Ngozi's Way A periodic column on Sanctum strategy, theory, and fun, by Ian Schreiber, Sanctum player name Gannon. You can reach Ian at ai864@yahoo.com.

Game Strategy: Probabilities (Part 1)
June 15, 2000

Fun With Randomness, Statistics and Probabilities

Sanctum has a lot of random effects in many spells, random town placement and terrain generation at the beginning of the game, and even some of the combat target selection is random. Yet most Sanctum players, even those well-versed in the field of Statistics, don't know many of the basics. What follows may not be the most interesting reading in the world, but it will certainly improve your game.

How Far Away Will The Nearest Town Be?

Myself and a few other players have been compiling statistics on this lately. While I can't guarantee the accuracy of this, it's the best guess I can give without reverse-engineering the code. You're out of luck there; I don't do assembly.

In this picture, some of the squares are colored in; the ones in blue are the places where two towns can appear on your side, red on the opposing side. The two squares in green are the two possible locations for the center town; in Veldt, at least, it is guaranteed to be in one or the other. Other than that, towns cannot be touching but I've found no further restrictions. Yellow outlines show Sancta and their bordering squares, of course.

Assuming town locations are entirely random (subject to above), then there are 88 distinct possible town layouts. Of those:

• 11 have one town that is two squares from your Sanctum (12.5% chance).

• 45 have at least one town that is three squares from your Sanctum but none that are two squares away (51.1% chance), and of those, 10 have both towns three squares away (11.3% chance).

• 31 of the remaining layouts have at least one town that is four squares away (35.2% chance), while only one town layout has both towns a full five squares away (1.13% chance).

• 10 layouts have no towns for either player that are on a direct walking path to the center town (11.3% chance).

• 16 layouts have no towns on a direct walking path to the center town for one player, while giving the other player at least one such town (18.2% chance). I'm sure you can relate to being on the unfavorable end of one of these more often than you'd like …

Now, when I refer to a direct walking path to center town above, I'm not counting terrain that might block an otherwise good walking path. As for terrain, that seems to have a wide variance. In my own games in Veldt, I've had as few as zero of each terrain type other than Forest or Plains, and as many as 3 Swamp or Desert and 7 Mountain or Water (!). Overall, though, it seemed as if there were an average of about two or three each of Mountain/Water per game, and about half a square each of Swamp or Desert (that is, you could expect to see one of each between every two games). Thus, any individual square has something around a 7% chance of being hazardous terrain that blocks movement from the beginning of the game.

So, what are the odds of terrain blocking your way to your nearest town? As a very rough estimate, I'd say somewhere around 5% of your games will involve you having to deal with terrain that blocks all direct walking paths to at least one of your nearest towns (if that seems low, note that this is only for one player; it means that about 10% of the time, either you, your opponent or both of you will have this obstacle). The majority of these, of course, will happen on town layouts where one of your nearest towns is in a straight line from your Sanctum … in those cases only one square need be blocked, while all other cases require two or more squares to be blocked. So, when including spells that get you past terrain, you must decide whether saving yourself from otherwise certain defeat in one out of every twenty games is worth a card that may or may not find use in the other nineteen.

What Are the Odds This Spell Will Work in My Favor?

Now let's look at some spells with random effects so that you know just how often they'll help you and how often they'll destroy you.

Chaos Feature has a fun list of spells associated with it. There are six beneficial spells (Regeneration, Rite of Growth, Retribution, Vicious Strength, Unholy Aura, and Beast's Embrace), nine harmful spells (Stupefy, Spectre's Curse, Weakness, Forsaken, Pantogar's Curse, Body Rot, Cannibalism, Curse of Khobai, Obeisance), and one spell that could work either way (Dire Portent). If you work out all the numbers, you get an average of about 7/16 mana (a little less than half of one point of mana) of a harmful spell, if you assume that mana from “good” spells cancels out mana from “bad” spells on a one-for-one basis. Moral of the story: the odds favor you if you cast Chaos Feature on an enemy recruit, but on its own it's still not worth the mana cost. You can improve your odds a bit by targeting an enemy recruit that's so pumped you can't beat it in combat normally; in such a situation, most of the beneficial spells will not make a difference, and a few of them will seriously hurt the target (Curse of Khobai, Forsaken, Dire Portent).

Bedlam is a different story, mainly because it hits a large group at once. It draws from the same pool of spells as Chaos Feature, but in this case the odds of getting something that royally hoses a group (Cannibalism, Obeisance or Forsaken) are much greater if you cast Bedlam in the path of the enemy Horde. For a group of four, the probability of the group being afflicted by at least one of these three is about 56%; against a group of five, about 65%… and that's just for one turn (the odds shoot up even higher if you can stall the enemy group's movement with monsters or terrain). The down side, of course, is that this can burn you in the long run: if allowed enough time, the opponent can eventually start marching individual recruits into the Bedlam and waiting for one to get Beast's Embrace followed by a few good combat spells; you can guard against this by keeping Disintegrate or Wrack in your deck, which will allow you to get rid of any nasty enemy recruits unless your opponent has Invisibility spells (note that a recruit cannot become Invisible from Chaos Feature or Bedlam).

I should note that for both of the above spells, the list they draw from has changed every time an expansion has been released, so the above probabilities are sure to change in the future.

Itrokos Gate chooses a random Monster from the following list of 52 summon spells: Blue Auk, Priodont, Minotaur, Giant, Ebon Guardian, Golem, Man of Iron, Makva, Gorgon, Trolls, Banshee, Centaurs, Undine, Obsidian Dragon, Pegasus, Stalking Blyk, Yfreet, Goldcaps, Simian Warriors, Tree Man, Huntsman, Wolf Pack, Salamander, Brimstone Dragon, Master Vampire, Sand Sphinx, Skeletal Horror, Legion of the Dead, Skeleton, Unicorn Herd, Sylph, Prismatic Serpent, Gnax, Cockatrice, Manticore, Werewolf, Entrail Eater, Bog Horror, Celestial Sphere, Revenants, Maloch Horror, Rakshasa, Great Bear, Furies, Phantom, Griffin, Gnome, Jade Dragon, Hellion, Wyrmling, Gatemouth, Boggart. Yes, it'll choose from the entire list (even terrain-restricted ones like Centaurs and Undine); if ineligible it'll pick again, and incidentally it'll quit without giving you a monster if it still hasn't found one after 50 tries. Of the monsters, ten of them will never be summoned by this spell unless they undergo actual functionality changes in the future; two of them will sometimes be summoned depending on board layout (Golem and Blue Auk). The average mana cost of a monster you'll get from this spell if you can't get Golem or Blue Auk is 6.15 (this average drops to 6.12 if cast on a square adjacent to a friendly town, 6.07 if cast within a diamond of a Water square, and 6.04 if both). Not a bad deal for a 5-cost spell! In case you're curious, the probability of not getting any monster (due to 50 unsuccessful tries) is 0.00000000000000000000000000000000041% (that's 4.1E-36 for you scientific types), which explains why no one has ever reported a bug that their Itrokos Gate didn't give them a Monster.

Bolt of Somersaults, Furies, Gatemouth, Lost Mine, Nihil, and Questing Beast all move or create a group in a random square on the gameboard. Groups cannot be bounced to or created in towns, colonies, Sancta, or squares already occupied by other minions. There are 94 total squares on the gameboard; of these, two are Sancta and five are towns (in Veldt, anyway) which leaves 87 squares; subtract one for every occupied square, and that's how many possible places there are for the group to land on. You can do further calculations on the fly, depending on the gameboard; for example, if there's 80 available squares and 4 of those are hazardous (Water, Void, etc.) then the odds of a single group appearing on one of those is 4 in 80 (if you prefer, 1 in 20, or 5%).

Figuring out how likely it is that the group will end up somewhere interesting (such as, “within one square of any town, Sanctum or enemy group”) is a bit harder, and your opponent might not want to oblige your request to take five minutes each game turn to count up the individual squares, pull out your calculator and compute your exact odds for that Bolt of Somersaults. I'm afraid I don't have any easy tricks for figuring the odds at a single glance of the gameboard for you, although I'm working on it. In the mean time, I'll give a general rule of thumb: for any given board layout, a group appearing in a random square will end up somewhere absolutely useless about half the time; it will end up somewhere in the middle of things about a third of the time; and it will end up somewhere extremely good (right next to an enemy town or Sanctum, for example) about a sixth of the time. Think of rolling a six-sided die; on a one, two or three the group is hosed, on a four or five it's okay, and on a six it's a huge threat to its opponent. Obviously, this is not true for all town layouts or gameboard positions; it's just a simplified way to think generally about the use of such spells, for those times when you don't have your calculator handy.

Geomancy is a fun spell, and those that cast it are no doubt dreaming that they'll manage to trap an enemy group in a Mountain or Water square. From the earlier discussion on hazardous terrain on the opening board, we can derive that the probability of any individual group outside of a town falling into hazardous terrain is about 7%, give or take a few percent. That may not sound like much, but if there's ten groups on the board, at least one of them is probably going to get nailed!

If a group can already handle Mountainwalking or Waterwalking, the “fatality” rate drops to a little more than 3% (about 0.5% if both).

Inspiration draws from all cards available, including “OOPs” (Bleak Isle, Apocalypse, Sword of Zana), “NIPs” (Waterwalking, Elven Piper, Organ Grinder) and Whispers… even unreleased Whispers. Much as I'd love to give you a way to figure the odds of getting a spell that's useful in any given situation, I just don't have that kind of time… and in the middle of a game, you probably don't either.

Changing Lands presents some interesting statistical problems. It chooses a random terrain from the following list: Plains, Barren Land, Mountain, Ice, Water, Void, Lava and Forest (the target square will never turn into a Swamp, Volcano or Desert). That comment in the card text about a 7 in 8 chance of changing simply means that it chooses one of those eight terrain types randomly each turn, even if it chooses the same one as last turn.

If a group without terrain-crossing ability attempts to walk through a Changing Lands square, barring other impairments such as summoned enemy monsters to block movement further, the probability of making it safely through in one turn (by Ice terrain) is 12.5%; in one or two turns, 39%. The probability of group death during entry (by Void) or second turn (Void or Water after already entering the square) is 25%. The rest of the time, some sort of uncrossable terrain will block the group's movement and it will take extra turns to either pass through or get killed. (Figuring out the exact probability of group death versus eventual crossing involves some infinite series that you'll just have to do yourself if you want to know that badly; be sure to account for death by Lava damage…)

If the group has either Mountainwalking or Waterwalking, the probability of making it safely through in one or two turns jumps up to 56%; for Waterwalking, the probability of group death by Void in the first or second turn drops to 22%.

Good luck!

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