Fastest way to ArrayFlatten a 2D array of 2D arrays, both for Sparse and Dense

I am using Mathematica 12 and want to know what is the fastest way to transform a 2D array of 2D arrays both for dense and sparse arrays. I remember looking for this a few versions of Mathematica ago, when ArrayFlatten was not that good. Since then I have been using option one below (which I found somewhere here, but I can’t find the question anymore – it had other suggestions as well). But a quick check shows that for SparseArrays that is not the case anymore. Is there anything better than ArrayFlatten

Also, why is the SparseArray version of 3 not the same as the other two, unless I make them dense?

dim = 5; eles = 50; sparse = Table[    KroneckerProduct[RandomReal[{-10, 10}, {eles, eles}],      IdentityMatrix[eles, SparseArray]], {ii, 1, dim}, {jj, 1, dim}]; Dimensions[sparse]  {5, 5, 2500, 2500}  RepeatedTiming[  jSparse1 =     Apply[Join[##, 2] &,      Table[Join @@ sparse[[All, ii]], {ii, 1, dim}]];]  {0.04, Null}  RepeatedTiming[jSparse2 = ArrayFlatten[sparse];]  {0.0094, Null}  RepeatedTiming[  jSparse3 =     SparseArray`SparseBlockMatrix[{{i_, j_} :> sparse[[i, j]]}, {dim,       dim}];]  {0.23, Null}  Dimensions /@ {jSparse1, jSparse2, jSparse3}  {{12500, 12500}, {12500, 12500}, {12500, 12500}}  jSparse1 === jSparse2  True  jSparse1 === jSparse3  False  jSparse2 === jSparse3  False  Normal[jSparse1] === Normal[jSparse2] === Normal[jSparse3]  True 

And here is the dense version with same conclusion

dim = 5; eles = 50; dense =    Table[KroneckerProduct[RandomReal[{-10, 10}, {eles, eles}],      IdentityMatrix[eles]], {ii, 1, dim}, {jj, 1, dim}]; Dimensions[dense]  {5, 5, 2500, 2500}  RepeatedTiming[  jDense1 =     Apply[Join[##, 2] &,      Table[Join @@ dense[[All, ii]], {ii, 1, dim}]];]  {1.251, Null}  In[5]:= RepeatedTiming[jDense2 = ArrayFlatten[dense];]  {0.61, Null}  RepeatedTiming[  jDense3 =     Normal[SparseArray`SparseBlockMatrix[{{i_, j_} :>         dense[[i, j]]}, {dim, dim}]];]  {2.3, Null}  Dimensions /@ {jDense1, jDense2, jDense3}  {{12500, 12500}, {12500, 12500}, {12500, 12500}}  jDense1 === jDense2  True  jDense1 === jDense3  False  jDense2 === jDense3  False  Max[jDense1 - jDense2]  0.  Max[jDense1 - jDense3]  0.  Max[jDense2 - jDense3]  0. 

and again the third version produces a result that apparently is not identical with the other two, though numerically they seem to be the same.

What is the fastest (non-spell) way to breach a castle wall?

A while back, my Pathfinder (1st edition) group was discussing late-game possibilities. I, playing a monk, noted that at 16th level my fists counted as adamantine for the purposes of overcoming damage reduction – I could literally punch holes in mountains, given a bit of time. I was reconsidering this conversation today and I got to wondering – what is the most efficient human siege weapon we can create?

So here’s the challenge: We are faced with a standard castle wall, 10 feet thick and made of stone (Per 3 feet of wall thickness: AC 2, hardness 8, 540 HP, Burst DC 50). We want to create a 20-foot-wide hole in the wall, in order to create a breach wide enough for soldiers to attack through. Since a wizard would have no trouble with this, we will be disallowing all spells and spell-like abilities (including those cast through an item). We’ll also ignore any siege weapons, excluding the PC themself. Everything else is valid, as long as it is official/published – supernatural and exceptional abilities, feats, magic weapons, and so on. How can we accomplish this in as few combat rounds as possible?

Fastest algorithm to find all the possible paths of length $n$ from a give node in a directed graph?

I am trying to find the fastest algorithm to find all the possible paths of length $ N$ from a given node in a directed graph.

My solution is to do a modification of breadth first search from the given node for $ N$ iteration. Its time complexity is around $ \theta(V+E)$ . But the problem is $ |V|$ & $ |E|$ becomes exponential because as long as there is an edge, the same node can be visited again.

Can there be a solution of this problem with polynomial time complexity? It seems this problem has optimal substructure solution. Is there any solution using dynamic approach?

What is the fastest path to an Ur-Priest Mystic Theurge

From the Complete Divine, the requirements for the Ur-Priest prestige class are:

  • Any evil
  • Base Saves: Fort +3, Will +3
  • Skill Ranks: Bluff 6, Knowledge (arcana) 5, Knowledge (the planes) 5, Knowledge (religion) 8, Spellcraft 8
  • Feats: Iron Will, Spell Focus (evil)

From the DMG, the requirements for the Mystic Theurge prestige class are:

  • Skill ranks: Knowlege (arcana) 6, Knowledge (religion) 6
  • Spells: Ability to cast 2nd level divine and 2nd level arcane spells.

The divine-side of the Mystic Theurge requirements can be fulfilled with 2 levels of the Ur-Priest class. Given at least 3 levels of an arcane caster class prior to entering Ur-Priest, what combination of base classes gives the fewest number of class levels in order to enter Ur-Priest? The 6 ranks of Bluff combined with a base Fortitude of +3 seem particularly difficult to acquire while still leaving room for at least 3 levels of an arcane caster.

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