
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, nonstructure
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 worstcase 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 dierolling 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 nonstructure 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 oneonone 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 twomana 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 managaining 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 threemana 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 threemana 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:
1 copy

Deck Size 
Turn

30

35

40

50

60

80

100

1

17

14

12

10

8

6

5

2

20

17

15

12

10

8

6

3

23

20

17

14

12

9

7

4

27

23

20

16

13

10

8

5

30

26

22

18

15

11

9

6

33

29

25

20

17

12

10

7

37

31

27

22

18

14

11

8

40

34

30

24

20

15

12

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):
2 copies

Deck Size

Turn

30

35

40

50

60

80

100

1

29,2

25,2

22,1

18,1

15,1

12,0

10,0

2

33,3

29,3

26,2

21,1

18,1

14,0

11,0

3

37,5

33,4

29,3

25,2

21,1

16,1

13,0

4

40,6

36,5

33,4

27,2

24,2

18,1

15,1

5

43,8

39,6

36,5

30,3

26,2

20,1

16,1

6

46,10

42,8

38,6

33,4

28,3

22,1

18,1

7

48,13

44,9

41,7

35,4

30,3

24,2

20,1

8

50,15

46,11

43,8

37,5

33,4

26,2

21,1

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):
3 copies

Deck Size

Turn

30

35

40

50

60

80

100

1

37,6,0

33,5,0

30,4,0

25,2,0

22,2,0

17,1,0

14,1,0

2

41,9,0

37,7,0

34,5,0

29,3,0

25,2,0

20,1,0

16,1,0

3

44,12,1

40,9,1

37,7,0

32,5,0

28,3,0

22,2,0

19,1,0

4

46,15,1

43,12,1

40,9,0

35,6,0

31,4,0

25,2,0

21,2,0

5

47,19,2

45,14,1

42,11,1

38,8,0

34,5,0

27,3,0

23,2,0

6

47,22,3

46,17,2

44,14,1

40,9,1

36,7,0

29,4,0

25,3,0

7

46,26,4

46,20,3

45,16,2

42,11,1

38,8,0

31,5,0

27,3,0

8

45,30,5

46,23,3

46,19,2

43,13,1

40,10,1

33,5,0

28,4,0

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):
4 copies

Deck Size

Turn

30

35

40

50

60

80

100

1

42,11,1,0

39,8,1,0

36,7,0,0

31,4,0,0

27,3,0,0

21,2,0,0

18,1,0,0

2

44,15,2,0

42,12,1,0

39,9,1,0

35,6,0,0

31,4,0,0

25,3,0,0

21,2,0,0

3

45,19,3,0

44,15,2,0

42,12,1,0

38,8,1,0

34,6,0,0

28,3,0,0

23,2,0,0

4

45,24,4,0

45,19,3,0

43,15,2,0

40,10,1,0

36,8,1,0

30,5,0,0

26,3,0,0

5

44,28,6,0

45,22,4,0

44,18,3,0

42,13,1,0

38,10,1,0

33,6,0,0

28,4,0,0

6

42,31,9,1

44,26,6,0

44,21,4,0

43,15,2,0

40,11,1,0

35,7,1,0

30,5,0,0

7

39,34,11,1

43,29,8,1

44,24,5,0

44,18,3,0

42,13,2,0

37,8,1,0

32,5,0,0

8

36,37,14,2

41,32,10,1

43,27,7,1

44,20,4,0

43,15,2,0

38,10,1,0

34,6,0,0

So, the next time someone asks you why they can't seem to draw that Armistice
from their 80card 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!
Read more
