Does doubling up on Rope and Manacles increase DC or just take multiple checks to escape?

My D&D group is new and we are all still getting used to playing the game and are running through the Starter Set adventure.

During this adventure we end up capturing a Wizard and we bound and gag him. My character being paranoid made sure to attach manacles to him and also to use two different lengths of 50ft rope to ensure that he would not be able to break free. During the return back to town the DM had him keep rolling strength checks against the manacles and rope and one by one he kept breaking through them and I ended up having to borrow the entire party’s supply of rope to keep him secure because he kept breaking through them. By the time I was able to deliver him he was wrapped in 200 ft of rope and looked like more rope than man after breaking my manacles and two different lengths of rope.

I know that the rules say that the DC of Hempen rope is 17 but I still think that in any real world sense it is ridiculous to think that this Wizard can Samson his way through each individual rope while also being held by several other lengths of rope and a set of manacles. One would think that the weight of the ropes alone would keep him in check.

I guess my question is that is there any precedent for the DC being raised when you are attaching multiple different factors into keeping a prisoner secure or are you just supposed to roll each rope individually?

Edit: My DM has responded to my post and it looks like I didn’t have all of the information originally

“I rolled 2 rolls per day of travel. This was his daily attempt to free himself from his bindings with disadvantage. He got 2 Nat 20’s the day he broke the manacles and a 19 & 20 to break free of the rope. His Str modifier is -1, but 18 still clears the rope’s DC. He had disadvantage due to the leather armor and sheer volume of rope around him He also had to break the manacles before he could even start trying to work at the rope, since you did put those on him first”

What are the statistical implications of doubling damage on crit instead of doubling the dice rolled?

I’ve been playing with a group that frequently allows players to double the value of dice rolled for crits (and other things) rather than rolling double the dice. Example, someone crits with 2d6 and rolls for 8 damage, which they then double for 16 total crit damage, rather than rolling 4d6.

This mostly just bugs me on principle, but I was curious what doubling the value rather than doubling the dice does mathematically. Does it actually make a difference? Is there a greater chance to hit extreme ends of the range of values (low and high)? Does the amount and type of dice create greater inconsistencies between the two scenarios?

The accounting Method analysis for table expansion by tripleling instead of doubling an array

If we double the array every time we get the amortized cost of 3n or 3$if you prefer. I was wondering what would it be if we tripled the array size instead of doubling it. The rational between the 3$ cost for every insertion is as follow:

• 1 dollar for the insertion of an element.
• 1 dollar is saved for when it will have to move itself to the new array with double the size
• 1 dollar paying for another element then itself when transfer will be required.

I can’t seem to find the cost for an array that triple each time.

Is the Spell Bombardment additional damage die subject to critical hit dice doubling?

Level 18 Wild Magic sorcerers have the following feature (PHB, p. 103):

Spell Bombardment

Beginning at 18th level, when you roll damage for a spell and roll the highest number possible on any of the dice, choose one of those dice, roll it again and add that roll to the damage. You can use the feature only once per turn.

When you score a critical hit with an attack (even a spell attack), you double the amount of dice you roll.

Does the additional damage die from Spell Bombardment also get rolled twice on a critical hit?

Will doubling the price & damage of grenades make them balanced compared to other weapons?

Context: I’m running a group through the Dead Suns adventure path. This is the first time any of us have used Starfinder. One of the party members is an ysoki envoy who would like to focus on grenades for damage (inspired by this question). It’s a party of 6, so I routinely give bonuses to named/solo foes and increase the number of mooks in encounters.

The group has reached level 5, and we’ve noticed that grenades just don’t do a whole lot, though. Our solarian routinely wallops enemies for 30+ damage per round, and can Supernova for 6d6 damage. Meanwhile the envoy is chucking around looted Mk1 grenades that do 1d6 or 1d8 damage or purchased Mk2 grenades that do 1d12 or 2d6 damage. In a recent fight, their enemy was able to make effective use of grenades only because 5 mooks threw grenades at the same time (and even then most of the party members struck took about 15 damage, since the saves were easy).

Proposal: I’m thinking about adding an “Elite” version of all damage-dealing grenades that costs twice as much and deals twice as much damage. As an example, the Mk1 Shock Grenade is a level 1 item costing 130 credits that deals 1d8 damage; the Elite Mk1 Shock Grenade would also be a level 1 item but would cost 260 credits and deal 2d8 damage. Foes important enough to have a name who carry grenades would carry the elite versions, while faceless mooks would use the normal versions.

Looking at later levels, an Elite Mk5 Frag Grenade would be a level 14 item costing 37.5k credits and dealing 20d6 damage. That sounds like a lot of damage, but almost any character could spend 72.3k credits and proficiently wield an Advanced Seeker Rifle, a level 14 longarm that deals 6d8 damage per shot. It seems like elite grenades would be highly effective from an action economy perspective, but too expensive to be used casually (which is how grenades probably should work, really).

Is this a viable change or am I overlooking anything?

Contractable and Simply Connected Doubling Spaces Homeomorphic to Euclidean Space

Is there a characterization of all simply connected, contractable doubling metric spaces which are homeomorphic to a simply connected subset of Euclidean space?

Maximizing the weighted average value of a set of items by doubling the weight of a subset of items

Given a set of $n$ items, each represented by $t_i=(w_i,v_i)$ for $1 \le i \le n$ , the weighted average value of those items is defined as:  \frac{\sum_{i=1}^{n}v_iw_i}{\sum_{i=1}^nw_i} 

The goal is to find a subset of items to double their weight, such that the weighted average value is maximized.

For example, suppose you have the following set of items, where $t_i = (w_i,v_i)$ :

 t_1 = (12, 1100000)\ t_2 = (12, 1000000)\ t_3 = (12, 850000)\ t_4 = (10, 800000) \ t_5 = (8, 1200000)  The weighted average value is 981,481. The best solution is to double the weight of items 1 and 5, which leads to a new weighted average of 1,024,324.

I am trying to come up with an algorithm to find the best subset of items to double, and so far I’ve tried using bruteforce. For each item, you can choose to either double it, or leave it as it is. This means that there are a total of $2^n$ possibilities to explore, which means exponential complexity. I’ve also tried a greedy algorithm to pick the highest value-to-weight ratio items, and determine the best number of items to pick, however this solution is not always optimal.

I am wondering, what is the most efficient algorithm to find the best subset of items to double their weight?

Let $$f(z)=\frac{z^2-1}{2z}$$ and $$\phi(z)$$ be a Mobius transformation such that $$\phi(0)=-1$$ and $$\phi(\mathbb{R})$$ is the unit circle in the complex plane.
How do I show that $$f$$ is topologically conjugate to the doubling map $$x\mapsto 2x\,(\text{mod}\,1)$$ on the unit interval?