# Cast your vote for “Best of PPCG 2016”!

### Final phase

You have chosen the categories and the nominees for PPCG's annual “Best of”. Now it is time to pick the winners!

Each of the ten categories is represented by an answer to this question, and each of these answers contains all nominations by the members of our community.

### Voting mechanism

Each nominee has been added as a comment to its category's answer. In each category, the nominee whose comment has the highest number of votes by January 24, 00:00 UTC will be declared the winner of that category.

### Further details

• Feel free to vote for multiple nominees of the same category, including your own posts.

## SGITW (Slowest Gun in the West)

For the best 2016 answer in a challenge which received no answers within 48 hours of posting. The challenge itself does not need to be from 2016 and the nominated answer doesn't need to be the first answer to the challenge, but the first answer to the challenge does need to be from 2016. Tips questions are not counted for this category.

It's very easy to get drawn to the fast-rising HNQ-hitting questions with a plethora of answers, so I thought this would be a good way to bring attention to answers that really needed to earn their love.

## Solve Grid-Tangram by Level River St

Nominated by flawr

I think we can agree that the challenge itself was not easy, however Level River St answered more than two weeks after the challenge has been posted. The answer itself includes a great explanation on how the program works. So far it still is the only answer to this challenge.

## “KNOT” or “NOT”? by Anders Kaseorg

Nominated by xnor

Anders Kaseorg solves a devilishly difficult challenge that remained without a valid answer for nearly 2 years. And then makes 20 edits to golf the code and improve the explanation, despite being the only answer. The result is 306 bytes of meticulously golfed Python for a problem where any solution is impressive.

## Unique Sudoku solver by Fatalize

Self-nomination

This answer was provided nearly 5 years after the challenge was posted. In an effort to reduce the number of unanswered challenges, I decided to tackle this one with the best tool for the job: a declarative logic language. This results in a fairly short answer to a seemingly difficult problem, where a lot of the code is here to conform with the tedious input format. This challenge still does not have any other answer as of yet.

## Given a table, place in the chairs - by WheatWizard

Nominated by DJMcMayhem

This was a very difficult challenge that remained unanswered for 15 months before Wheat Wizard finally answered it. And it remains the only answer. If that's not impressive enough, consider that this answer got golfed down by Fifty-three percent with 36 edits over a 5-month period. That is some serious golfing dedication there, for a very impressive answer.

• – Dennis Jan 16 '17 at 19:18
• – Dennis Jan 16 '17 at 19:19
• – Dennis Jan 16 '17 at 19:19
• – Dennis Jan 16 '17 at 19:19

## Kansas City Shuffle

or Best mathematical insight

On this site we often see answers in languages specifically designed for short code, or designed to be fast. Sometimes, a nice golfing trick or speed-up technique surprises us with its ingenuity, beyond the standard use of that language.

And occasionally an answer shows up that uses an unexpected approach to greatly simplify the problem, and makes us wonder how the author could ever think of that. This usually involves some far-from-obvious mathematical equivalence, or a particularly simple approach to the problem that was not evident at all (once revealed, other answers often follow the same approach).

This category is for the answer with the best mathematical insight or unexpected approach that led to greatly simplifying the problem, in any challenge type (code golf, fastest code, or others). The insight should have led to a significant improvement according to the challenge's metric (code length, run time, or whatever applicable).

Nominations:

## Difference of three input integers by xnor

Nominated by TimmyD

An easy enough challenge (that xnor even helped tighten up before answering), but practically every answer posted says something like "Uses xnor's algorithm". The algorithm itself is a simple conclusion of the challenge setup, but approaches it from an unintuitive angle, resulting in an elegant approach that many others utilized.

## Calculate a string's cumulative slope by Dennis

Nominated by wat

Dennis simplified this whole challenge by showing that the numbers form a telescopic series, making answers much easier than completing the original problem as intended.

## Is it a Proth number? by Dennis

Nominated by DJMcMayhem

Is it a Proth number is one of my favorite challenges I've posted. The premise is very simple:

Can a number N be expressed as

N = k * 2^n + 1

Where k is an odd positive integer and n is a positive integer such that 2^n > k

Which is a fairly straightforward thing to test for. Dennis flipped this on it's head, and threw in a bunch of bit-shifting magic, and almost an entire paper of mathematical analysis to prove it worked. I remember watching answers pouring in after his, all claiming

Uses Dennis's algorithm

## Goodness Giza Golf! by Gabriel Benamy

Self-nomination

This answer tackles the decision problem "Is this number a Giza number?" by observing a beautiful mathematical property of all Giza numbers, and applying this insight into the solution. Instead of analyzing the problem purely from a programmatic standpoint, the code, and the explanation, highlights the relationship between Giza numbers and repdigits, and takes full advantage of it in the code.

## Absolute Sums of Sidi Polynomial Coefficients by Dennis

Self-nomination

It's quite common to reduce a "real-world" problem to a single mathematical formula, but this answer expands a mathematical formula to an elaborate real-world problem. By doing so, a simple pattern becomes apparent that would be hard to detect in the mathematical definition of the sequence. This allows to replace the combinations in the original formula (which would require a lengthy implementation in Python and other languages that lack the built-in) with a doubly-recursive function that uses only elementary operations.

## Coprimes up to N by Dennis

Self-nomination

While determining coprimality is straightforward in programming languages with a GCD built-in and computing the GCD with the Euclidean algorithm is also quite easy, filtering all coprimes from a range requires two function definitions, which take a toll on the byte count. This answer does something different: it uses multiplicative inverses modulo n to detect coprimes. Not only makes this computing the GCD unnecessary, it manages to solve the problem with a simple loop. The algorithm from the original Python was successfully ported to JavaScript, which also lacks a GCD built-in.

## Add and multiply perplexing numbers by xnor

Self-nomination

This is an insight I'm proud of that didn't get much visibility.

The problem is to evaluate an arithmetic expression in a variant of the complex numbers where j^2 = 1. It would seem like you need to parse the expression and resolve how each operator acts on this new type of number.

Instead, I use the fact that both the numbers j=1 and j=-1 satisfy j^2 = 1. So, the resulting equality must hold for them as well. Evaluating the input expression as Python code on these values gives a system of two equations that can be solved for the output. This allows a 62-byte solution, far shorter than anything with parsing.

## Compute the Carmichael function by Dennis

Self-nomination

Computing the Carmichael function can be a straightforward loop or recursive function in languages that have coprimality or GCD built-ins, but some languages (e.g., Python) lack both. This answer defines a different mathematical function which definition uses only elementary arithmetic operations; determining coprimality or calculating GCDs is not required. The answer also contains a formal proof that this function is identical to the Carmichael function and provides a concise, recursive implementation.

## How many steps does it take from n to 1 by subtracting the greatest divisor? by Dennis

Self-nomination

Superficially, the problem at hand seems entirely unrelated to coprimality or Euler's totient function. Yet, this answers uses a unique approach based on the latter, which has a particularly concise implementation in the chosen language. While that approach saved only one byte over the straightforward solution, the approach is most unexpected and achieved a 10% reduction of the score.

• – Dennis Jan 16 '17 at 19:05
• – Dennis Jan 16 '17 at 19:05
• – Dennis Jan 16 '17 at 19:05
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• – Dennis Jan 16 '17 at 19:07
• – Dennis Jan 16 '17 at 19:07
• – Dennis Jan 16 '17 at 19:07

## Pro tip

Best answer to any general question. That is, candidates should be drawn from "Tips for golfing in X" questions (as opposed to specific "how do I shorten this piece of code" questions).

Some of the most useful content that we generate for other golfers is in our tips questions, so it would be nice to reward an exceptional golfing trick this way.

Candidates should be tips which show deep insight into the language, yet are applicable in a wide range of situations. The exact amounts of bytes saved by the tip is not relevant.

## Julia: Redefine operators to define functions by Dennis

self nomination

This tip discovered an undocumented way of re-defining Julia's operators, which is applicable to all function definitions with one non-optional argument or two (optional or non-optional) arguments. Not only does this alternate way of defining functions save bytes in most cases, the special status as operators obliterates the needs for parentheses in multiple or recursive calls as long as the operator precedence is correct. Since its discovery, this tip has been used in countless Julia answers. It also inspired the Operators as Functions Mathematica tip.

## Octave: Anonymous functions with more than one statement by rahnema1

Nominated by Luis Mendo

In this tip, @rahnema1 shows how to overcome the limitation that an anonymous in Octave can only contain one statement. This is done by exploiting the facts that an assignment can be done anywhere and indexing can be chained. It is a brilliant tip with many potential uses. One example, illustrated in the tip, is returning a function's output other than the first.

## Mathematica: Lists with repeated values by Martin Ender

Self-nomination

This is a seemingly simple trick which combines a couple of the more obscure features of Mathematica in order to shorten a wide variety of list literals with repeated values. It took me a couple of years of golfing in Mathematica to come up with it and really shows that Mathematica's golfing can be surprisingly subtle. The tip can be used a variety of situations as it applies to some fairly commonly needed lists but can also be applied to lists of arbitrary structure.

## Best Showcase of a New Language

This is for new languages (golfing or otherwise) created in or after December 2015 (meaning, the first commit on GitHub or the like was from that date). The category is designed to highlight the answer that best shows the tricks, features, and benefits of using this new language. The author of the answer does not necessarily need to have been the author of the language.

Answers from the Showcase of Languages or challenges are ineligible.

A not necessarily exhaustive list of eligible languages: MATL, Jelly, AnnieFlow, Pyke, IPOS, Stack Cats, V, Sesos, S.I.L.O.S, Jellyfish, Brain-Flak, Logicode, Cubix, 7, Turtlèd, Charcoal

Nominations:

## A keyboard so real you can almost TASTE it by DJMcMayhem - V

Nominated by Kritixi Lithos

This answer showcases many of the interesting features of this Vim-based language: ranges, recursive macros and shorter regex statements to name a few. Not only that, this answer wins the challenge by beating the second best answer at a whopping 30 34 bytes!

## Sylvester's Sequence by Leaky Nun - Brain-Flak

Nominated by Wheat Wizard and DJMcMayhem

According to its github Brain-Flak is a "A minimalist esolang designed to be painful to use" but in the last year it has gathered a user-base on PPCG. It now has over 80 answers written in it across several users. Nowhere is the elegance and surprising power of Brain-Flak better demonstrated in Leaky-Nun's Sylvester Sequence solution. Not only is it one of the best Brain-Flak golfs I have ever seen, it also demonstrates how Brain-Flak loops can be used to create polynomial expressions. This submission is surprisingly terse for a language as limited as Brain-Flak and takes full advantage of everything Brain-Flak has to offer.

## Draw an ASCII-O'-Lantern for Halloween by DLosc - Charcoal

Self-nomination

Charcoal is a language that specializes in ASCII art, designed by ASCII-only and me. This answer showcases its canvas-based approach on a moderately complicated problem, using many of the core features of the language (automatic line characters, ASCII literals, reflection, and overwriting previously drawn characters). It's not every day you get to outgolf Jelly, Pyth, and V on the same challenge!

## Add two numbers by Sp3000 - Stack Cats

Nominated by Martin Ender

Stack Cats (SKS) is a language that Sp3000 and I created this year, whose unique structure makes programming in it quite a challenge, even though the language is Turing-complete. At the same time it doesn't reach Malbolge levels of difficulty, so programming in it is actually possible, if tricky. "Add to numbers" is of course quite trivial, but it provides a surprising challenge for a language like SKS. So far, we haven't been able to figure out how to program in it without using the <(...)*(...)> template which skips half the code, circumventing SKS's main restriction. This answer showcases both such a (well-golfed) handwritten program, as well as several solutions which were found by automated search, which don't use the template and beat the handwritten one by two bytes. The answer also contains a short introduction to SKS as well as detailed explanations of both solutions which should make the answer accessible to people who are not familiar with the language.

## Numbers divisible by the sum and product of their digits by Martin Ender - Jellyfish

Self-nomination

Jellyfish, created by Zgarb, is my favourite language of the year. Its semantics are loosely inspired by J, in that the language is functional and you'll work a lot with higher-order functions, but it has a very unique two-dimensional syntax, where each function takes an argument from below and an argument from the right (based on this challenge), which lets you reuse arguments and makes the layout a very important component of golfing.

To my knowledge, this answer is the most elaborate Jellyfish program that was written so far and it shows off a lot of interesting features for constructing higher-order functions to solve complex problems and how the layout can be relevant. The answer also contains a fairly detailed explanation of how the program works.

## 1, 2, Fizz, 4, Buzz by Dennis - Jelly

self nomination

This answer combines several useful Jelly abilities that are uncommon in other languages: pre-defined variables (³), splitting a link into chains (µ), automatic vectorization (ḍ), advanced array-manipulation built-ins (Ṭ), and in-built dictionary-based compression (“¡Ṭ4“Ụp»). Not only does the answer illustrate these rather unique abilities, but it combines them into a 20-byte solution to this classical programming problem. For comparison, the next submission that does not use a FizzBuzz built-in requires 27 byte for the same task (GS2), followed by a 30-byte Pyth solution.

• – Dennis Jan 16 '17 at 19:02
• – Dennis Jan 16 '17 at 19:03
• – Dennis Jan 16 '17 at 19:03
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## Rookie of the year (answer)

Awarded for the best answer by a user who had not posted an answer before 2016. Note that the user may have posted challenges before 2016. SEDE query for such answers, ordered by score.

Nominations:

## Radiation Hardened Quine by ais523

Nominated by Martin Ender

ais523 started contributing in mid-November and has since posted 139 answers and 10 challenges as well as very valuable contributions on meta. The overall quality of their content is absolutely outstanding, and I'm having a really hard time picking a specific answer to nominate (this Cubix quine would also be a decent candidate). I've chosen the answer above because a third-order radiation hardened quine in a non-esoteric language is a very impressive achievement and the answer also comes with a great in-depth explanation.

## Golf you a quine for great good by udioica

Nominated by DJMcMayhem

Not a lot of people on this site golf in vim, so I used to think I was the king of vim-golf on this site. I wrote a 17 byte vim-quine and challenged anyone to outgolf me, thinking I was safe. Boy was I wrong.

Udioica noticed the challenge, and rose to it, golfing a whopping 6 bytes off of my solution on the first day they ever used PPCG.

## Build a digital clock in Wireworld by niemiro

Nominated by Peter Taylor

This is niemiro's only answer on PPCG so far, and claims to be their first Wireworld circuit too, but it is most definitely ambitious. The answer includes not only an animated GIF of the circuit in action but also a detailed breakdown of its construction.

## A rather knotty conundrum by ais523

Nominated by Peter Taylor

The only answer at present to a rather difficult question, and an excellent illustration of the importance of choosing the right language for the task.

## A Mouse with Dynamite by Ton Hospel

Ton Hospel is someone who needs no introduction. He is one of the best Perl golfer in the world, has been so for pretty much as long as Perl golfing is a thing, and his solutions are always innovative, unexpected and short. And as it happens, he only joined PPCG this year. His code to solve A mouse with Dynamite is a good example of his skills, and definitely one of the best golfings I've ever seen (and it comes with a very detailed explanation).

• – Dennis Jan 16 '17 at 19:21
• – Dennis Jan 16 '17 at 19:21
• – Dennis Jan 16 '17 at 19:22
• – Dennis Jan 16 '17 at 19:22
• – Dennis Jan 16 '17 at 19:22

## It's not a Bug, It's a Feature

Answer with the most clever use of a bug, error, or undocumented behavior.

# Palindromizing the strings (Pyth) by isaacg

Self nomination

In this answer, I exploited a bug in Pyth's I (invariant) operator, where if a program used an auto initializing variable (J in this program) inside a use of I, an extra token would be produced, and not consumed by the I operator, leaving it sitting around. This is caused by the fact that Pyth parses J differently the first and second times it appears in a program.

The use of I was within the condition of and if statement, and the extra token left sitting around ended up in the body of the if condition, allowing it to printed out. If you look at the code directly, it's extremely unintuitive: There's a loop, whose body is an if statement, and nothing else. The body of the if statement has a break (B), and nothing else. There's nothing to even get printed without the bug.

## Draw an asterisk triangle by udioica

Nominated by DJMcMayhem

This answer is a beautiful abuse of weird and undocumented vim behavior, using "autoindent" to glitch out and create a cascade of increasingly indented lines, which is pretty much exactly what the challenge was asking for. In all my years of using vim, it's something I never would have noticed otherwise.

## List all ordered partitions of n by Martin Ender

Nominated by Peter Taylor

As I commented at the time, my first thought when I saw the program was that it must be buggy because it would clearly produce an error. It was only after I added debugging that I realised that the error was being deliberately exploited to break the loop early.

## Write a program to elasticize strings by Adnan

Nominated by Dennis

In this answer, Adnan used the P atom to perform string multiplication. This is not supposed to be possible; the × atom is intended to multiply numbers, and P (which just reduces by ×) is intended to do the same. In doing so, they managed to create a type Jelly is not even supposed to have; Python's str is intended solely for characters, and Jelly's strings are merely (Python) lists of characters. While that may cause all sorts of problems, it manages to solve the task at hand in merely three bytes, giving the answer the needed edge to win over two other four-byte answers. It also inspired the Jelly tip Abuse string bugs.

• – Dennis Jan 16 '17 at 19:07
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• – Dennis Jan 16 '17 at 19:08
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## Not as simple as it looks

This award goes to a high quality challenge (not necessarily ) that is easy to understand and looks simple at first glance, but where the best solutions are actually quite difficult to find.

• The difficulty should be intrinsic to the problem, and not come from messy edge cases or strict formatting rules.
• The challenge should be fun to solve, and not impossibly hard. Unanswered challenges may not be good candidates.
• The challenge may allow for simple solutions, but a competitive solution should be very intricate and/or not obvious at all.

## Bake a slice of Pi by Copper

Nominated by TimmyD

A simple challenge, right? Not so fast. Turns out the compression is decidedly difficult, such that even getting a few bytes shorter than the boring "print string" solution is deemed a victory, with only 1/3rd of the answers approaching that threshold.

## Upgoat or Downgoat by Downgoat

Self-nomination

This is a challenge which started out as kind of a joke but turned out to be one of the most popular challenges in 2016, is a very, very simple challenge. To determine whether or not an image of a goat has been flipped. Various answers have tried all sorts of techniques, checking which side is darker, checking contrast in various parts, or simply detecting which direction is more "goaty". The challenge is a simple problem but is a tricky problem to properly solve with many, many different approaches.

## Can my 4-note music box play that song? by apsillers

Nominated by Martin Ender

Conceptually, this is a very simple problem that can be easily visualised. However, the structure of the problem doesn't map to any common data structures or algorithms which results in a wide variety of interesting approaches that also differ substantially for languages with different paradigms. There are 15 answers and almost as many different ideas.

• – Dennis Jan 16 '17 at 19:08
• – Dennis Jan 16 '17 at 19:08
• – Dennis Jan 16 '17 at 19:09

## Rookie of the year (challenge)

Awarded for the best challenge by a user who had not posted a challenge before 2016. Note that the user may have posted answers before 2016. SEDE query for such challenges, ordered by score.

## Futuristic Gun Battle by Frenzy Li

Nominated by isaacg

King of the Hill challenges are some of the hardest challenges to write, and require a great deal of programming to implement the controller as well as ongoing management of the system. Frenzy Li pulled off a great king of the hill challenge around a well-thought-out, fun game, simple enough to specify and complicated enough to be fun, and managed the submissions well, closed loopholes where needed, and generally made an excellent challenge that's been fun for lots of people.

## Surrounded Countries by Sisyphus

Nominated by Martin Ender

This was a great challenge by a user who hadn't posted anything else before (and unfortunately not a lot since). The problem is interesting, not too contrived, well explained, and the specification is tight and has a good number of test cases. It led to various interesting approaches in the answers. There isn't a lot more you could ask for in a challenge.

## 5 Favorite Letters by carusocomputing

Nominated by TimmyD

The perfect blend of challenge, allowing for many different approaches while simultaneously being narrow enough that each approach makes sense. The specifications are clear and unambiguous, and the added challenge of meta-golfing to determine which five letters to use adds a very unique twist.

## The letter A without A - TheBitByte

self-nomination

Display the first letter of the alphabet. You can't use the letter itself. Or it's unicode value. Or its HTML value. Sounds easy?

But it's gotten nice solutions. No plus signs for Brainfuck means you have to be creative. Or in Javascript, grab the second letter of "NaN" (among other ways), or literally grab the first letter of the alphabet, boring, but hey its less than 10 bytes!

## Add a language to a polyglot - ais523

Nominated by DJMcMayhem

This is a very fun challenge that perfectly blends two more obscure challenge types: and . The beautiful thing about this challenge is that it started out extremely easy, but with each new answer it gets significantly harder. Still, even after a month, this challenge continues to produce incredible answers, and many new users have joined the site to post an answer. This challenge has produced so much great content, it even prompted a meta discussion about whether we could repost this particular challenge in the future, so it's clearly well-loved by the community.

• – Dennis Jan 16 '17 at 19:09
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Every once and a while, an answer takes the challenge to the extreme. This prize will be awarded to an answer which went far beyond the expectations of the challenge. This could include

• a code golf answer that brute-forced/proved the shortest program in some language
• a graphical-output popcon answer of extreme size and quality
• a KOTH answer of high complexity which absolutely dominated the competition

## Patch the Image by mınxomaτ

nominated by flawr and TimmyD

The challenge itself was certainly not easy, however mınxomaτ implements an algorithm developed in 2003 and provides a very detailed explanation. His submission shows excellent results, and also includes real-world application of the techniques.

## Radiation Hardened Quine by Martin Ender

nominated by Kritixi Lithos

The first non-deleted answer was a 1-order radiation hardened quine. Then came an exceptional 3-order one. But this answer by Martin Ender goes beyond that past the skies into space with a 14-order radiation hardened quine. That's almost 5 times more than the previous best solution! Not just that, his answer has a very detailed explanation and how he came up with this solution.

## Calculate the number of primes up to n by Dennis

self nomination

This answer combines a lot of different techniques to implement the π function as fast as possible. Measured by the official scoring algorithm, it is at least one order of magnitude faster than the second-best answer; it would be several orders of magnitude accounting for start-up times. Achieving this took roughly 60 revisions and two hours of work. In the end, the answer managed to outperform not only its competitors, but also kimwalisch's primecount (GitHub). Some ideas from the answer were ultimately implemented in the GitHub repo.

## Golf you a quine for great good! -- by WheatWizard

Nominated by DJMcMayhem

For those of you who are unfamiliar with brain-flak, it's a maddeningly difficult turing-tarpit designed by me with help from Wheat Wizard and 1000000000. In a lot of ways, it's similar to brainfuck, although it tends to be longer since it has no explicit forms of IO. I didn't think a brain-flak quine was possible, but WheatWizard proved me wrong. The original version was too long to actually fit on a computer in the known universe, but had a python script to verify that the answer worked in theory, given infinite time and memory.

2 months later, it was golfed down to 12K bytes to fit into the observable universe. Over the month of december, it was further golfed down to its current state of 3855 bytes. This answer is the first brain-flak quine, and includes a novel approach of encoding large amounts of data in a single number, and then decoding it to print certain values. It's also the only answer I can think of that has been golfed by over 10*e576 bytes.

## Hello, World! by primo

nominated by Martin Ender

primo kicked off the year with one of the most impressive answers I've seen on this site. As far as we know, this answer contains several world records for various spelling variants of "Hello, World!", which are based on a form of recurrence relation that is easy to implement in Brainfuck, and which yields a lot of useful values to print the string at hand. primo subsequently spent several weeks crunching the numbers, bringing down the score by another 11 bytes. I find this extremely impressive, considering that Brainfuck is one of the oldest and most thoroughly studied esolangs out there. A year later, the world records still stand.

• – Dennis Jan 16 '17 at 19:10
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## Best Explanation

This category is for the answer with the best explanation accompanying it. Ideally, the winner will be an answer with a very detailed explanation that is accessible to anyone, regardless of the amount of relevant knowledge already possessed.

## Evaluate the aspect ratio of a triangle by Lynn

Nominated by Luis Mendo

The coloured diagrams and accompanying explanations make Jelly so much simpler to understand, and were totally unexpected by many of us. The comments to that answer reflect the impression caused by the graphical explanation.

(I could have nominated this one as well, but I preferred to nominate the one I did because I think it was the first to include those diagrams.)

## Radiation Hardened Quine by ais523

Nominated by TimmyD

Over 1kb of Perl code meticulously explained in very readable English. Each code choice is gone through in detail, explaining why certain elements were included, and what happens when those elements are removed by the radiation. Additionally, the explanation also includes a verification script (and explanation thereof!) so people can run the code themselves.

## Absolute Sums of Sidi Polynomial Coefficients by Dennis

Self-nomination

It's quite common to reduce a "real-world" problem to a single mathematical formula, but this answer expands a mathematical formula to an elaborate real-world problem. By doing so, a simple pattern becomes apparent that would be hard to detect in the mathematical definition of the sequence. Thus, this answer does not simply have an explanation, but the explanation is what made the answer possible in the first place!