up very high.

We could make our fights last longer by adding +1/+2/+3/+4 onto the wound categories, but I prefer shorter combats.

How much better do you get for going up one level?
We'll measure this by asking how many opponents of one level lower can you kill on average before you are killed.

Level	Explore	AD&D
0	1.4
1	1.3
2	1.3	2.2
3	1.2	1.5
4	1.2	1.3
5	1.1	1.1
6	1.1	1.0
7	1.1	0.9
8	1.1	0.8
9	1.0	0.8
10	1.1	0.8

Thus in AD&D, there's a big jump between 1st and 2nd level, and then it rapidly tails off. In Explore the improvement tails off only slightly.

Conclusion

We have the following useful rule of thumb:

You can convert an adventure between D&D and Explore by keeping the level/HD of the PCs and monsters the same.

The rate of improvement is roughly the same in both systems, but Explore is more smooth (particularly in the jump between 1st and 2nd level).

At high levels in AD&D improvements slow down, but in Explore it stays the same, so it scales up to high levels.

Since the exponential growth in Explore is similar to FASERIP I am interested at some point in exploring the possibilities of this as the basis for a Superhero game, or perhaps whether you could have characters going to super high level and being able to wrestle down a Storm giant barehanded.

As a final note, you may have noted I tend to over analyse things. At least I sometime think I do. Perhaps I should analyse less? On the other hand possibly it's just a feeling I've got. I'm going to have to go away and think about this one...