"A puzzle for your intellectual enjoyment" Topic

I intend to sit down and work this out, but for those who enjoy maths problems I present it here for those tired of suduku.

In PBI by Peter Pig the side that becomes the defender has to put a certain number of units in reserve- determined by the 'pregame game', at least 1, upto total units-1 (he is always allowed 1 on table at the start). The process is this (NB for purposes of the answer A is the most desirable unit, then B etc, and a defender will have an army of 3-6 units)

He goes through each unit in order determined by him, and rolls 1d6. If a 5 or 6 is rolled that unit is late arriving. He does this until the correct number is put into reserve. Note it is the number put into reserve that is counted. If all are checked without the requisit number he rolls again in the same order.

What is the best order to check units?

Does this vary with number of units?

Does this vary with number to be put into reserve?

Corrorally question.
If he is only allowed one unit, he may risk a second unit to be on table, but it will usually lose 1/3 of its strength (hence the 'risk'). Assuming the same desirability, does the order of checking change?

I will check back in a few weeks to see if you have come up with the same as me- definately an improvement on Suduku!

Lost me on the very first line. I get the ague just seeing the word "math" in print.

What is the best order to check units?

Put the units you want in order of how much you want them, and then roll from the bottom up: so Z first, to A last.

Does this vary with number of units?

No.

Does this vary with number to be put into reserve?

No.

The strategy is most successful when the number put into reserve is small compared to the number of units tested. It's not the absolute numbers that matter, but the ratio of reserve to the total.

If he is only allowed one unit, he may risk a second unit to be on table, but it will usually lose 1/3 of its strength (hence the 'risk'). Assuming the same desirability, does the order of checking change?

No.

So 1-4 the unit is safe and 5-6 its into the reserve? And the goal is to get the more desirable units onto the table, not in the reserve?

The units rolled earlier are more likely to go into the reserve, because for later units to end up in the reserve the first ones have to get one result, then the later ones have to get a particular result. So if the odds are the same as you progress, earlier is more likely to go into reserve.

So if only one were going into reserve 1-4=safe, 5-6=reserve…

#1 – 1/3 chance of being in reserve
#2 – 2/9 chance (2/3 being opposite of #1's roll, times 1/3 on its own).
#3 – 4/27
#4 – 8/81

If more than one are going into the reserve the same principal applies.

Ideally, you want the reserve to fill up before you even have to roll for the more desirable units.

Put another way, you have to risk each unit until the required loss is satisfied, you want to risk the less desirable units first, which also results in them being risked more often, since you will likely end the sequence in the middle of your list, and those at the end will have to endure fewer rolls.

I'm not even going to tack the corollary. I think there's insufficient info. If the more desirable unit loses 1/3 its value, is it still more desirable than the other units?

And I assume the reserve units still get into the game, so in this particular game, which I know nothing about, how easy is it to stall until your reserve gets here? If your reserve all show up on turn 3 of a 12 turn game I'd rather wait and have them show up at full power. If they have a 10% chance of showing up each turn in a four or five turn game, I'll risk the 1/3 value in a hot second.

Andrew

"So 1-4 the unit is safe and 5-6 its into the reserve? And the goal is to get the more desirable units onto the table, not in the reserve?"

Yes. The reserve units may not appear due to the method of bringing them on. They do not appear in any particular order- the question is about getting the units you want on. I'd guessed it was the reverse method- Z first, but as it was late when thinking about it I think subconciously I'd got the Curse of the Wargamer- believing the dice had memory, so the 5/6 became more likely. I intend to sit down and see what the actual odds are for the 15 combinations (1 reserve from 3, 2 from 3, 1 from 4 etc). Won't affect the game- just for the maths hell of it.

Beware, the calculations are long and fiddly if done by theory for large numbers of reserve units. Technically, they are all infinitely long, but in practice one can stop a bit early.

I would do it as simulations. It's much less work.