Thursday, 20 November 2014

A funny thing about dice

O.K. Let me start by saying this is not the one about when you say "anything but a one". That is just coincidence made large by the strange way our minds work with coincidences and probabilities. This is about the mathematical quirk of a rule system and probabilities.

In classic Piquet virtually all die rolls are opposed. That is to say that both player's roll a dice and the result is determined by the difference in the dice. In a morale challenge roll, for example, the attacker rolls a dice against the defenders dice and there are four possible outcomes:

  • The defender beats the challenge by rolling higher.
  • The defender fails the challenge by rolling the same or less (disordered).
  • The defender fails the challenge by rolling less and his score is doubled (routed).
  • The defender fails the challenge by rolling less and his score is tripled (routed unralliably).

Last night, following the Wednesday war gaming session, I said to the guys as they left "You know there is more chance of rolling higher and tripling than there is of doubling." Both Graham and Peter were unconvinced, in fact I think they thought I'd gone a bit crazy. I thought I'd worked this all out in the past and discovered it to be true but, with these guys being pretty good with maths, I started to doubt myself. Surely, it is easier to double?

When I went to bed I got to work with the Sumerian abacus. Stop tittering. Hold your hand palm up. Stop tittering. Take the thumb of that hand and place it on the first bone of your index finger, that is 1. Moving up the next bone is 2. Moving up each finger and across you can count to twelve on one hand (I've heard this is why the dozen came about). Using the other hand to keep track of dozens you can count to 144, though I work left to right on both hands. It's useful for keeping a track when doing this kind of thing in your head.

Well, lets look at cases. D8 Vs D8.

  • There twenty ways to beat the defender without doubling or tripling.
  • There are seven ways to double the defender.
  • There are nine ways to triple the defender.
Hang on! 

This all comes about because of the double / triple quirk on low defender rolls. In the rules, triples always take precedence over doubles, so when the defender rolls a one, there is only one way he can be doubled - the attacker rolls a two - but there are six ways the defender can be tripled on a one - the attacker rolls three, four, five, six, seven, or eight.

Now I haven't worked out all the combinations of opposed dice, but a I'm fairly sure a similar thing happens when any attacking dice is rolled Vs D8. It's quirky.


Epictetus said...

I think the problem arises from your own lack of rigour in framing the question last night. It isn't easier to triple than double on opposed dice roles; in fact quite the opposite. It is however, easier to triple or more than it is to double or more but not triple. If you had expressed yourself in a clearer fashion then we wouldn't have looked at you with such disdain. Or perhaps we would. Any chance of a blog post on the length of the day at the equator?

Epictetus the Stoic BSc Mathematical Sciences


Now, you know I can't get my head around that one. Couldn't yesterday, can't today, will not tomorrow.

And, you always look at me with disdain. It looks well with your pedantry.

Sgt Steiner said...

Tripled or doubled all I know is I always suffer as the low roller ! Except of course when rules choose a low roll as good/successful then its high high higher :-)

oldschool said...

Interesting, but you also have to factor in that the defender can only be tripled on a roll of 1 or 2, but can be diubled on a roll of 1-4. So can only be tripled 25% of the time, but doubled on 50% of his rolls.

Rolling low as the defender in an opposed roll is really bad news.

oldschool said...

Apologies if this appears twice, but I forgot to prove I am not a robot the first time. Anyways, just in case the first one ended up in the spamosphere.

Interesting observation, but not the whole story. You also need to factor in that the defender can only be tripled on a roll of 1 or 2, but doubled on a roll of 1-4. In effect only 25% of a defender's rolls can be tripled, but 50% can be doubled.

In game terms this means that the outcome of the opposed roll is not linear. low rolls are significantly worse in an opposed roll mechanic.

Moral of story - don't roll a 1.