On Astragalomantic Probabilities

Using the astragaloi, or knucklebones, for divination has really intrigued me lately, as if you couldn’t tell from my last two posts on the subject.  Something about them feels different from other divination tools I’ve used; it could be that they’re actual bones taken from a living creature once, or that they just feel more arcane and ancient than my divination dice or cards I’m known to use.  All the same, they’re quickly becoming my favorite divination tool (besides geomancy generally), and I’m struck by their power and potency in getting answers.  The method is overall simple: take five astragaloi, throw them, and find the oracular verse associated with the combination of the sides that come up.  It’s simple, but elegant and straightforward.

However, they’re also different from my other divination tools in that they have really weird statistical properties.  Consider a die: every side of the die has (approximately) an equal chance of coming up when thrown.  Thus, on a six-sided die, throwing a 1 comes up as often as throwing a 2, 3, 4, 5, or 6.  Knucklebones, however, are different: they’re not ideal Platonic solids, nor are they regularly shaped in any sense.  Their organic and geometrically awkward shape results in there being different probabilities in throwing an astragalos on any given side.  Of course, the probabilities will differ slightly based on the individual knucklebone used and how hard it’s thrown, but based on an analysis by Phil Winkelman, we can approximate throwing an astragalos onto a particular side as follows:

  • Khion (1): 10%
  • Hyption (3): 40%
  • Pranēs (4): 40%
  • Kōon (6): 10%

It struck me that, because of the statistical probabilities associated with each number, not all oracular verses associated with each throw of the astragaloi will come up equally.  Some verses might be relatively common, while others would be extraordinarily uncommon to obtain, whether for good or evil.  Having some free time on my hands, I decided to run a short statistical analysis on how common different throws of the astragaloi would come up and how that would affect divination using astragaloi as compared to my other divination methods or suggested ways to use the astragalomantic oracular verses.

For instance, consider the use of astragaloi for grammatomancy.  Grammatomancy is my expanded version of the Greek alphabet oracle, and traditionally you would use five astragaloi for obtaining a Greek letter by throwing the bones and summing up the sides of the astragaloi.  So, for instance, if you threw (1,1,6,4,3), the sum would be 1 + 1 + 6 + 4 + 3 = 15.  The minimum sum you can get is 5 (1,1,1,1,1) and the maximum is 30 (6,6,6,6,6); based on how the numbers add up, you could not obtain a sum of 6 which requires (1,1,1,1,2) nor a sum of 29 which requires (6,6,6,6,5).  Between the numbers 5 and 30 inclusive, excluding the numbers 6 and 29, there are 24 possible sums.  Thus, we can associate each sum with one of the 24 Greek letters, starting with 5 = Ω and 30 = Α.  However, because the probability of an astragalos rolling on a 1 or 6 is 0.1, and on a 3 or 4 is 0.4, we get different possibilities for rolling different combinations of astragaloi and, further, obtaining different sums.  Below is a table that maps each letter of the Greek alphabet with its corresponding astragaloi sum (presented both in Arabic numerals and Greek numerals) and the probability one will obtain that letter from rolling five astragaloi.  The more extreme (higher or lower) the sum, the more rare the throw.  Thus, it’s extraordinarily unlikely that one will obtain Α or Ω with astragaloi (0.001% of the time), but comparatively common to obtain Μ and Ν (15.48% of the time).

Letter Astragaloi Sum Probability
Numerical Greek
Α 30 Λʹ 0.00001
Β 28 ΚΗʹ 0.0002
Γ 27 ΚΖʹ 0.0002
Δ 26 ΚϜʹ 0.0016
Ε 25 ΚΕʹ 0.00325
Ζ 24 ΚΔʹ 0.008
Η 23 ΚΓʹ 0.02
Θ 22 ΚΒʹ 0.0328
Ι 21 ΚΑʹ 0.0624
Κ 20 Κʹ 0.09674
Λ 19 ΙΘʹ 0.12
Μ 18 ΙΗʹ 0.1548
Ν 17 ΙΖʹ 0.1548
Ξ 16 ΙϜʹ 0.12
Ο 15 ΙΕʹ 0.09674
Π 14 ΙΔʹ 0.0624
Ρ 13 ΙΓʹ 0.0328
Σ 12 ΙΒʹ 0.02
Τ 11 ΙΑʹ 0.008
Υ 10 Ιʹ 0.00325
Φ 9 Θʹ 0.0016
Χ 8 Ηʹ 0.0002
Ψ 7 Ζʹ 0.0002
Ω 5 Ε 0.00001

For me, being used to my divination dice, this is shocking.  I use a dodecahedron die (d12, 12-sided die) for grammatomancy, where I roll the die twice.  The first roll gives me an odd or even number, which refer to the first 12 or last 12 letters in the Greek alphabet, while the second roll gives me the letter within that set according to its rank.  So, if I roll a 5 and an 8, I end up with the Greek letter Theta (eighth letter of the first half of the alphabet).  Using a 12-sided die where every side has an equal chance of turning up (approximately 8.333% of the time), every letter of the Greek alphabet has an equal chance of occurring (4.1667% of the time).  The statistical difference between getting the same Greek letter with a 12-sided die used in this way compared to using five knucklebones is huge; we’d get Α on the die 4.1667% of the time, but on the astragaloi only 0.00001% of the time.  It’s not impossible, just far more unlikely.  Then again, another classical method of grammatomancy was the method of ψηφοι, psēphoi or “pebbles”, where one has a jar of stones each marked with a different letter.  By reaching into the jar and pulling out a random stone, you get approximately an equal chance of obtaining any single Greek letter, which gets us the same results as using a 12-sided die in my fashion of using one.  Whether the use of astragaloi or psēphoi was more common for grammatomancy isn’t clear to me, but both methods work.

So what about the actual throw for proper astragalomancy, where we’re looking at the combination that results instead of the sum that’s formed from the combination?  We know that:

  • There are four sides (1, 3, 4, 6) on each astragalos
  • There are five astragaloi
  • Order of the dice doesn’t matter

Thus, although there are 1024 possible combinations of astragaloi, we only end up with 56 possible throws of the astragaloi when we disregard the order and only consider unique combinations of the bones.  Below is a table that shows the probability for each possible throw of the astragaloi; remember that order doesn’t matter, so (1,1,3,4,6) is equivalent to (1,3,6,4,1) and (6,3,1,4,1).  Generally, the more 3s and 4s there are, the more likely a particular throw is.  Thus, we end up with a probability of 0.0001% for (1,1,1,1,1) and (6,6,6,6,6) as our most unlikely throws, and a probability of 10.24% for (3,3,3,4,4) and (3,3,4,4,4) as our most likely throws.

Throw Sum Probability
A B C D E
1 1 1 1 1 5 0.00001
1 1 1 1 3 7 0.0002
1 1 1 1 4 8 0.0002
1 1 1 1 6 10 0.00005
1 1 1 3 3 9 0.0016
1 1 1 3 4 11 0.0032
1 1 1 3 6 13 0.0008
1 1 1 4 4 11 0.0016
1 1 1 4 6 13 0.0008
1 1 1 6 6 15 0.0001
1 1 3 3 3 11 0.0064
1 1 3 3 4 12 0.0192
1 1 3 3 6 14 0.0048
1 1 3 4 4 13 0.0192
1 1 3 4 6 15 0.0096
1 1 3 6 6 17 0.0012
1 1 4 4 4 14 0.0064
1 1 4 4 6 16 0.0048
1 1 4 6 6 18 0.0012
1 1 6 6 6 20 0.0001
1 3 3 3 3 13 0.0128
1 3 3 3 4 14 0.0512
1 3 3 3 6 16 0.0128
1 3 3 4 4 15 0.0768
1 3 3 4 6 17 0.0384
1 3 3 6 6 19 0.0048
1 3 4 4 4 16 0.0512
1 3 4 4 6 18 0.0384
1 3 4 6 6 20 0.0096
1 3 6 6 6 22 0.0008
1 4 4 4 4 17 0.0128
1 4 4 4 6 19 0.0128
1 4 4 6 6 21 0.0048
1 4 6 6 6 23 0.0008
1 6 6 6 6 25 0.00005
3 3 3 3 3 15 0.01024
3 3 3 3 4 16 0.0512
3 3 3 3 6 18 0.0128
3 3 3 4 4 13 0.1024
3 3 3 4 6 15 0.0512
3 3 3 6 6 17 0.0064
3 3 4 4 4 18 0.1024
3 3 4 4 6 20 0.0768
3 3 4 6 6 22 0.0192
3 3 6 6 6 24 0.0016
3 4 4 4 4 19 0.0512
3 4 4 4 6 21 0.0512
3 4 4 6 6 23 0.0192
3 4 6 6 6 25 0.0032
3 6 6 6 6 27 0.0002
4 4 4 4 4 20 0.01024
4 4 4 4 6 22 0.0128
4 4 4 6 6 24 0.0064
4 4 6 6 6 26 0.0016
4 6 6 6 6 28 0.0002
6 6 6 6 6 30 0.00001

These probabilities are still different from the coin-toss method Kostas Dervenis gives in his Oracle Bones Divination.  Dervenis suggests one uses three coins flipped to obtain one of four results (T = tails, H = heads), each with the following probabilities:

  • Khion: HHH (12.5%)
  • Hyption: THH (37.5%)
  • Pranēs: TTH (37.5%)
  • Kōon: TTT (12.5%)

Thus, using coins as a substitute for astragaloi, we’d have a 0.0000305% chance of obtaining a (1,1,1,1,1) or (6,6,6,6,6) roll and a 7.41577% chance of obtaining a (3,3,4,4,4) or (3,3,3,4,4) roll.  These are pretty big changes in the probabilities of particular rolls, and all the other rolls would be affected similarly.  In either case, however, we have a situation where some results will come up far more regularly than others; then again, the oracle overall seems designed to have common outcomes assigned to the common fates, and extraordinary news to uncommon throws.  After all, it’s not every day you have the help of Zeus, King of the Gods and Men at your side, but far more common that you should wait a bit longer since your right time to act in the cosmos isn’t yet here.

So where does this leave us?  Should we forsake the use of dice and coins in favor of authentic knucklebones for astragalomancy since the probabilities of a given outcome are so different based on the tools used?  I don’t think so.  If we were playing a game of chance, then yes, the tools definitely matter, just as weighting a particular die to come up more on a given side would.  However, we’re not simply gambling with the gods here.  Divination is a sacred art and profession, and it helps the gods communicate with us so that we can ascertain their will as well as understand our own fates and our place in the divine order of creation.  Sure, it may be our hands that throw the bones, but it’s the hands of the gods that determine the outcome and how they land.  We’re not just rolling dice on our own, no more than things in the cosmos happen according to pure chance and nothing else.  This is why it’s important to invoke the gods of divination, like Hermes and Apollo, so that they’re involved in the throw of the astragaloi and can help guide them to fall on the proper sides so that we have a proper understanding of their wills and knowledge based on the result of the throw.  In that sense, using dice or bones or coins wouldn’t really matter, since it’s ultimately up to the gods to determine the outcome, and nothing is impossible for the gods.  Although they may have a preference for the system and tools used (hence the consecration and divination ritual from the previous post), they’re pretty handy when it comes to the myriads of tools used for divination.  So long as you’re letting the gods answer when you ask, the tools and their statistical qualities don’t matter in the long run.

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