Devising an algorithm to remove “complete” sets of numbers from a list of numbers

Suppose I have a list of numbers consisting of numbers 1 through 5, where some numbers repeat. Define a set of numbers to be "complete" if it consists of each of the numbers 1-5 exactly once. Now, I want to pull out as many possible complete, sorted (lowest to highest) sets from this list. How can I do this (efficiently)?

Here is the solution I have in mind so far. Traverse through the list, and place each number into a pile of numbers that is the same number. For example, place all the 1s into a pile, all the 2s into a pile, and so on. Then, go through all of the piles and determine the minimum number of elements in a pile. For example, if the pile of 2s has five 2s and all other piles have greater than or equal to five elements, then 5 is the minimum number of elements. This minimum, call it min, will give us the number of complete sets in the list. Finally, go through the piles in order and pull out the elements one by one until you form a complete set, and again to form another complete set, and so on until min number of complete sets. Tada! You now have pulled out as many possible complete, sorted sets from this list.

Here is an example:

Let’s say the list is 3 2 1 2 2 3 4 5 5 4 4 3 4 1 1 2 5. We traverse the list and put them into piles:

333 2222 111 4444 555

The minimum number of elements in a pile is 3, so we have 3 complete sets. Now just go through the piles in order and pick up elements one by one:

12345 12345 12345

And tada! We have pulled out as many complete, sorted sets as possible.

Where I am lacking in my solution is figuring out how to efficiently go through the piles in order or make an efficient way of ordering the piles so that I can easily go through them in order.

I’d love to hear any feedback, especially other algorithms you might devise or ways to go about ordering. Thanks!

examples for languages of natural numbers

I need to find examples for language $ L_i$ $ i\in[1,3]$ of natural numbers that is:

  1. $ L_1\in$ $ RE \backslash R$

  2. $ L_2\in$ $ coRE\backslash R$

  3. $ L_3\in$ $ \overline{ R \cup RE}$

My idea was in any case to take a language in the desired language class, and find some sort of function from the language to the natural numbers. But I do not know if this is the right way to approach the problem, and how exactly to find such function.

What target numbers would be a certain level of difficulty under this system?

I’m writing a homebrew game system, and I found that I have an action resolution mechanic but not a good system for target numbers (I call them Success Thresholds, or STs, in this game, and from now on I’ll use that term to refer to the minimum number a player gets that can succeed).

To resolve an action, most of the time players roll 2d6 and add a modifier ranging from +0 to +3, depending on the stat. With Advantage, it is (3d6 drop lowest)+mod, and Disadvantage is (3d6 drop highest)+mod.

There are also 4 (well, 5, but one auto succeeds) levels of difficulty. The Trivial tasks are automatically successful. Easy tasks should succeed about 75-80 percent of the time, Moderate tasks should be successful 50-60% of the time, Hard tasks should be successful between 25 and 40 percent of the time, and impossible tasks shouldn’t succeed more than 25% and often more like succeeding below 10-15% or the time even with Advantage and a +3 mod.

This is an anydice program with the base probabilities for a +0 mod. I want to know what number should the ST be for each level of difficulty? I had initially considered 7 as a base difficulty for Moderate tasks, before I added modifiers to rolls.

Order posts alphabetically with numbers but some of the posts has numbers in the title

I have this query:

 $  args = array(        'post_type' => 'alerts',     'post_status' => 'publish',     'posts_per_page' => -1,      'orderby' => 'post_title',      'order' => 'ASC', );  $  alerts_loop = new WP_Query( $  args );  

The problem is that the order breaks in certain point. look at the imagen any ideas why it breaks. I have used:

'orderby' => 'title', 'suppress_filters' => true, 'ignore_sticky_posts' => true 

enter image description here

Find the smallest group of numbers with sum bigger then $X$

Given a list of numbers $ S$ where $ 0 < s_i < 100$ , find the smallest group of numbers with sum bigger than $ X$ .
Each number can be used multiple times.

Ex: for $ S = [3,4.1], X = 10$ the solution is $ [3, 3, 4.1]$

Is it a known problem? What will be the best way of solving it?

For now, my best solution is to randomly pick numbers and repeat the process multiple times.

Collection of online tools and utilities for text, numbers, dates, and various other data types.

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I've build this as a challenge for myself, and accomplished as much. Unfortunately, I'm tight on cash and would like to pass it on to someone else who may want or need it — either for SEO or otherwise. It has great potential in the right hands.

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Collection of online tools and utilities for text, numbers, dates, and various other data types.

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Why do we consider enumeration up to $w$ instead of leaving it to as many ordinal numbers?

A few minutes ago I asked a question about a "proof" that $ \mathbb{R}$ is enumerable that crossed my mind: What's wrong with this "proof" that $ \mathbb{R}$ is enumerable?

I was told to look into ordinal numbers, and that after crossing $ \omega$ we stop considering something to be an enumeration.

Why is this the case? Are there negative consequences if we don’t put this limitation?

Edit: I always thought of $ \mathbb{N}$ as the "counting numbers" – but… when we cross over to ordinals like $ \omega$ , $ \omega+1$ , etc, aren’t we still effectively counting?

Largest set of 10-digit numbers where none have Hamming Distance = 1 with any other

I’m working on a system that will require manual data entry of 10-digit numbers (Σ = 0123456789). To help prevent data errors, I want to avoid generating any two strings that have a hamming distance of 1.

For example, if I generate the string 0123456789 then I never want to generate any of these strings: {1123456789, 2123456789, 3123456789, …}

What is the largest set of unique strings in the universe of possible strings that satisfy the constraint where no two strings have a hamming distance of 1? If this set can be identified, is there any reasonable way to enumerate it?

Can TLS defeat the manipulation of TCP sequence numbers?

Assuming there is a powerful adversary who can arbitrarily manipulate the sequence number of each tcp packet, then the following packet-reorder attack should be possible, right?

Assuming the packet the attacker wants to disorder has the tcp sequence number n, he first allows the n+1, n+2, …, n+m packets to be sent out but modifies the sequence-number fields to use numbers n, n+1, …, n + m -1. Finally, the attacker uses the sequence number n+m to send the detained packet.

Is the attack still possible when TLS/SSL is used?