Simulating these special dice on more regular dice

So I’ve got a game that uses special dice for its resolution. Basically when making an opposed check, each side rolls a bunch of them based on the relevant trait, and whoever rolls most points wins. The margin by which you win matters for resolution, usually.

But I’d like to simulate the resolution on regular dice. (Or rather; I’d like to be able to play, and let others play, without buying more of these special dice).

But I’m stumped because the dice symbols don’t line up easily. Effectively, the special dice are marked [-1, 0, 0, 1, 1, 2] with a minimum score of 0 for any roll.

Because the margin of victory matters, I can’t just replace it with “whoever rolls higher”, and because of the -1 on the die there’s a bigger chance of rolling nothing than normal.

Can anyone see a way to match (or at least get very close) to the resolution mechanic used here, while using easy to get dice? I’m okay with them not being 6-sided, but I would like something with a regular distribution.

How many rings that have special effects can one character wear at a time while still having all of the special effects [duplicate]

This question already has an answer here:

  • Is there a limit to the number of magic rings a character can wear on one hand? 3 answers

So i was wondering how many magical rings can one character wear at once while still retaining their special effects.

Disjoint Set Union-Find Special Case

I was doing reading on weighted quick-union using path compression. I have a clear understanding of why this is $ O(1)$ with amortized analysis for union and find operations but do not understand how to address this special case.

Suppose we have $ x$ items to start with, (all disjoint) and we perform a sequence of $ m$ operations such that all calls to find come after all calls to union. I am trying to determine what kind of data structure could be used, because in my previous understanding of disjoint sets, the union operation is always dependent on the find operation.

Despite having this information about the order of operations, I do not quite see how it is useful- or, how it can be used to achieve an amortized analysis of $ O(1)$ . Also, not sure of how to approach the union operation without relying on calls to find.

Are these special (one production) Context-Free Grammars always unambiguous?

Consider the following (Context-Free) Grammars with only one production rule (not including the epsilon production):

  • $ S \rightarrow aSb\;|\;\epsilon$
  • $ S \rightarrow aSbS\;|\;\epsilon$
  • $ S \rightarrow aSaSb\;|\;\epsilon$
  • $ S \rightarrow aaSaaSbb\;|\;\epsilon$
  • $ S \rightarrow aSbScSdSeSf\;|\;\epsilon$
  • etc…

Grammars like these uphold 4 basic rules:

  1. The non-terminal symbol $ S$ can never appear next to itself.
    • e.g. $ [\;S \rightarrow aSSb\;|\;\epsilon\;]$ would not be allowed.
  2. The non-terminal symbol $ S$ can appear at the beginning or end of a production but not on both sides at the same time.
    • e.g. $ [\;S \rightarrow SabaS\;|\;\epsilon\;]$ would not be allowed.
    • e.g. $ [\;S \rightarrow Saba\;|\;\epsilon\;]$ or $ [\;S \rightarrow abaS\;|\;\epsilon\;]$ would be allowed.
  3. The sequence of terminal symbols that exist between each non-terminal $ S$ cannot all match.
    • e.g. $ [\;S \rightarrow aSaSa\;|\;\epsilon\;]$ , $ [\;S \rightarrow abSabSab\;|\;\epsilon\;]$ , etc. would not be allowed.
    • e.g. $ [\;S \rightarrow aSaSb\;|\;\epsilon\;]$ , $ [\;S \rightarrow abSabSaf\;|\;\epsilon\;]$ , etc. would be allowed.
  4. The sequence of terminal symbols at the beginning and end cannot match.
    (i.e. $ [\;S \rightarrow ☐_1SaS☐_2\;|\;\epsilon\;]$ s.t. $ ☐_1 \neq ☐_2$ where $ ☐_1$ and $ ☐_2$ are a sequence of terminal symbols)
    • e.g. $ [\;S \rightarrow aSbSa\;|\;\epsilon\;]$ , $ [\;S \rightarrow aaSbSaaS\;|\;\epsilon\;]$ , etc. would not be allowed.
    • e.g. $ [\;S \rightarrow aSbSb\;|\;\epsilon\;]$ , $ [\;S \rightarrow aaSbSaxS\;|\;\epsilon\;]$ , etc. would be allowed.

Are (Context-Free) Grammars, that follow these 4 rules, always unambiguous? It would seem so. I really don’t see any conceivable way that such Grammars could be ambiguous.

(Note: To show why Rule 4 is necessary consider the grammar $ S \rightarrow aSbSa\;|\;\epsilon$ with the string $ aababaababaa$ . A visual derivation can be found here.)

I’ve spent a lot of time thinking about this question and have had a hard time finding any way of either proving or disproving it. I’ve tried showing that Grammars like these are $ LL(1)$ . However, it seems only Grammars of the form $ S \rightarrow aSb\;|\;\epsilon$ are $ LL(1)$ . Grammars like $ S \rightarrow aSaSb\;|\;\epsilon$ are not $ LL(1)$ . Maybe I need to use a higher $ k$ for $ LL(k)$ ?

(This question is a follow-up/reformulation of a previous question.)

I would really appreciate any help I could get here.

By RAW, does jumping in combat need any special treatment?

Jumping is a special kind of movement. There doesn’t seem to be special combat rules for jumping, so it behaves the same as walking. Is any special consideration needed when jumping in combat?

These prompts may help you form a response:

  1. Do you have to declare how far you are going to jump?
  2. Can you attack mid jump?
  3. Can you end your turn mid air?
  4. How does jumping over an enemy interact with attack of opportunity?
  5. How does jumping vertically higher than 5ft + an enemies height interact with attack of opportunity?
  6. How do actions such as Disengage or Dash interact with jumping?
  7. Are there any other quirks?

Please note I am asking about RAW only, and I understand the consequences of doing so. No frame challenges please. Do not assume this question is useless. Assume I want to jump, have considered alternates, and that it is useful to me.

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Recursion from a special case of quick sort

I have the following special case for merge sort:

“John wishes to perform a quicksort on an array, where he picks the partition as the $ \left \lfloor{\frac{2n}{3}}\right \rfloor th$ least element in the array (If the array has 10 elements, it will pick the element at the 6th position in the sorted array) such that this pivot selection procedure takes $ \Theta(n)$ time.

What is the worst case running time?”

I have tried forming a recurrence, but I end up having floors/ceilings and don’t know how to deal with them. I would like to reduce the case down to the recurrence:

$ T(n) = T(2n/3) + T(n/3) + \Theta(n) $

Where I’m then able to produce a recursion tree I can easily analyse.

However, when producing an upper bound I’m getting a ceiling on the $ n/3$ and a floor on the $ 2n/3$ .

For the purpose of producing a recursion tree, can I just assume these floors/ceilings are negligible for a very large n?

Thanks

restrictions on the bone knight special skeletal mount

Paladins becoming a Bone Knight lose the services of their paladin mount and gain a skeletal mount instead. The wording seems to restrict the type of mount, allowing only a skeletal warhorse or pony:

Summon Skeletal Steed (Sp): At 2nd level, you gain the services of a skeletal steed: a heavy warhorse with the skeleton template applied (or a war pony with the skeleton template applied for Small bone knights). You may call this steed in the same fashion as a paladin whose level equals your paladin level plus your bone knight level, and the steed gains the same special abilities as a paladin’s special mount at the same effective level. A skeletal steed cannot be turned while its bone knight master rides it.

But previous paladin levels are taken into account in almost every aspect and the wording seems not really extremely explicit about the restriction. So.. could this be interpreted as maybe the warhorse and pony being the default but also allowing different types of mounts (provided the skeletal template is applied), such as a skeletal Pegasus for example? and could allowing that, considering the strengths and weaknesses of the Bone Knight class, still be considered fair and balanced?

PS: personally, a skeletal warhorse or skeletal pony sound rather useless to me, especially for a paladin previously owning, say, a unicorn. more than anything the skeletal template only ruins whatever it is applied to (well, special abilities anyway).

Check subset sum for special array

I was trying to solve the following problem.

We are given N and A[0]

N <= 5000 A[0] <= 10^6 and even  if i is odd then  A[i] >= 3 * A[i-1] if i is even A[i]= 2 * A[i-1] + 3 * A[i-2] element at odd index must be odd and at even it must be odd. 

We need to minimize the sum of the array.

and We are given a Q numbers

 Q <= 1000  X<= 10^18 

We need to determine is it possible to get subset-sum = X from our array.

What I have tried,

Creating a minimum sum array is easy. Just follow the equations and constraints.

The approach that I know for subset-sum is dynamic programming which has time complexity sum*sizeof(Array) but since sum can be as large as 10^18 that approach won’t work.

Is there any equation relation that I am missing ?