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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