Monday 21 September 2015

A Puzzle and a Code

Last weekend the players found a dead messenger. On his body they found some mysterious symbols on a strange device: two wooden disks attached together so that the middle one turns...

I devised this as a simple puzzle for my youngest, but in practice the off-centre centre of rotation and the extra marks meant it took a few minutes to get it oriented correctly...


The box took a bit of searching to find...

It's great sometimes when the players find something, to simply hand it to them and see what they do. One of them's bound to pick it up before they realise they should cast Detect Curse on it! Shaking the box revealed it contained something, but what?

A letter sealed (rather badly I'm afraid) with a Z (as per the Z in the middle of the puzzle wheel, which they had already noted). Quickly opening the letter they found...
Obviously mirrored writing, but my daughter impressed me by simply holding it up to the light to read. I don't think that trick is going to work for an LCD screen, so here it is again:
Evidently Zanquibar was the wizard they had located earlier, holding a pair of doors shut, both the doors and the wizard turned to iron. Below was the puzzle with the key to setting him free:

I promise you the solution is simple, though it kept the players happy for an hour. I gave them a hint at one point not because they needed it but as a little encouragement.

If you try and solve it you will see how it keeps your interest whilst you search in vain for the answer, it isn't a case of staring at it for an hour - you keep on finding things which might be the answer... but aren't... but which help you to the solution without you realising it.

(Note: it helps if you print this out to solve it)

Thursday 17 September 2015

Low Ceilings Revised

I've had some helpful feedback on my Effect of Low Ceilings on Range post, and this has lead me to revisit the subject and revise the rule.

Q: Gaston’s Hat contrasted my max range for an of Axe 80ft with competitions over very short distances e.g. IKTHOF  (3/4/6/7/9m) and AKTA (13/15/21/30ft).
A: Although you can throw an axe 80ft, this is close to the record (and in Explore you could only achieve this with a high Strength and Athletics skill). Also it is quite hard to be accurate at 80ft with an Axe (in Explore you would get -9 on to hit for range, and axes are harder to aim in the first place). Both of these facts would imply that you'd have competitions at much lower distances than max range, just as you do in archery, so this range for competitions is as expected.

Q: Gaston’s Hat observed that in this video the axes are clearly being thrown quite high in the air to hit a target only 60ft away.
A: To achieve max range, which it looks like those throwers are at, you have to throw at 45 degrees. In this case the projectile reaches a height of one quarter the range, which would be 15ft in that video (plus 6ft for the thrower’s height). Reduce the distance only slightly, and if you throw with the same velocity, you can reduce the angle massively, and hence the maximum height. At a range of 50ft the max height is only 7ft (+6ft).

A picture is worth a thousand words, so here's a diagram of the trajectory of a projectile with max range 80' launched at targets at ranges 10' to 80'. As you can see, the max height drops rapidly as the distance to the target reduces.

Q: Thiles Targon commented that throwing a ball only 20-25ft he was often hitting a 9ft ceiling.
A: Evidently if you shoot someone with a gun at this range, you can ignore the 9ft ceiling, similarly with a bow. The effect of a low ceiling depends upon the velocity of the projectile, not the distance to the target. Hence, if you throw the ball slowly, or mis-throw it, or throw it over someone, then you’ll likely hit the ceiling. If you only throw the ball fast enough to go 20ft, in a 45 degree arc, then the ball would go 5ft+6ft = 11ft high and bounce off the ceiling. On the other hand, if you’re world class you’d have a range of 480' and hence the ball would go in almost a straight line, and only rise by 1.3 inches.

Here's a diagram of the trajectory of a projectile with max range 20' at a target 20' away, and then thrown at the same target with sufficient velocity for max range 40' and 80':

Revised Rule
Note that in these answers I have invoked the height of the thrower - in my original results I stated that I was ignoring this. I thought the effect wasn't large enough to worry about, but on second look it is, as these discussions show.
In addition, I was always rounding to the nearest category - but this is often rounding up quite a lot. I'd say no reduction when it was actually 60' reduced to 51'.
Time for a re-examination of the figures....

Firstly by introducing half-way values I stop the rounding up issue (so 51' is now 50').
Secondly I noted that the effect of introducing the height of the thrower is almost the same as reducing the height of the ceiling by the height of the thrower.
The solution has to be a compromise; I've tried several approaches but in the end settled on a single table for all races, but you use different columns depending upon your height:

For example, with a 10' ceiling and 160' max range, a 6' human uses the middle (white) line for ceiling height, so has the max range reduced to 70', whereas a 3'6'' halfling or kobold uses the bottom row, hence uses the column one to the left and has it only reduced to 100', giving them a big advantage in missile combats in corridors.

The heights are 2'+/4'+/8'+ rather than 2.5'/5'/10' as this is what gives the closest fit.

Performing the Calculations
My calculations were initially done with a computer program – but as I'm ignoring wind resistance I should have just solved the equations! When ignoring the height of the thrower the equations aren't too complicated, and the result is:

Let R be the max range, H the height of the roof, r the reduced range (due to the low roof) we can derive the following results:

So it is a quarter-ellipse!

For example, with max range R=120, height in the X axis, r in the Y axis:

From this you can calculate my results above, but not the effects of the height of the thrower. Unfortunately when you try and calculate the range of a projectile thrown at 6' high, with a 10' roof, aiming at a target 3' off the floor, the maths becomes rather complex and I reverted to my simulation program.

Friday 11 September 2015

The Effect of Elevation on Range

In my recent series of posts on missile weapons, I’ve been promising an analysis of the effect of the archer being at a different elevation to the target. This was a bold statement as my initial investigations had yielded nothing and I was hoping for inspiration to strike. Fortunately it has, and I'm very surprised at the result.

Note that as the Referee I'd do these range calculations, and just tell the players if they’re in range or not and what the relevant bonuses are!

Imagine you have max range R and are shooting at a target horiz distance X away, height H below you.

For example:  throwing an Axe with max range 80' from a 20' high wall at something 75' away.

Here's a diagram to accompany the example:

Firstly, guesstimate the direct distance D by adding half the height H to the horiz distance X. If H is bigger than X then do the reverse - add half the horiz distance X to the height H. From this direct distance you get your to-hit penalty for range.

In our example it is 75'+10'=85' direct distance, so -9 to hit.

Secondly we check that it is within range: direct distance must be less than Range + Height.

In our example range + height is 80'+20'=100' which is bigger than 85' so it is within range.

Lastly you calculate the damage bonus/penalty: Take Range + 2*Height and see how many categories that shifts the range, you get +1/-1 for each shift.

In our example Range + 2*height  = 100'+20'=120' and this is +1 range category so +1 damage.

Apart from the approximation for the direct distance, this is a 100% correct model (ignoring wind resistance) - no simplification needed. These are the actual equations when targeting a missile at something with a different elevation!

As a second example, consider throwing an Axe with max range 80' up at the top of a 20' high wall, 75' away. The direct distance is 75'+10'=85' as before. However 80'-20'=60' so it's out of range.

I've added one extra "limiting rule" (to mimic terminal velocity) –

You can add at most 960' to the height when calculating the damage.

This is unlikely to ever be needed!

As a third example, consider throwing a javelin with 240' max range off a 2000' cliff at a target 400’ away. It is clearly within range. Damage is capped at 960'+240' = 1200' which is +4. The direct distance is 2000'+200' = 2200', which means you'd be severely unlikely to hit anything, as the to hit penalty is now -36!

Example #4: Throwing the Axe (max range 80') from the top of a 100' high cliff at a target 100' away. Direct distance = 150' which is -12 to hit. This must be less than 80'+100'=180', so it's in range. Damage is @80'+100'+100'=280' which is +3.

Finally we have two observations – with these rules both of the following happen to be true:

The max rise is half the range.
The horiz distance is capped at twice the range.

This second one is a happy accident, which I discuss at the bottom of the post.

Example #5: Throwing the Axe (max range 80') from the top of a 200' high cliff at something 160' away. Direct distance is 200'+80' = 280' which is -18 to hit. This actually the max, which is 80'+200'=280'. Damage is @80'+200'+200'=480' which is +4.

We'll apply these rules to a common in-game situation:

Attacks by Flying creatures
For simplicity we’ll assume diving attacks are always at 45 degrees. This means the diving speed of a bird should (for simplicity) be given in stat blocks as the speed direct / speed horizontal & vertical (which is two-thirds).

For example, a Roc dives at speed 440'/rnd, which is 2/3*440' = 293'/rnd both vertically and horizontally. Hence we'd specify Roc Dive:440'(290').

When attacked you’d shoot at point blank range, and then one round before that, and one round after that, so distances are always a simple multiple of the movement speed.

Continuing our example, we have a bow (range 960') to protect us from the Roc. Is it in range at 440'? The max is 960'-290'=670', so it is in range (just). To hit penalty is -24. Damage is @670'-290'=380', so -2. In actual play the Roc swooped in, all shots at range missed, the next round (at point blank range) it successfully grabbed Elanor but got badly wounded, the subsequent round it got shot out of the sky due to the bonus to hit from it being wounded – Elanor was extremely grateful for having earlier polymorphed to grow a pair of wings!

So how did I arrive at these rules? This involves a bit of maths – sorry – but it’s purely by way of explanation; it’s completely irrelevant when playing the game and certainly will not appear in the rulebook!

My initial thoughts were that if:
1) shooting a distance R requires an initial KE of mg x 0.5R
2) shooting up height H requires kinetic energy of mg x H
Then shooting at range R and height H would impact with the KE of (1) minus (2), i.e. mg x (0.5R-H), i.e. the same KE as shooting a distance of R-2H. Hence we arrive at the damage reduction rule.

I had hoped that the max horizontal range would be X-2H, but results from my simulation program showed the max range was a straight line close to X-H, but above a 30 degree angle or with big drops the line curved away and it was difficult to have a simplification without wacky edge cases. By complete accident I added in an extra row to my results showing the direct distance, and found it was an exact straight line which seemed rather too good to be coincidence. Going back to this standard formula, I calculated the max direct distance for a given height and range...

and was astonished that everything cancels! Gosh, I haven’t written out an equation using Latex since I finished my thesis in 1997…

An Accidental Range Limit
One consequence of using an approximation for the direct distance isn't clear at first: the horiz range maxes out at double. With a correct calculation for direct distance you would have the rate of increase slowing down but never stopping, there would be no such maximum, but in reality you’d also have wind resistance, so we want a cut-off, so it is a happy accident.

For example with 100' range, 200' drop, we have max horiz range as 200', but it should be 224’, which is still very close. But if we increase the drop to 400’ our equation still gives 200' when it should have increased to 300'.

Monday 7 September 2015

The Worst: Hey look! A Dungeons and Dragons ride!

I've seen several people commenting recently that when they reread the rules of AD&D they discovered they were doing it wrong. This may be so, but not quite of the same order of magnitude as the makers of the Dungeons & Dragons cartoon. What a fantastic opportunity this was to introduce the game to a whole new audience. After watching a few episodes of this series, new players would flock to the game, eager to join in with the latest craze...

So, let me see if I understand this correctly - you play Dungeon Master and tell us what to do, in riddles.

What do you mean my Cavalier needs a horse? You're wrong – I've seen the show and he ain't got no horse. Do you know nothing? What makes them different from Fighters isn’t having a horse – what makes them different is not having a sword as they abhor violence.

I want to play an Acrobat. Fifth level? I need to wait until I’m fifth level???

Can I have a baby unicorn as a pet? I think they're so cute, the way they bleat like a sheep. So what if I'm male?

I need a hat. Why? Duh – so I can cast spells.

So what magic items do I start with? What do you mean I don't get any???

Orcs? I can deal with them. I scream at them and in their confusion we run through the middle of them.

Perhaps it's a magic cloak – I put it on and try taking the hood on and off a few times to see what happens.

We've had retroclones of every other version of D&D – surely now the time is ripe for a retroclone of "AD&D – TV edition". I warn anyone brave enough to take on this challenge that it'd take a lot of work - it'd mean an almost complete rewrite of OSRIC.

Saturday 5 September 2015

The Best: Wainwright's Pictorial Guide

Continuing my irregular series on the Best and Worst things I've ever seen in RPG products, it's time for another "Best" post (despite the fact that the "Worst" posts get twice as many hits!).

This time I move from an actual RPG product to one that should be an RPG product. One that every RPG product should aspire to be like. One that is a work of art to behold, is of eminent practicality both for planning adventures and when you're having them, and is inspirational when you just pick it up to read it. I am, of course, talking about the Pictorial Guide to the Lakeland Fells, written by
This series of seven volumes published between 1955 and 1966 details all 214 fells of the Lake District, dedicating a chapter to each. It is thoroughly comprehensive, starting with an overview of the fell, followed by all routes of ascent, points of interest, the summit and the views, and all ridge routes to other nearby tops.

It threw typographical conventions to the wind, and is entirely written in pen and ink. This enables AW to freely mix text and maps and illustrations. Each ascent route is given a dedicated page, which the author describes as
"The routes of ascent…are depicted by diagrams that do not pretend to strict accuracy: they are neither plans nor elevations; in fact, there is deliberate distortion in order to show detail clearly: usually they are represented as viewed from imaginary ‘space-stations'."
The resultant map-come-route descriptions are incredibly easy to use in practice. There's no back and forth between the map and the route description; everything you need is to hand where you need it.

AW self published the first volume in 1955, paying for 2000 copies. For thirteen years he worked on his labour of love at a rate of one page per day.

What particularly appealed to me was that you could climb one hill using one route (often with variations), then link together summits with the ridge routes, descending back to the start; thus discovering your own route.  Definitely no rail-roading here!

All my childhood holidays were spent in the hills, most of them in the Lake District, and those mostly "climbing the Wainwrights". When I met starting dating my wife in 1990 we set about climbing them all together, which took us until 1996 (we had no car and were living in the south of England so it took a while). When we married in 1994 we did his "Coast to Coast Walk" as our honeymoon. We still have the bug - we spent the week before last in a cottage in Eskdale with my parents and our three children, still finding new routes to try and hidden spaces to discover.

If only every RPG product was as great to look at, inspirational to read, easy to prep, and easy to use in play as this.

Thursday 3 September 2015

The Effect of Low Ceilings on Missile Fire

I've been busy creating a new adventure locale for my group (and the adventure went in an unexpected direction, which is good) so for the blog I've chosen an easier topic to resolve: Does the height of the ceiling significantly affect the range of missile weapons indoors?

Now in D&D Gary Gygax had this odd “distances are in yards outdoors, feet indoors” rule (AD&D DMG p39), which reduces all ranges and areas of effect by a factor of three in dungeons. To me this seemed firmly based upon reconciling the awesome power of Fireball in Chainmail with the desire to reign in the power of Magic Users for dungeon crawls, but it also applies to missile fire with the justification of low ceilings. Is this drastic universal range reduction justified? In particular it has a big effect on short range missile weapons like axes. The reduction on range for bows seems plausible (apart from his odd statement that Crossbow bolts don't follow a parabolic flight), but the 3" for throwing axes and spears from Chainmail (p10) means 30 yards outside, but only 30 feet indoors. The ranges are carried forward with minor changes through to AD&D where it's become 30ft for Axes, 60ft for Javelins (PHb p38).

I’ve run some simulations for missiles of different maximum ranges over various maximum ceiling heights, and calculated what the maximum range is reduced to. For my purposes I’ve made several simplifications: I’ve lumped all missiles with the same range together (ignoring different trajectories for different types of projectile due to wind resistance – it has little effect). I’ve ignored the height the missile was thrown at (which should be about 5 foot – it has little effect, and I ignored it when calculating ranges in the first place). I’ve omitted ceiling heights under 10’ high as that’s unlikely to be relevant, though note that it follows the same pattern.

The results are as follows: (all heights and ranges as per my standard "double every other category")

I’ve shaded the ranges which are reduced. I then simplify this table (below) by altering the reduced range to be the nearest category, allowing myself latitude in a couple of cases (marked red) where I round up for some results just less than halfway between to make the results fit a regular pattern.

Now the pattern in that table is easy to see but difficult to extrapolate from, but it's probably an acceptable simplification to drop every other column (who ever heard of a 15' ceiling anyway) and the resultant cut down table is much simpler, especially if we reverse the order of the columns:

So we note that unless you’re aiming at something more than 80’ away, the low ceiling has zero effect on range. Furthermore, if you’re shooting at more than 80’ you get a -12 range penalty on to hit. So range restrictions would only ever come into play if the target you’re aiming at is so far away that you’re unlikely to hit it in the first place!

So in summary, the range reduction can be quite easily reduced to a simple pattern which can be summarised in one short table. However, I think that it is extremely unlikely that this range reduction will ever actually have an impact on play (since it only ever takes effect over longer distances than are usual). Note that this result is absolutely nothing like “reduce the range threefold” except for long ranges with a 10' ceiling, and in fact ignoring ceiling height altogether would be a more accurate simplification for most situations!

Comparison with Delta's Results 
Delta did some similar analysis using D&D’s fixed ranges,  but in Explore the range varies according to the archer so these results are not much use to us except for comparison purposes to check the results.
Delta shows thrown weapons (range 90') having a reduced range only with a 10’ ceiling – when it is reduced to 63'. The nearest equivalent 80’ in Explore would only be reduced to 69’, which in this case I rounded up not down, hence in our case there’s no range reduction at all.
Similarly the three bows have fixed ranges 150/180/210/240 yards or 450/540/630/720 ft. In Explore we don’t have this level of granularity in ranges, but we get similar results: approx. one third range with a 10' ceiling, one half with a 20' ceiling, two-thirds with a 40' ceiling.