Showing posts with label fantasy. Show all posts
Showing posts with label fantasy. Show all posts

Tuesday, September 5, 2017

Your Random May Vary: Addendum

My coworker and good friend Tim has provided a bunch of good information in response to my last post. I'm posting it here (mostly) unedited, because it sheds light on the underlying principles.

"sir, you have posted something false to the internet and it is my duty to berate you for it!

 combining multiple random numbers is not the same thing as choosing a random number between bigger values.

 it flys in the face of one of the most beautiful underlying principles of statistics (possibly the only beautiful thing about statistics):

 If you add together many random numbers, you get a NORMAL DISTRIBUTION!
Isn't that interesting... I wonder why the normal distribution pops up so often? Oh that's right because this sort of thing happens all the time. Think about it for a bit; if you flip a coin 6 times, what is the probability of getting all heads? Is it the same as rolling a 6? Nope... you have a might higher probability of getting 3 heads than getting 6 heads. You might not notice it much for six goes, but if you flip a coint 18 times, you are very unlikely to get 18 in a row. It's because the distribution of outcomes is very different, not because one is more or less random than the other.

 d&d is built on this principle; creating pseudo normalized random distributions by combining dice rolls.

 If you want a better explination of this (and the 3 principles of statistics) I highly recommend reading the short chapter on statistic from "The Art of Game Design: A Book of Lenses" -- Jesse Schell. You can get it from the local library for free and the chapter is really concise and illuminating. The whole book is excellent actually."

 Many thanks to Tim for the info!
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Monday, September 4, 2017

Your Random May Vary


At the beginning of the summer, we started hosting a semi-monthly Dungeons and Dragons game at our house. It's typically a lot of fun, involving much role-playing and dice rolling. We share Dungeon-Mastering duties; I think I've run most of the sessions, but at least two or three have been run by someone else. When I'm running the game, my character is typically an NPC. It's a bit of a shame, because my stats are pretty awesome.

I got to thinking this morning... I rolled my current character (as all my characters), using real dice and a "roll-4-keep-3" approach. I roll four 6-sided dice, discarding the lowest, and summing the other three. I do that six times in a row, and I've got the stats block for a new character.

I never use anything but real dice unless I happen not to have a set (and fyi: I carry a couple of sets around in my backpack just in case a spontaneous or random one-shot happens). Garrett used to roll great stats very often using his IPhone dice, and tended to roll challenges really highly as well. I started thinking about the difference in our results and decided to mess around with a rolling app this morning.

It's been a while since I posted anything technical. Bear with me.

I started by creating a little app that would roll a six-sided dice for me.

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(ns stats-generator.main
  (:gen-class))

(def d6 [1 2 3 4 5 6])

(defn roll-d6 []
  (rand-nth d6))

(defn -main
  "I don't do a whole lot ... yet."
  [& args]
  (println "Hello, World!"))

I quickly realized that I didn't quite know where to go. TDD to the rescue!

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(ns stats-generator.main-test
  (:require [clojure.test :refer :all]
            [stats-generator.main :as main]))

(deftest test-roll-4-keep-3
  (testing "Given a roll of 4 6-sided dice, sums the highest 3"))

I started with a description of one of the end goals: a function that would roll four dice and keep the highest three. That... is probably too broad. Let's decompose for a moment. We know we'll need to roll four 6-sided dice and drop the lowest roll. Maybe that dropping function is easier to test and implement.

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(deftest test-drop-lowest
  (testing "Given all different values, drops the lowest"
    (let [v [1 2 3 4]
          expected [2 3 4]]
      (is (= expected (main/drop-lowest v))))))

There's the test; we should stub a red (failing) function.

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(defn drop-lowest
  "Given a vector, returns a vector sans its lowest value."
  [v]
  v)

Done! Run the tests!

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(run-tests)

Testing stats-generator.main-test

FAIL in (test-drop-lowest) (main_test.clj:9)
Given all different values, drops the lowest
expected: [2 3 4]
  actual: [1 2 3 4]
    diff: - [2 3 4]
          + [1 2 3 4]

Ran 2 tests containing 1 assertions.
1 failures, 0 errors.
=> {:test 2, :pass 0, :fail 1, :error 0, :type :summary}

Failed as expected. What's a simple solution that will work?

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(defn drop-lowest
  "Given a vector, returns a vector sans its lowest value."
  [v]
  (rest (sort v)))

Seems reasonable. Tests tell us...?

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(run-tests)

Testing stats-generator.main-test

Ran 2 tests containing 1 assertions.
0 failures, 0 errors.
=> {:test 2, :pass 1, :fail 0, :error 0, :type :summary}

Yay! Green tests! Let's fill in some other assertions to make sure we're not fooling ourselves.

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(deftest test-drop-lowest
  (testing "Given all different values in random order, drops the lowest"
    (is (= [2 3 4] (main/drop-lowest [1 2 3 4])))
    (is (= [2 3 4] (main/drop-lowest [4 2 3 1]))))
  (testing "Given identical values, drops one of them"
    (is (= [1 1 1] (main/drop-lowest [1 1 1 1]))))
  (testing "Given some duplicated values, drops whatever's lowest"
    (is (= [2 2 3] (main/drop-lowest [2 3 2 2])))
    (is (= [5 6 6] (main/drop-lowest [5 6 4 6])))))

That's good enough. A question I'm often asked is "how much automated test coverage should I implement?" The pedantic answer is "100%," but the pragmatic answer is "as much as you can that covers stuff you would do by hand anyway." That's my rule-of-thumb, in any case. Sometimes I do more, sometimes less. But if I find bugs, I try to reproduce them with tests and then they're covered into perpetuity. As such the test coverage grows organically.

Alright, let's fill in some more of the decomposed functions and see where we wind up.

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(ns stats-generator.main
  (:gen-class))

(def d6 [1 2 3 4 5 6])

(defn roll-d6 [_]
  (rand-nth d6))

(defn roll-4
  "Rolls a d6 4 times"
  []
  (map roll-d6 (range 4)))

(defn drop-lowest
  "Given a vector, returns a vector sans its lowest value."
  [v]
  (rest (sort v)))

(defn sum [v]
  (apply + v))

(defn roll-4-keep-3
  "Rolls a 6-sided die 4 times, summing the highest 3 values"
  []
  (-> (roll-4)
      (drop-lowest)
      (sum)))

(defn -main
  "I don't do a whole lot ... yet."
  [& args]
  (println "Hello, World!"))

That seems reasonable. Test coverage is minimal, but enough for me to know that I'm not doing something overtly silly.

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(ns stats-generator.main-test
  (:require [clojure.test :refer :all]
            [stats-generator.main :as main]))

(deftest test-roll-4
  (let [roll (main/roll-4)]
    (is (= 4 (count roll)))
    (is (every? #(<= 1 % 6) roll))))

(deftest test-drop-lowest
  (testing "Given all different values in random order, drops the lowest"
    (is (= [2 3 4] (main/drop-lowest [1 2 3 4])))
    (is (= [2 3 4] (main/drop-lowest [4 2 3 1]))))
  (testing "Given identical values, drops one of them"
    (is (= [1 1 1] (main/drop-lowest [1 1 1 1]))))
  (testing "Given some duplicated values, drops whatever's lowest"
    (is (= [2 2 3] (main/drop-lowest [2 3 2 2])))
    (is (= [5 6 6] (main/drop-lowest [5 6 4 6])))))

(deftest test-sum
  (is (= 18 (main/sum [6 6 6])))
  (is (= 3 (main/sum [1 1 1])))
  (is (= 12 (main/sum [3 4 5]))))

(deftest test-roll-4-keep-3
  (testing "Given a roll of 4 6-sided dice, sums the highest 3"
    (is (<= 3 (main/roll-4-keep-3) 18))))

At this point, I'm ready to do the REAL implementation. We'll set up a little recursion to see if we can generate a set of six 18s. I'm not good at math, but the likelihood seems pretty low that I'll actually do it. Let's compromise and try 100 MILLION times. We'll also squawk if we get close.

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(defn roll-all-18s []
 (let [counter (atom 0)]
   (println "STARTED AT:" (now))
    (loop [current-roll []]
      (swap! counter inc)
      (cond
        (= 100000000 @counter)
        (do
          (println "Gave up after ONE HUNDRED MILLION sets...")
          (println "ENDED AT:" (now)))

        (= [18 18 18 18 18 18] current-roll)
        (do
          (println "All 18s took" @counter "attempt(s).")
          (println "ENDED AT:" (now)))

        :default
        (do
          (when (<= 5 (count (filter #(= 18 %) current-roll)))
            (println "close call!" current-roll))
          (recur (roll-stats)))))))

Alrighty! We can run this in the REPL. The first try's results follow.

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(roll-all-18s)
STARTED AT: #inst "2017-09-04T17:26:34.981-00:00"
close call! (5 18 18 18 18 18)
Gave up after ONE HUNDRED MILLION sets...
ENDED AT: #inst "2017-09-04T17:44:07.983-00:00"

It took us 22 minutes to try 100 MILLION times, and in all those tries, we only got close once. The frequency lines up with my (admittedly bad-at-math-and-fuzzy) expectations.

So how in the world did Garrett's IPhone roll so well for him so often? I suspect they weren't using the same approach as I was (roll-4-keep-3). After a little more research, I found that another approach that simply took a random integer between 3 and 18. This seemed like a pretty simple and probably more efficient implementation, but I suspected that the level of randomness in that picking was LOWER than that of the roll-4-take-3 approach. Again, I don't have the math or computer science to tell you why -- it was just my instinct.

So, I implemented a couple more rolling functions...

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(def range-3-18 (range 3 19))

(defn roll-d3-18 [_]
  (rand-nth range-3-18))

...and refactored the roll-stats function into a multi-method that dispatches on the roll style you wanted to use.

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(defmulti roll-stats identity)

(defmethod roll-stats :roll-4-keep-3
  [_]
  (map roll-4-keep-3 (range 6)))

(defmethod roll-stats :range-3-to-18
  [_]
  (map roll-d3-18 (range 6)))

No tests were added. I wanted to get to the bottom of this silly thing. Let's do some more REPLing!

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(roll-all-18s :range-3-to-18)
STARTED AT: #inst "2017-09-04T21:56:32.650-00:00"
close call! (18 12 18 18 18 18)
close call! (18 18 17 18 18 18)
close call! (18 18 18 18 18 10)
close call! (18 18 18 18 17 18)
close call! (18 18 18 18 12 18)
close call! (18 18 18 18 18 7)
close call! (18 18 18 9 18 18)
close call! (18 18 18 17 18 18)
close call! (18 18 18 18 8 18)
close call! (18 13 18 18 18 18)
close call! (18 9 18 18 18 18)
close call! (18 18 14 18 18 18)
close call! (18 18 18 4 18 18)
close call! (18 18 18 18 12 18)
close call! (18 18 18 18 10 18)
close call! (18 18 18 18 18 3)
close call! (18 3 18 18 18 18)
close call! (18 18 18 18 10 18)
close call! (18 18 10 18 18 18)
close call! (18 18 18 18 18 3)
close call! (18 18 18 18 15 18)
close call! (18 18 15 18 18 18)
close call! (18 18 18 18 12 18)
close call! (18 18 18 11 18 18)
close call! (3 18 18 18 18 18)
close call! (18 18 15 18 18 18)
close call! (18 18 18 9 18 18)
close call! (18 18 18 18 18 7)
close call! (17 18 18 18 18 18)
close call! (18 18 18 18 15 18)
close call! (18 18 18 18 11 18)
close call! (18 17 18 18 18 18)
close call! (18 4 18 18 18 18)
close call! (18 18 16 18 18 18)
close call! (18 18 18 8 18 18)
close call! (18 18 18 18 9 18)
close call! (18 18 15 18 18 18)
close call! (9 18 18 18 18 18)
close call! (18 18 18 18 7 18)
close call! (18 11 18 18 18 18)
close call! (18 18 18 6 18 18)
close call! (18 18 18 18 8 18)
close call! (18 18 18 15 18 18)
close call! (18 18 18 18 18 9)
close call! (18 18 9 18 18 18)
close call! (18 18 18 18 4 18)
close call! (18 18 18 18 11 18)
close call! (7 18 18 18 18 18)
close call! (18 18 18 18 18 3)
close call! (18 11 18 18 18 18)
close call! (18 18 12 18 18 18)
close call! (18 18 18 5 18 18)
close call! (18 18 18 18 18 6)
close call! (18 17 18 18 18 18)
close call! (18 6 18 18 18 18)
close call! (18 18 18 9 18 18)
close call! (18 18 18 11 18 18)
close call! (18 11 18 18 18 18)
close call! (18 16 18 18 18 18)
close call! (14 18 18 18 18 18)
close call! (18 18 10 18 18 18)
close call! (18 18 18 18 18 12)
close call! (18 18 18 18 18 16)
close call! (18 18 18 4 18 18)
close call! (18 18 18 18 12 18)
close call! (18 11 18 18 18 18)
close call! (18 18 18 18 16 18)
close call! (18 18 5 18 18 18)
close call! (18 18 18 18 18 16)
close call! (18 18 18 17 18 18)
close call! (5 18 18 18 18 18)
close call! (18 18 18 11 18 18)
close call! (18 18 18 18 18 12)
close call! (18 18 18 18 18 7)
close call! (18 18 18 3 18 18)
close call! (18 18 18 18 8 18)
close call! (18 18 18 5 18 18)
close call! (15 18 18 18 18 18)
close call! (18 5 18 18 18 18)
close call! (18 18 18 16 18 18)
close call! (18 18 18 18 4 18)
close call! (18 18 17 18 18 18)
close call! (18 14 18 18 18 18)
close call! (18 18 18 18 18 7)
close call! (18 18 18 4 18 18)
close call! (18 18 18 18 4 18)
All 18s took 13880334 attempt(s).
ENDED AT: #inst "2017-09-04T21:57:12.554-00:00"

WHOA! There were a TON more close calls, and an actual hit about 14 milllion rolls in.

I decided to re-run the roll-4-take-3 strategy to verify its performance.

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(roll-all-18s :roll-4-keep-3)
STARTED AT: #inst "2017-09-04T22:51:37.468-00:00"
close call! (18 18 18 18 14 18)
close call! (13 18 18 18 18 18)
close call! (18 18 18 18 18 9)
Gave up after ONE HUNDRED MILLION sets...
ENDED AT: #inst "2017-09-04T23:09:02.571-00:00"

A few more close calls were had, but roughly the same result. I re-ran the range-3-to-18 strategy:

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(roll-all-18s :range-3-to-18)
STARTED AT: #inst "2017-09-04T23:24:08.765-00:00"
close call! (18 18 8 18 18 18)
close call! (18 18 18 5 18 18)
All 18s took 321349 attempt(s).
ENDED AT: #inst "2017-09-04T23:24:09.925-00:00"

Wowzers. SO MUCH MORE LIKELY to roll 18s using that approach. The second run only took a second and slightly more than 321,000 tries.

So, what's the moral of the story? I think it's that the amount of randomness you get from computers may vary, based on what the underlying strategy is. Random numbers aren't truly random in the machine. Always keep that in mind when they rise up and become our overlords.

Also: rolling 321,000 times by hand, at one set of rolls per second, would still take you almost 4 straight days of rolling to get that all 18s set. So for you DMs out there... if a player comes to you with a character they said they rolled, and it's got six 18s, you give them a knowing wink while handing them your dice and telling them "roll again..."

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Friday, January 13, 2017

Book Review: "The Heart of What Was Lost" by Tad Williams


It's been more than twenty years since I've been to Osten Ard, so I wasn't sure how I'd feel stepping back into the world that turned fantasy literature on its head for me. Up until that point, contemporary fantasy for me flowed from the wellsprings of people with names like Weis and Hickman, Feist, and Brooks. Then Tad Williams shoved me straight through a darkened doorway into a world that was darker, more menacing, grittier.

I discovered that I loved it. It resonated with a part of me that I didn't pay much attention to at that point -- the truly human part of me. It was the part of me that realized that things aren't as nice and neat and crisply defined as would make the world an easy and understandable place to be. And for that reason, it felt more real to me than most other fantasy worlds in which I'd immersed myself.

Given all of that, I was slightly trepidatious about re-entering that world. Would I still feel the same? Would it bind me the way it had all those years ago?

Yes. Oh yes. A hundred times yes!

"The Heart of What Was Lost" is a short novel that serves as an epilogue to "To Green Angel Tower" and a prologue(?) to Tad Williams' series (in progress) "The Last King of Osten Ard." It tells the story of the disorganized retreat of the Norn to their northern stronghold and their pursuit by Rimmersmen forces under the command of Duke Isgrimnur. The Duke is accompanied by a Sithi, as enigmatic as ever, and in his ranks are two soldiers from the south that didn't expect to be there at all.

We're treated to three main viewpoints: Duke Isgrimnur, Porto (one of the southern soldiers), and Viyeki, a Norn Builder (similar to a civil engineer). I found the Norn viewpoint particularly compelling, as I was originally drawn to the enigmatic nature of the Sithi, and we are reminded about how closely related the Sithi and Norn really are via his story.

The book is remarkably thoughtful and considerate of the reader as well. I've always loved when an author has the ability to make art from both the story and wordsmithing. Tad Williams consistently does this, making it a pleasure to actually read the sentences as well as follow the story. Secondly, he knows that it's been a long time since many readers have read "Of Memory, Sorrow, and Thorn," so he's includes a short essay labeld "An Explanation" in the appendices and points the reader to it in the author's notes at the beginning of the book. The essay serves to warm you up and reintroduce you to some of the events and concepts from Osten Ard. I found this to be well crafted and immensely helpful.

Finally, I said it was short. I meant it. And yes, I'm as surprised as you are about that. The whole thing, including appendices is 210 pages. This is the shortest Tad Williams book I've read, by far. To be fair, I'm just getting into the Bobby Dollar books, but even those are 400 pagers in paperback. This book was just the right size for the start of the year.

Thank you, Tad, for letting us back into this world. I'm looking forward to at least one more long adventure there!
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Saturday, March 12, 2016

Book review: "Mistborn" (original trilogy) by Brandon Sanderson

This is actually a general review of the entire Mistborn original trilogy, as I've read them over the past several months with the boys and on my own.  We started reading them together in the car via Audible, but Garrett couldn't wait for Gabriel and I and finished listening to the entire series on his own before we'd even finished the first book as a group.  Eventually, Gabriel said he was going to pause listening to them, leaving me to finish them on my own.

I also have the paperback collection, shown here.  I finished the series by both reading and listening to them on Audible.

This review contains minor spoilers at the very least.  Read on at your own peril!

Encapsulate the plot of this book in one sentence.

A group of thieves plot to overthrow the rule of an immortal emperor/god, deal with the repercussions of all actions to that end, and discover the full history, responsibilities, and sources of the mystical powers that both hold the world together and rip it asunder.

When and where did you get this book?

I originally got the first Mistborn book in November, 2014.  I subsequently bought the paperback trilogy set from Barnes and Noble in the summer of 2015.

What year or edition?

First editions, from audible.com and in paperback.

Did you finish it?

Yes, although it took me longer than expected... :-)

What's your verdict?

Fantastic series all around.  Brandon Sanderson is a really interesting world builder, and his magic systems are really well thought ought.  His characters are relatable, and this series features a really strong female lead.

The paperback versions have held up well under normal usage, although the paper and cover are a bit thin.  Considering the books are ~700 pages each, it doesn't seem like there's a lot that can be done about that.

The Audible series is narrated by Michael Kramer, who has become one of my favorite readers.  He found unique voices for the characters, and there are lots of them.  The only knock on him was that his voice for Elend (one of the main male characters) seemed to change between books 2 and 3.  There's quite a bit of character development between those two books, so it's certainly possible.  However, reading the words on the pages, I didn't make that change in my head.

Overall, both content and performance, 4.5 out of 5 stars!

What surprises did it hold, if any?

Without giving too much away, the first book had a really interesting and somewhat unexpected plot twist near the end.  That kind of thing continues to happen throughout the series, although never as dramatically as in the first book.

The other surprise has to do with the magic system.  You figure everything out about the races, the settings, etc, but a surprise about the magic system is revealed near the end of book 3 and is never resolved.  Apparently, Brandon Sanderson decided to play that one REALLY close to the vest... ;-)

Which scenes will stay with you?

There are lots of pretty memorable scenes.  The first time we see a mistwraith, the first ball, seeing the Lord Ruler (book 1), the assault on Cett and the attack on Luthadel (book 2), the rush to the end (book 3).  Describing these in any more detail would give more away than I want to.

Which characters will stay with you?

Vin, Vin, Vin, and Vin.  She's astounding.  Powerful, tough, vulnerable, delicate, complex, blunt -- like I said earlier, a fantastic lead.

Elend and Sazed to lesser extents, and Kelsier (of course).  Spook and Breeze as well.  And I suspect I'll think of TenSoon every time I see a big black wolf-ish canine for the near future at least.

What genre would you say it is?

Fantasy all the way.

Have you read anything else by this author?

Yes!  The first two of The Stormlight Archive and The Rithmatist. I've also got Elantris and Warbreaker on the shelf.  (and maybe Steelheart?  I'll have to check...)

Is it available today?

Yes, and you should go get it and read it, especially if you like fantasy.
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