Math Attack!

I know it's been a long while since there has been a meaningful update to the information here, and the characters have unearthed a veritable library of secrets, rumours and innuendo. I will delve into those next, but to get back into the practice of writing about Bellstone Keep, here are some math (yes!) problems that the players have been dealing with.

A while back I was marvelling at the concept of advantage/disadvantage as a game mechanic.
(Hey, Jonathan Tweet! Advantage dice were an awesome facet of Over the Edge. Why didn't they make it into D&D 3.0?) Advantage or disadvantage as a general rule applies only to d20 rolls like attacks, saves or ability checks. For advantage, you roll two icosahedra and take the greater of the two results. Disadvantage is similar, but taking the lower number from the two dice. It seems reasonable to assume that results with advantage are better than average, but is it possible to prove that conjecture mathematically? Is there some way to quantify what numeric benefit advantage confers? Conversely, can we compute the penalty incurred when rolling a penalty die?

Well, yes! If you want all the details, the raw calculations are on the spreadsheet here. To calculate the average result, all possible results were added and that total divided by the number of outcomes. For example, a standard d20 has twenty possible outcomes: 1, 2, 3, 4...18, 19, 20. The sum of all those numbers is 210. 210/20 = 10.5 as shown in the chart below. The other results were obtained in a similar fashion.

The basic overview is that you gain better than a +3 bonus on your result when rolling with advantage. With disadvantage, your result is impacted worse than if you rolled d20-3.

Out of curiosity, I extended my computations to allow for rolling three twenty-sided dice and taking the best (or worst) result. As expected, the results were more pronounced. The benefit from advantage is just shy of +5 (Paging Holy Avenger! Mr. Avenger, will you please pick up the white courtesy phone?) The penalty also computes to roughly -5 as well. I don't know if I'll ever call for a "double advantage" roll but at least I know what to expect.

Two sessions ago, the party encountered a burning house. In an effort to douse the conflagration, our cleric Willow cast create water inside several rooms within the house. Without crunching the numbers, I allowed the 30' cube of water that spontaneously sprang into the room to extinguish the flames therein. But was that snap decision accurate?

A 30' cube is 30 x 30 x 30 = 27,000 cubic feet of water. The rooms targeted were 40' wide and 35' long with a 15' high ceiling. 40 x 35 x 15 = 21,000 cubic feet of water. So each room could be filled to the brim with water, leaving an extra 6,000 cubic feet of water to drain onto fires elsewhere in the house.

If the room had been 20' high instead, would the spell have been as effective? 40 x 35 x 20 = 28,000 cubic feet, so there would have been a thousand cubic feet of space left untouched by the spell. Arguably the fire might exhaust all the oxygen in that air pocket and extinguish itself, so the desired outcome of the spell probably would have been just the same.

(Brief aside: so how much weight did this spell add to the already damaged structure? At room temperature, a cubic foot of water weighs more than 62 pounds. The full spell summons over 1.6 million pounds of water into existence, similar to inviting 140 Asian elephants to march in and make themselves at home. Even the 6,000 cubic feet rushing back into the hallway weighs over a third of a million pounds. Did I mention that the cleric casting this spell from outside the room is a halfling?)

(Last brief aside: When will the United States switch over to metric? A cubic meter of room temperature water weighs a ton. 30 feet is roughly 9.144 meters. That length cubed comes to just under 765, so that's 765 metric tons of water. Finito!)

So last session, the party ventured under the saved if smouldering edifice to search a secret area mentioned by Rosazco the Rapscallion. (That's not slander; the nobleman will happily answer to that moniker!) The entrance was blocked by a stone slab 10 feet long and 5 feet wide resting over a depression and concealing a staircase. With great feats of strength and engineering, the party breached the subterranean vault. For visual reference, I stated that the slab was angled up at 30 degrees, allowing the party access to the descending stairs.

On the way home, Robin pensively asked, "Would everyone in the party fit under the slab if it was only raised 30 degrees? That would only be five feet of clearance."

I quickly answered, "Well, the staircase also started down at the lip of the opening, so there was more than five feet." That reply satisfied Robin but I wondered how accurate I was.

As Robin correctly computed, a ten-foot slab lifted at 30 degrees rises five feet above the ground. If it helps, think of a 30 degree angle as part of the right triangle formed when an equilateral triangle is cut perfectly in half.

Before even figuring how quickly the staircase descends, remember that the ten-foot slab is rotating at one end to form that 30 degree angle. So the base of the triangle formed is less than ten feet long, but the opening still stretches for the full ten feet.

 If only there were some way to mathematically determine the length of the base of this triangle.

Pythagoras is judging you with his dead, dead eyes.

Okay, so yes, the Pythagorean Theorem teaches us that the sum of the squares of the lengths of the two legs of a right triangle equals the square of the length of the hypotenuse. Simple, right?

Rounding, we find that the base of this triangle would be 8.66 feet. But the slab and the hidden aperture were both ten feet long. So where would the stairs begin?

Lifting up the slab also moves that edge away from directly above the rim of the gap. Subtracting the two values shows that the stairs run a full 1.34 feet before coming under the cover of the lid again. So how much staircase is that?

For simplicity's sake, I judged that the staircase would slant at a brisk 45 degrees. The benefit of this presumption is that the two legs of a 45 degree right triangle are congruent, so the length of one equals the length of the other. If the stairs go across 1.34 feet before reaching the slab, then the stairs go down 1.34 feet in that same space.

With a total of 6.34 feet of clearance, all but the most colossal half-orcs should easily get below the house without banging their heads.

I felt a 45 degree incline was reasonable on a staircase, but modern construction yields a less demanding flight. Today stairs are commonly built with eleven inch landings and seven inch risers, making the slope easy to figure out. (We don't need the angle of decline for our calculations, but it works out to roughly 32.47 degrees if you were curious.) We know how big the gap is (1.34 feet) and we also know that for every 11 inches over, the stairs drop 7 inches. Multiply 1.34 by 7/11 (or 0.63 repeating as a decimal) to get roughly 0.85 feet of decline for that gap. That's another 10 inches, so only the tallest characters would have to stoop to get into the secret tunnel. (Given our reflex to NOT bump our heads, probably some characters under 5'10" would still have ducked a wee bit.)

One last math question came to me while working on another campaign with a sailor turned druid. Before taking to adventuring, he was a fisherman. So could he wield a harpoon or trident or another nautically themed weapon?

Druids are proficient only in simple weapons. This restriction covers the spear but not the trident. (A "harpoon" does not exist specifically in the rules currently.) While bristling at this exclusion, I started comparing the statistics on the two weapons. What I saw was confusing.

Other than being half as heavy but twenty-five times the cost(!), there is no difference between the two weapons. None. They throw the same distance, do the same damage whether using one or two hands to attack. So maybe the answer to this question isn't found in math, but what makes the trident a more selective weapon?

~Tidwin

 12/29/16