Can an evil character cover there alignment?

Is there anyway for an evil NPC to cover up there alignment if a player attempts detect evil?

I am working on a Silver Dragon antagonist. This NPC will appear to the players as a number of human NPC’s setting tasks, missions, appearing as a friend. The dragons goal is to attempt to prevent an ancient doomsday prophecy coming to fruition. Each spell it learns will become branded to its scales in a series of runes which in human form will take the form of tattoos so over time the players may be able to work out these different humans are connected in some way.

The Dragon will determine the only way to stop this prophecy is to enact an ancient ritual involving it sacrificing itself to destroy all intelligent creatures with evil or chaotic alignment in all the planes of my world therefore becoming the very prophecy it was seeking to prevent. At first I thought the dragon would remain good as it believes its aims are good, but, I now see it will have to have its alignment shift as the campaign progresses and it becomes more convinced that mass genocide is the only way to save the world from lawful good at the start, to chaotic good and then chaotic evil.

Is there a current way for this dragon to hide its true alignment from any magical check or am I going to have to create a way for it to do this? Possibly one of the spells the party hunt out for it.

Is the cover art of the 1e AD&D module Sinister Secret of Saltmarsh a deliberate homage to the opening credits of Scooby Doo?

The 1e AD&D module U1 Sinister Secret of Saltmarsh has a plot that…

Coincidentally, the cover art of the 1e version of the module by David De Leuw depicts the mansion with bats flying out of it that is quite reminiscent of the spooky mansion with bats in the opening credits of Scooby Doo.

Are there any sources that verify this was a deliberate homage by the artist based on the plot of the module?

Cover art:

Cover of the 1e module

Scooby Doo opening credits:

Scooby Doo opening credits.

How can we formulate an anti-knight Sudoku as an exact cover problem?

Formulating standard Sudoku as an exact cover problem is easy and well documented. All of the constrained groups contain every digit which makes it natural to express the problem this way. Wikipedia claims without citation that:

Although other Sudoku variations have different numbers of rows, columns, numbers and/or different kinds of constraints, they all involve possibilities and constraint sets, and thus can be seen as exact hitting set problems.

How would one formulate other variants as exact cover problems with an incidence matrix? I’m focusing on the anti-knight variant, but with the goal of learning about expressing arbitrary variants.

The anti-knight Sudoku variant includes all of the standard rules, and additionally requires that no two squares which are a knight’s move (in chess) apart may contain the same digit. I don’t know how to approach this because a constraint between two squares cannot be expressed as requiring all digits to be present in some permutation. I can imagine a version of Algorithm X that solves this by removing additional rows by manual application of the anti-knight rule, but I believe Wikipedia is claiming this constraint should be expressible in the matrix alone without modifying the algorithm.

Is Wikipedia’s claim that all variants can be expressed this way correct, and how would one approach the anti-knight example?

Cover rules and spells that move as bonus action

The rules of cover are as follow:

A target with half cover has a +2 bonus to AC and Dexterity saving throws

Let’s say a wizard casts flaming sphere. He can then move the sphere as a bonus action to ram it into an opponent. If there is a cover in between the wizard and its target, but there are no obstacle between the sphere and the target, does the target benefits from the +2 bonus to the Dexterity Saving Throw ?

Minimum vertex cover and odd cycles

Suppose we have a graph $ G$ . Consider the minimum vertex cover problem of $ G$ formulated as a linear programming problem, that is for each vertex $ v_{i}$ we have the variable $ x_{i}$ , for each edge $ v_{i}v_{j}$ we have the constraint $ x_{i}+x_{j}\geq 1$ , for each variable we have $ 0\leq x_{i}\leq 1$ and we have the objective function $ \min \sum\limits_{1}^{n}{x_{i}}$ . We say such a linear programming problem LP. Note that it is NOT an integer linear programming problem.

We find a half integral optimal solution of LP that we say $ S_{hi}$ . For each variable $ x_{i}$ that takes value 0 in $ S_{hi}$ , we add the constraint $ x_{i}=0$ to LP.

For each odd cycle of $ G$ , add to LP the constraint $ x_{a}+x_{b}+x_{c}+…+x_{i}\geq \frac{1}{2}(k+1)$ where $ x_{a},x_{b},x_{c},…,x_{i}$ are the vertices of the cycle and $ k$ is the number of vertices of the cycle. We find a new optimal solution of LP that we say $ S$ .

If $ x_{i}$ is a variable that takes value $ 0.5$ in $ S_{hi}$ and value $ \gt 0.5$ in $ S$ , can we say that there is at least a minimum vertex cover of $ G$ that contains the vertex associated to $ x_{i}$ ?

The idea behind the question: in an odd cycle $ c$ with $ k$ vertices, the number of vertices needed to cover the cycle is $ \frac{1}{2}(k+1)$ , therefore for each odd cycle we add to LP the constraint $ x_{a}+x_{b}+x_{c}+…+x_{i}\geq \frac{1}{2}(k+1)$ . If in $ S_{hi}$ the sum of the variables of $ c$ is $ \frac{k}{2}$ (that is all the variables of $ c$ take value $ \frac{1}{2}$ ), then in $ S$ at least a variable $ x_{i}$ of $ c$ takes vale $ \gt \frac{1}{2}$ and the vertex associated to $ x_{i}$ belongs to at least a minimum vertex cover of the given graph.

Greedy algorithm for vertex cover

Given a graph $ G(V, E)$ , consider the following algorithm:

  1. Let $ d$ be the minimum vertex degree of the graph (ignore vertices with degree 0, so that $ d\geq 1$ )
  2. Let $ v$ be one of the vertices with degree equal to $ d$
  3. Remove all vertices adjacent to $ v$ and add them to the proposed vertex cover
  4. Repeat from step 1. until in $ G$ there are only vertices with degree $ 0$ (no edges in the graph)

At the end the removed vertices are a vertex cover of the given $ G(V, E)$ , but is it a minimum vertex cover? Is there an example where the algorithm does not find a minimum vertex cover?

How to resolve multiple sources of cover?

The cover rules in the PHB say:

There are three degrees of cover. If a target is behind multiple sources of cover, only the most protective degree of cover applies; the degrees aren’t added together. For example, if a target is behind a creature that gives half cover and a tree trunk that gives three-quarters cover, the target has three-quarters cover.

However the DMG (250/251) appears to have an alternate method for resolving multiple sources of cover:

To determine whether a target has cover against an attack or other effect on a grid, choose a corner of the attacker’s space or the point of origin of an area of effect. Then trace imaginary lines from that corner to every corner of any one square the target occupies. If one or two of those lines are blocked by an obstacle (including another creature), the target has half cover. If three or four of those lines are blocked but the attack can still reach the target (such as when the target is behind an arrow slit), the target has three-quarters cover.

The only case I can think where these might collide is something like the following:

multiple character cover

Individually T would not get cover from D1 or D3 and only get half cover from D2. The DMG seems to indicate T might get full cover since all 4 lines would be blocked. However if you use the guidance from the PHB (as shown above) T would only have half cover.

Seems to me it’s just a judgement call which interpretation is correct. Anyone know of any rules that clarify this other than a judgement call?

Is it mathematically possible to only have 3/4th cover?

In Is the DMG 3/4 cover diagram in 5e incorrect? the question asker points out the diagram shown in the DMG implies 3/4th cover, but it’s possible to get a better angle on the defender meaning they only have 1/2 cover.

Diagram showing various approaches to getting cover, from the DMG pages 250 and 251

Is it mathematically possible to construct a scenario where only 3/4th cover is possible? Intuitively I can’t think of a place to put the attacker in the diagram where one corner isn’t able to get 1/2 cover by picking a better square, whilst still having some attack on the defender.

Is the DMG 3/4 cover diagram in 5e incorrect?

From page 250 and 251 of the DMG:

To determine whether a target has cover against an attack or other effect on a grid, choose a corner of the attacker’s space… trace imaginary lines from that corner to every corner of any one square the target occupies. If one or two of those lines are blocked by an obstacle (including another creature), the target has half cover…

grid diagrams taken from the DMG. The final diagram shows the construction lines provided in the DMG to determine cover (showing 3/4th cover) as well as the question askers own construction lines, superimposed showing it should be half cover.

It looks like they just chose the wrong corner. If the attacker was moved one square to the north it would be 3/4 cover. Or do you think they meant "choose the closest corner"?