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 2)
June 22, 2000

If you missed the first article on Probability, check out Part 1.

Tindelhunden appear in a random, unoccupied, non-structure square at the edge of the board. The four squares adjacent to the Sancta are considered structures for this purpose (so are towns and colonies, of course). The edge of the board, in this case, means any square that's not adjacent to four other gameboard squares; there are 26 edge squares on the gameboard, 4 of which are adjacent to Sancta, so Tindelhunden can appear in up to 22 different squares.

If summoned at an enemy group on the edge of the board, they'll appear close enough to attack on the same turn they're summoned about 9% of the time, and on the turn after that about another 9% of the time; the worst-case situation of appearing a full 12 squares away will happen 27% of the time. Against an enemy group in the center of the board, of course, anywhere on the outside edge of the board will take pretty much the same amount of time to reach it. Moral of the story: if you want your hounds to stay on the board for as long as possible, summon them at the enemy group closest to the edge of the board and cross your fingers; if you want them to reach the target group as fast as possible (on average), pick a group near the center of the board.

Blue Auk, Giant, Griffin, Pegasus, and Undine all have nasty terrain restrictions, but just how bad are they? From our discussion on Bolt of Somersaults, we guessed that about half of the squares on the board were at least somewhere near a strategic target of some kind (and about a sixth were directly adjacent to something important). From our discussion of automatically generated terrain at the beginning of the game, we know that there's an average of about two or three each of Water or Mountain squares. So, the odds are surprisingly good that you'll have some useful place to summon these Monsters onto (the odds are far better for Blue Auk, of course, since it can be summoned in the diamond around a Water square, and is thus the only one of these that can normally be used to block enemy movement). Likewise, both Giant and Undine will be slightly worse off for being Territorial, so they have to be closer to the action than the others.

So, as yet another (very rough) rule of thumb based on the above, Blue Auk, Griffin and Pegasus will be useful in combat about 4 out of every 5 games; Blue Auk will be useful as a roadblock, Undine as a detour and Giant as a combat hulk about 1 out of every 3 games.

Cessao Rift, of course, takes groups that fall through Void squares in other games, so the odds are entirely dependent on who else is playing. That's what my contacts at DA tell me, anyway.

Amok, Disorient and Fingle (for the sake of completeness, even though you probably knew this one already) will send an enemy group in a direction other than the one it was ordered 75% of the time; if the group has two equally favorable directions it could have walked in anyway, the probability this will hurt your opponent drops to 50%. So, try not to use these on groups that might move in any of several directions.

Faerie Circle and Gnax both involve a random number between 2 and 6; this is entirely random and not weighted towards any other number (you die-rolling types can think of them as 1d5+1). So, on average, you'll get four Gnaxen and a Faerie Circle'd group will stay away for four turns.

Forbidden Ichor is a nicer spell than many give it credit for. It always changes the recruit to one of the other eleven Nations, never leaving it as it was.

When cast on a +1 hand damage recruit (Imp, Keeper, Human) it has a 2/11 chance of keeping the same ability, 3/11 chance of getting 9 HP instead, and 6/11 chance of losing its combat bonus entirely (assuming you don't cast it on an enemy Archer).

When cast on a 9 HP recruit, it has a 2/11 chance of staying the same. If cast on a Swordsman, there's a 3/11 chance of it gaining +1 hand damage (on first swing in combat) instead, and 6/11 chance of it having no combat bonus; if cast on an Archer, 2/11 chance of it gaining +1 missile damage and 7/11 chance of it having no combat bonus.

When cast on a Waterwalking, Mountainwalking or +1 missile recruit, there's a 1/11 chance that it will stay the same, 2/11 chance that it will become each of the other ones, and 3/11 chance each that it will gain 9 HP or +1 hand damage. This means you have a 10/11 chance of stranding a Mountainwalking group on a Mountain, and a 10/11 chance of drowning a target Waterwalking recruit.

Assuming each House has an equal chance of being played in a game, Forbidden Ichor will either be immediately useful as a minor combat modifier or else a good situational threat card (such as drowning a recruit or stranding its group) about 77% of the time, and the net average effect of the spell is somewhere around 2 mana. Not a bad deal considering it only costs 1… made up for by its randomness, and the fact that it's usually no better than any other cheap combat filler like Weakness. But when it works even better than that, oh what a bargain!

Jumping Land has a 60% chance each turn of moving, period; assuming it can move to any of the four adjacent squares (i.e. there's no towns or board edges in the way), then, it has a 15% chance of squashing a specific adjacent group each turn… if it can only move to three of the four adjacent squares, as when there's a town on one side, this probability jumps to 20%.

Lode Star has a rather complicated effect. Basically, it takes a random square of the 5x5 diamond centered on the target town, ignoring structures, and turns it to lava, and repeats the process a total of five times. If a single square is randomly chosen more than once, then you'll end up with fewer than five squares turned to Lava at the end of the spell. Thus, in a situation where there are no “illegal” squares within range other than the target town itself (i.e. there's 24 squares that can be turned to Lava), there's about a 19% chance that it will dump Lava into any given specific square… such as the one your opponent's group is standing in.

Lava Flow is similarly complicated in how it works, and technically the card text isn't correct right now to top it off. Here's what actually happens: every turn, there's a 1 in 3 chance that an eruption will happen. In an eruption, it will choose a random non-structure adjacent square twice (i.e. it'll choose two squares, but they might be the same one, in the same way as Lode Star). Those squares will turn to Lava, and then on the turn of “casting” +1 (that is, the turn after the turn after they turn to Lava) they turn to Barren Land.

Mirrored Armor will break, on average, after being struck 9 times (that is, on the tenth strike it will break, and the recruit will take damage). Realize that this is an average over a huge range; it might break on the first swing or it might last for the rest of the game past hundreds of swings. At any rate, considering the average, it means you can reasonably expect it to last for a typical one-on-one combat (although it'll probably break very quickly if several enemies gang up on it). One warning: if you want to prove this statistic mathematically, the steps involved are not pretty.

Prophet is more of a metagaming issue than a gaming one, as far as whether or not you'll help out your opponent (even if you do, Hope generally has more use for large quantities of mana than any other House).

But that aside, let's assume that you're equally likely to play against any of the twelve Houses. If your opponent plays a two-mana deck, you have a 1/12 chance of giving them two additional mana (i.e. they're playing Hope, too), a 5/12 chance of giving them nothing useful at all, and a 6/12 chance of giving them 1 mana they can use. Thus, on average, you're giving yourself +2 mana and your opponent +2/3 of a single useful mana, for a net gain of 1 1/3 mana over your opponent (or a net gain of 2/9 mana per mana point spent on the Prophet spell, which is actually the worst ratio of any mana-gaining spell in the game). On the other hand, as noted above, four Prophet spells will probably help you a LOT more than your opponent.

If your opponent plays a three-mana deck, assuming the third mana type is equally likely to be any of the mana types not associated with the opponent's House, you have a 5/24 chance of giving them two useful mana, 14/24 chance of giving one useful mana, and 5/24 chance of giving no useful mana at all. In this case, you're giving yourself +2 mana and your opponent (on average) +1 useful mana, for a net gain of 1 mana over your opponent (a net gain of 1/6 mana per mana point spent on the Prophet spell). Moral of the story: if you're afraid of the opponent playing Prophet, play a three-mana deck.

How Often Will I Draw This Card?

And finally, some basic odds on how soon you'll draw a specific spell that you put in your deck. The following tables assume that you'll be casting or discarding once per turn, so they are just a general guideline; if you have lots of cheap spells in a deck so that you can cast and discard early on, your odds will improve. The numbers given are the percentage of the time that you'll draw the card by the given turn, rounded to the nearest whole number.

If you have one copy in your deck:

If you have two copies in your deck (the second number is the percentage of the time that you'll draw both by that time):

If you have three copies in your deck (the second number is how often you'll draw two copies; the third number is how often you'll draw all three copies):

If you have four copies in your deck (the second number is how often you'll draw two copies, third number three copies, fourth number all four copies):

So, the next time someone asks you why they can't seem to draw that Armistice from their 80-card deck, calmly explain that they have less than a 50/50 shot at even seeing one of them by the time their opponent is attacking their first town.

Good luck!

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