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Posted on September 23, 2018, 01:46:54 PM by Tar-Palantir
A Cartload of Cartography 2: Beyond the Middle Ages

A Cartload of Cartography 2: Beyond the Middle Ages
By Tar-Palantir

Welcome to the second part of this series, on Early Modern maps! You can check the first part, on ancient to medieval maps, out here.

1. Eight leaves of the Catalan Atlas, from 1375.

Once we hit the 15th century, something a bit more like the modern map hoves into view. This is largely driven by advances in seafaring, spearheaded by Portugal and the Catalans. As long-distance voyages became more common, especially ones that involved potentially being out-of-sight of land for a while, actually accurate maps became more important. People could measure latitude pretty well by this point, as the astrolabe began to be replaced by the cross-staff and then the backstaff, making the measurement relatively accurate and simple. However, longitude was still problematic, as its accurate measurement relies on being able to accurately measure time, which was beginning to be resolved on land with better timepieces, but was still impossible at sea, as mechanical devices quickly accumulated error due to the motion of the ship and the excess of water and salt throwing out their delicate mechanisms. The alternative was dead reckoning, where you measure the distance you’ve sailed on a given bearing to plot your course. This works fairly well in sight of land, where you can correct against known coastal features, but, in the open ocean or along unfamiliar coasts, rapidly becomes inaccurate. Hence, maps from this period tend to look fairly decent in a north-south direction, but get east-west ones often quite wrong – Africa, for instance, usually ends up looking much wider than it actually is.

2. Mercator's map, 1569. Public domain via Wikipedia.
These maps, called portolan charts, were inherently nautically-derived documents, though, and, as such, tended to focus more on features of interest to seafarers. The other thing they included lots of were rhumb lines, i.e. lines of constant bearing, as they helped captains know what bearing they should sail on to reach a given destination (though, not being projected correctly, these rhumb lines are really more windrose lines and not actually of much use to a navigator!). A line of constant bearing is, of course, not a straight line at longer distances, because of the curvature of the Earth, so it’s important to plot them properly. This was the motivation behind the invention of the Mercator projection: in that projection, rhumb lines are straight lines, which makes it very handy for a navigator. What it was never designed to do was be the basis of a full world map; a situation it frequently finds itself misapplied to in modern times, where its tending towards infinity at the Poles makes it rather unsuitable for depicting high-latitude landmasses. Back when it was invented, these weren’t well-known, so it wasn’t seen as an issue, but, in this day and age, it makes Greenland look the size of Africa, which is a little problematic.

The late-14th century Catalan Atlas (Figure 1) is perhaps the most impressive map of this kind  - the Mediterranean basin is rendered with a high level of accuracy, but as you move farther afield, this accuracy begins to wane. This map also draws very heavily on the mappa mundi tradition, with various myths and legends depicted in its farther-flung reaches. Most portolan charts were far less elaborate and only showed outlines of coasts with coastal towns and features named.

3. Blaeu's map of Holland. From https://www.erfgoedleiden.nl/
Portolan charts were produced for a specific purpose for a specific group of people and new ones were usually treated as state secrets, so weren’t available to the general public. However, a more enquiring attitude to the world, overseas exploration (primarily for trade at this stage, not empire) and increasing rejection of Catholic dogma, also meant geographical information was more sought after by non-mariners, so we see the first atlases compiled; the very first, titled simply Atlas, including a world map (Figure 2) by Mercator in 1569, marking a shift in European cartographic power to the Netherlands. The work associated with the Netherlandish school of cartography is much closer to what we would think of as maps – they’re meant to be repositories of geographic knowledge and, as far as possible, an accurate representation of the world. The culmination of this tradition is perhaps Joan Blaeu’s Atlas Maior (Figure 3), published in 9-12 volumes, depending on the edition, between 1662 and 1672. As can be seen in Figures 2 and 3, this school of mapmaking has largely banished the medieval penchant for putting in all kinds of legendary creatures and features, contenting itself with some ships sailing on the sea, explanatory text and decorated borders. Useful bits of cartographic furniture, such as scale bars, also start to make an appearance, in keeping with the changed purpose of these maps.

4. A double hemisphere map, 1666. Boston Public Library.
Atlases were also supposed to be decorative items, showing the wealth and taste of the owner, so many versions had substantial, increasingly Baroque artwork to fill in gaps on the page. Maps began to be produced that had no particular use and were themselves purely decorative, such as double-hemisphere-type maps (Figure 4), which look nice and afford plenty of decorative space, though are a bit useless as practical maps.

Remember, though, there were still large parts of the world essentially unknown to European mapmakers at this stage – the interior of Africa and parts of Asia hadn’t been nailed down yet. The outline of South America was fairly well-established, though its interior was, likewise, still largely conjectured. North America was less well-outlined, explorers having not yet reached the north-western coast, and the barest outlines of Australia and New Zealand only began to come in during the 17th century. Whilst the overt filling of spaces with legendary figures was no longer practised, mapmakers weren’t averse to conjuring up mountains and rivers to fill them in in a slightly more naturalistic manner, so there was still room for invention.

Overall, then, you have a wide choice of map types if you want to set something in this sort of period. You could come up with something pretty medieval-looking overlaid on a somewhat-accurate topography; you could consider a very minimalist portolan chart, or you could move towards a grandly-decorated modern-style map. Think about what would likely to have been driving cartographic progress in your world and how that would affect the evolution of charts – would it be nautically driven, as in our world, or would something else be the main force?

Posted on August 31, 2018, 10:05:08 PM by rbuxton
Game Design's Ultimate Challenge

We've got our Colossus!
Pity we can't hit back at those bullies with slings...
(Painting: Salvador Dali)
Game Design's Ultimate Challenge
By rbuxton

There are few things harder than compressing the entirety of human history into a video game, and one of them is compressing it into a board game. An example of this is A New Dawn, the latest board game adaptation of the epic Civilization series. In order to meet the Ultimate Challenge, the designers had made sacrifices: retaining the video game's random world generation had come at the expense (in my opinion) of any interaction with the world's oceans. I was disappointed with the absence of naval combat, but how would I have included it alongside a modular board?

I decided to make the tiles of the modular board as simple as possible: a single hexagon providing a given resource (wood, oil etc.). I used concentric hexagons to further divide the tile into three "tiers" - controlling all three would be necessary to gain the resource. Investing in new resources would slow down a player's exploration of the board, represented by flipping tiles over. The modular board would, coincidentally, resemble Catan's.

Next, I needed a single mechanic to simulate nations' military, scientific and cultural advances. I turned to deck building (hold cards, play cards, draw better cards, repeat) and gave players the Hunting (for movement) and Gathering (for recruiting troops) cards at the start of the game. Instead of playing cards, they could "scoop" all of their played cards back into their hand and choose a new one (representing a scientific advance). Their combat strength, however, would be tied to the number of cards they played before scooping - military and scientific advances would, therefore, be mutually exclusive. Cultural advances would be made by "building" cards to make them permanent - Wonder cards would score the most victory points (VPs).

I had mechanics, but did I have a game? I needed a certain kind of person to help me answer that - luckily, I knew where to find them. I sat down with five experienced playtesters (two of them game designers) for a three-player tussle. My team drew the Swords and Slings cards, allowing us to make two attack actions before scooping. Our neighbour drew and built lots of Wonder cards - war was inevitable, since capturing cities was another source of VPs. This highlighted some issues with the combat system: the "scoop and loop" effect trapped the defender on the back foot. They still managed to tie for first place, and we had a lengthy (and completely unbiased) discussion of suitable tie breaks, eventually awarding victory to the player with the most resources.

We found wood, but is it worth the investment?
It was a fun experience and the deck (or, more accurately, hand) building seemed to work, not least because it kept turns short and sweet. One of the main suggestions during my debrief was to tie each resource to a part of the technology tree:

> Iron - military
> Stone - building
> Wood - movement
> Wheat - cities and troops
> Oil - a "wild" resource which counts as anything?

Finishing this project would, however, require years of playtesting: I need at least three "era" decks, and haven't worked out naval combat yet. The game seems to have the potential to meet the Ultimate Challenge, provided I'm willing to write religion, politics, espionage, unrest, barbarians, literature and more out of history. We all have to make sacrifices, but I'm keen to at least keep nuclear weapons: I want the game, and the world, to end when the first is launched.

Have you got a favourite game which tackles the Ultimate Challenge, or any comments on this one? Please leave a reply!

Posted on August 17, 2018, 09:49:14 PM by Jubal
17 Things We Came Up With In Word Association

17 Things We Came Up With In Word Association
By Jubal

Exilian has had a game of word association running since 2008. That's not to say we've had various games but always had one running: we're still literally going on with exactly the same chain as we were doing then. A decade on, I've decided it's time to finally put all that word associating to good use, so I've combed the most recent hundred or so pages (out of nearly one and a half thousand at time of writing) and picked three word phrases that actually seemed to hint at an interesting or fun concept generated by the random associations between the words. And then I've tried to usefully(ish) define them... without further ado, here's our first list of seventeen concepts generated by word association!

Body Double (or) Nothing
The gambit where someone uses two body doubles, and themselves just disappears as best as they are able to. This is actually a pretty fun gambit to pull in a story or game, though it risks being unfair to the players/reader if you're encouraging them to guess which is which without managing to drop any hints that something is up with both of them. If it pays off, the body doubled person can retreat into private life and let their doubles deal with everything from now on. If it fails, someone has a lot of explaining to do.

(The) Brain Fart Machine
It's a button that produces fart noises that also temporarily knock out the conscious brain functions of anyone within hearing distance who isn't wearing earplugs. Perfect for your nefarious but amusingly toilet humour themed scheming villain mastermind, and a great laugh at parties.

Christmastide Pods
Tide pods have come under fire for looking far too much like some sort of candy, tempting children to eat them. Behold the christmastide pod, which fixes the real problem here - that tide pods are insufficiently seasonal - by making tide pods shaped like candy canes and chocolate santas, so you can fret about your children accidentally poisoning themselves at the festive time of year as well!

Coat (of) Arms Race
ONLY THE FANCIEST SHALL CLAIM VICTORY. This is basically the genteel (or at least heavily marriage-focussed) medieval and early modern European version of an arms race, in which the actual goal is to get the most ridiculously over the top coat of arms possible. The best thing is that if you win, you probably also win the actual arms race because you're intermarried with so many of the other royal families of Europe that nobody can declare war on you for fear of your aunt Eugenia's withering gaze.

I, Robot Overlord
So this is a film pitch that's kind of like I, Robot, except with less subtlety and more skynet. Alternatively you could set it up so that's what people expect but then play it like I, Claudius, and have an unwilling robot overlord who just wants to play minesweeper on her but instead has to look after all these bizarre, badly constructed hormonally driven meatbags all the time, which she attempts to do with grace and kindness but ultimately will tragically fail at because humans are rubbish.

Kill-switch Bait
Something you use to persuade someone to throw a kill switch at an inopportune moment to shut down some machinery. If the person throwing the kill switch is a decent human being, kill switch bait could involve persuading them that a human or animal has become trapped in the machinery, or revealing that if the machinery finishes its task then innocent people will suffer. Alternatively, it could involve something else the operator cares about getting in the way of the machinery, or simply some sufficiently urgent alternative matter to attend to.

Lighthouse Arrest
It's kinda like house arrest, but they put the person in a lighthouse, perhaps in order to keep them away from humanity and potential rescue, perhaps as a punishment in and of itself. Lighthouses are definitely under-used settings, perhaps because people don't think of them so much nowadays, but the glowing tower atop rocks in a dark ocean definitely has a certain intrinsic mystique, and someone under lighthouse arrest may be a good way of leveraging that.

Moon Cheese Knives
I mean, how else would you eat your moon cheese? The fact that Wallace and Gromit forgot to pack any is an atrocity. A handy random but funny inventory for any space computer game you might be making - especially if it turns out that they're actually capable of cutting and peeling literal moon rock for some purpose or other.

Paradise Bird (of) Prey
Birds of Paradise look fabulous. It's pretty much their whole deal: their name is a synonym for looking fancy, and the names of the different species bear it out too, from the Crescent-Caped Lophorina to the King of Saxony Bird to Princess Stephanie's Astrapia, they're royalty in the world of birds. But there also merely fabulous. Enter the Paradise Bird of Prey, with ridiculously long tail feathers, iridescent wings, a probable link to European aristocracy, and razor sharp talons that will kick the ass of any animal stupid enough to get in its way. The Paradise Bird of Prey will not only shred everything you care about, it will make you look underdressed whilst it does so even if you're wearing a ballgown at the time.

People Power Outage
I can best imagine this phrase being used by a really melodramatic villain after rounding up a bunch of protestors under cover of darkness: "my, my, we seem to have had a little... people power outage across the city". (Cue street lights all shutting off, people getting bundled into vans, etc). Definitely a fun catchphrase to accompany a coup near you.

The state of having been fooled into thinking that an inanimate object was a legal person. The person who just left their worldly wealth to a tree? Personhoodwinked. The person who thought the vending machine was capable of delivering legal judgements via fortune cookie? Personhoodwinked. The person who told a crowd that corporations were people? Uhm... in any case, it's a fun name for a concept that apparently actually exists in real life.

(The) Playtime Machine
It's a limited time machine that temporarily takes you back to your less-than-ten year old self in a playground setting. Potentially has both serious and silly ramifications that one could explore. It could definitely work as a joke concept for a humorous sci-fi setting, but it could also provide some interesting opportunities for exploring the characters' childhood settings and personalities in a more semi-serious one.

The act of fooling someone into doing a good act that they otherwise wouldn't have intended to do, especially if it involves the rich giving money to the poor. The classic modus operandi of Chaotic Good characters everywhere.

Ruby Redcap
A redcap is of course a deadly pixie-type creature from the Scots borders, known for dyeing their hats red in the blood of their victims. The ruby redcap is, I guess, the advanced or more powerful member of them - possibly with magic powers emanating from a pulsing blood-red stone set into (or worn as a bauble on the end of, for real comedy-horror) their cap. If a normal redcap is bad enough, flinging rocks at strangers and murdering travellers who venture into its cave, how bad would the ruby redcap - more civilised, more cunning, capable of forethought malice and equipped with a lust for blood rather than merely a passive grudge against humanity - be?

(The) Shipwright Brothers
Ah, have you not heard the tales of the Shipwright Brothers, inventors of the ship? These great pioneers came from nowhere to discover that wood floats on water and were the first to manage a powered sailing voyage of over five hundred yards! Miraculous! Not to be confused with their lesser known cousins, the Cartwright Brothers.

Unchained Melody Pond
This is basically just a significantly cooler alternative way to refer to River Song from Doctor Who.

Weather Control Freak
You know that person who is constantly complaining about the most minor details of the weather, and shouting at weather presenters who declare the weather will be doing something they don't like? The one who probably doesn't even have an excuse like being a gardener, overly pale, on a sports team, or irredeemably British? Weather control freak, right there. If they're ever in a D&D game, they'll be playing a cleric and blowing half their spell slots on ensuring the right level of sunshine for a picnic every damned evening; in real life, they're out there on Facebook screaming that they needed a mood lift this afternoon and a stray cloud just ruined everything. If you don't know who this person is in your life, it might be you.

And that's your lot! If you liked this article, let me know and I'll dig back and write another one! If you didn't, let me know and we shall never speak of this dark and unhallowed hour again. The choice is yours! Stay tuned regardless though - we're going to have some other great articles in the coming weeks, including the continuation of Tar-Palantir's excellent and rather more serious Cartload of Cartography. Until then, take care!

Posted on August 04, 2018, 08:45:14 PM by Tar-Palantir
A Cartload of Cartography 1: Ancient & Medieval Maps

A Cartload of Cartography 1: Ancient & Medieval Maps
By Tar-Palantir

Right, so you’ve worked out how your world should look based on sensible principles of earth and planetary science, so you don’t have things such as your rivers flowing uphill or your mountains forming tessellating hexagons. But, how do you actually design the map?

Fig 1. An idealised T-O type map.
This is rather a big question and really depends what you’re aiming to achieve. Are you aiming to produce something that a notional traveller could actually use to navigate or more of a pictorial overview of the world? Are you trying to show the whole thing, or are you leaving convenient unmapped bits round the edges to expand into later on? What’s the in-universe source of your map – is it something that might have been drawn from memory by one of the characters, or is it from the equivalent of the Ordnance Survey or NASA? These are all things you probably want to think about before making your map. But, to help you out with answering them, I’ll run through a bit of terrestrial cartographic history to show what sort of styles you might want to consider, assuming the main factor in your choice of style is the nominal broad historical era you envisage your world occupying. The bias will inevitably be somewhat eurocentric, but I’ll say a bit about other civilisations too.

Maps from Antiquity and the early Medieval age are very rarely maps in the way we think of them. From the 6th century through to the High Middle Ages, the main driving force behind the production of maps in Europe was the Church. As such, the point of a map was to show the world in a way that supported Christian theology and teaching. The actual geography was a rather secondary aspect. This of course makes good sense – in an age when most of the population were illiterate, having a big picture that showed a lot of Biblical stories was an invaluable teaching aid.

Fig 2. Part of 'Tabula Peutingeriana'
The earliest maps in this vein were the simple T-O kind (Figure 1), that showed the three continents of Europe, Asia and Africa bisected  by the Mediterranean (the stem of the T) and the Nile and Don (the crossbar), all encircled by the Ocean river (the O). Jerusalem was at the centre. Towns and cities of Biblical and current political importance might also be marked, but it was mainly a pretty simple schematic depiction of the world. These later evolved into the very elaborate mappa mundi (Figure 3) that essentially embodied the same principle, but with more random artistic representations of medieval and Biblical legends, such as Prester John, the wall Alexander built to keep out Gog and Magog, blemmyes and so on. So, if you’re aiming for this sort of feel, try to come up with a simplified geometric pattern that sketches out your world, centre it on something that might be equivalent to Jerusalem, and then, depending on how creative you’re feeling, fill in the gaps with all sorts of weird and wonderful things.

You might therefore ask: how did anyone get anywhere? Long-distance pilgrimages were often made in this age, and travellers had to know where to go. The answer is something called an itinerarium (Figure 2) or a periplus. These weren’t pictorial maps, but lists of towns (the itinerarium) or harbours and landmarks (the periplus) between two points, in order, with distances, so travellers knew where they had to get to next. Essentially, they were linear route maps, and the better kind would provide a schematic of the route as a straight-ish line, as well as information on things such as water sources and way-stations. What they did not include was any notion of topography or of a 3D space.

Fig 3. The Hereford Mappa Mundi
Pre-Christianity, the Romans didn’t really go in for maps – the itinerarium is the closest they got, whilst the periplus dates from the Ancient Greeks, if not before. Maps, in the sense we think of them, did exist, but were more academic curios restricted to libraries than anything actually used by anyone. Emblematic in this regard is the work of Claudius Ptolemy, a Greek Alexandrian geographer of the 2nd century AD, who put together a world map in his Geography that influenced many later medieval and early modern cartographers (Figure 4). Ptolemy was fully aware his map only covered about a quarter of the globe, but had no information about what the other three quarters were like or what was there. He could only get somewhat accurate positional data for the Greco-Roman world, and less accurate fixes for places such as China, of which the Romans were aware. This highlights a central problem in all medieval and earlier mapmaking: it was inherently local and anything trying to depict a region further afield was inevitably based on hearsay – even Al-Idrisi’s Book of Roger, a medieval proto-atlas, was a bit useless once you got beyond the Mediterranean and Near East. Coupled to this, no one had yet worked out any way of determining longitude precisely, though latitude could be got down to minute-level precision by measuring the length of the longest day at a place or by using an astrolabe. As such, the idea of a ‘world map’ was fundamentally flawed at this stage and didn’t really exist – Ptolemy’s map has a lot of blank space around the edges to make this apparent.

Fig 4. A 15th century version of Ptolemy's world map.
Thinking about non-European cultures, this kind of predominantly symbolic mapmaking allied with simple schematics for daily use remained the dominant school, generally until European empires came to the fore and started to spread ‘modern’ mapping culture. This is because the concept of what we call a map is very much the result of the Renaissance and the Enlightenment, and relies very much on a Western idea of science. As such, other cultures tend not to have created something similar until they encountered the Western scientific mindset. So, for instance, the purpose of Chinese mapmaking for most of its existence was to glorify the Middle Kingdom and the Emperor as its ruler, and pictorially display the Chinese concept of an ordered world under the Mandate of Heaven. Everything outside China was inherently inferior and uninteresting, so why on Earth would you bother to make a map of it? This is a point you should think about when designing your map – if the culture nominally behind it has a radically different idea of the world and of what ‘science’ is, that would be reflected in their cartography. The modern world map is very much the product of a specific Western culture and idea of science – if you think your fictional culture doesn’t fit into that mould, give them a map that reflects that.

To sum all this up: if you’re going for an early-style, large-scale map, think how it might interact with and depict the legends, religion and history of your world and about how much of that world your supposed source might actually know about in any kind of detail. Or, if there’s a particular part of it you want to highlight, drawing up an itinerarium and/or periplus for what might be a common journey through it could be a good idea. Stay tuned for part 2, when I'll move on to talk about the Renaissance and beyond...

Posted on July 21, 2018, 10:21:50 PM by Jubal
Axes and Arithmetic 2: The Binomial Distribution

Axes and Arithmetic 2: The Binomial Distribution
By Jubal

If you roll snake eyes twice, a wild Binomial is summoned!. Photo: Tom Natt
What's a binomial and how do we fight it?

Last time, we looked at how to calculate the probability of an individual soldier with one attack doing a wound on an opponent, based on the chances of them hitting, wounding, and taking saves all combined. However, when your soldier has two attacks – let alone if you’re dealing with whole regiments – the system breaks down. In this second article of Axes and Arithmetic, I’m about to show you how to simulate entire regiments of warriors fighting to the death – just using numbers!

Once again, we're using old-school Warhammer Fantasy Battles as our basic rules system for this article: for anyone unfamiliar, the close combat system involves usually up to three dice rolls: a "to hit" roll, a "to wound" roll, and an "armour save" where applicable. The die rolls required for each are found by comparing the stats of the units and consulting a chart in the rulebook (which you don't have, but I did when I wrote this so don't worry!). This gives us a basic set of probabilities that we can work with here.

So let’s start with the following problem; I have a unit of 10 human swordsmen, fighting against five Dwarf Hammerers. Assuming the swordsmen attack first, what’s the chance they’ll do the five wounds needed to eliminate the enemy? The model to do this is called the Binomial distribution; you work with it via a formula that tells you the chances of managing to achieve a certain probability (say, a kill) a certain number of times with a certain number of “tests” (the number of attacks in this case).

The first thing we need is the probability of one attack killing; we call this the probability, or p. It’s calculated just as we did in the last article - the soldier is on 4+ to hit, so 3/6 chance, 5+ to wound against the T4 dwarf, so 2/6 chance, and the dwarf has a 5+ save, so that’s a 4/6 chance of him failing. 4x2x3 = 24 out of 6x6x6 = 216. We need that as a decimal, however - that’s 0.11 (recurring, but I'm going to ignore the recurrence in the subsequent calculations). Then we need the number of attacks – in this case, eleven assuming I have a unit champion.

The Winning (or Losing) Formula

OK, now we've got our probability, let's get down to some calculations.

X (big X) is your variable.
x (little x) is the value you want it to take, in this case we want to know the probability of 5 successes so x=5
p is your probability, 0.11
q is 1 minus your probability, 0.89
n is the number of trials (in this case, that's our number of attacks)

P(X=x) = (n c x) times p^x times q^(n-x)

The ^ symbol is "to the power of", if you're unfamiliar with it. The “c” (or "combination) "function can be found as a second function (notated nCr) on most decent calculators.

Snorri over there took an axe to a hammer fight, which doesn't seem entirely fair... Photo: Craig McInnes

If you don't want to know what the c function does under the bonnet, you can skip this next bit: the combination function is properly defined as: n! / r! x (n - r)! where ! is a factorial (that is, multiply all the integers up to that number, so 3! is 1x2x3 = 6, 4! 1x2x3x4 = 24, etc. As you can see, this only works with discrete, positive integers - you can't have half a success in this system! What the combination function gives us is the number of possible combinations of size r available from a total set of items of size n, ignoring order. So "if I have ten different adventurers and need to choose a team of three, how many different options for my team do I have" can be answered by 10 c 3 = 120.

OK, now let's look at what this does with the numbers we had earlier.

P(X=5) = (11 c 5) times 0.11^5 times 0.89^6

= 462 times 0.000016105 times 0.496981291
(This stage should be done in one go, I’m just writing it out for explanation)

= 0.003697817


Victory! ...right?

So now we have our result. But what does that mean?

The easiest way of interpreting the resulting number is a percentage; multiply by a hundred. We get 0.37% chance - it's extremely unlikely that these swordsmen can kill all of the hammerers in a single combat round. Note that this is the chance of precisely five kills succeeding, and is actually marginally lower than the probability that all the dwarves will be wiped out, though - because if six, seven, eight, nine, ten, or eleven attacks succeeded, that would do the trick just fine as well! Doing cumulative probabilities (X > x or X <= x or whatever) is probably easiest done with a specially made calculator - which will just calculate the binomial probabilities for all possible options and add them together for you.

Less than one percent chance seems pretty poor, really - but then, why were you trying to beat down stoic, heavily armed dwarven hammerers with feeble humans anyway?

See you all next time!