Merge pull request #2 from bennn/day22

day22 solution

day22-input.txt

@@ -1,2 +1,2 @@
 Hit Points: 58
-Damage: 9
+Damage: 9

day22.rkt

@@ -3,16 +3,261 @@
 
 @aoc-title[22]
 
-
-@link["http://adventofcode.com/day/22"]{The puzzle}. Our @link-rp["day22-input.txt"]{input} tells us our hit points and damage.
+@link["http://adventofcode.com/day/22"]{The puzzle}. Once again, our @link-rp["day22-input.txt"]{input} tells us the boss's stats.
 
 @chunk[<day22>
-       ;;
-       ]       
+       <day22-player>
+       <day22-spells>
+       <day22-dp>
+       <day22-q1>
+       <day22-q2>
+       <day22-test>]
+
+@section{You're a Wizard, Henry}
+
+The rules of the game are different this time.
+Instead of items, attack power, and defense power, we have mana and spells.
+The boss still has a fixed attack power, so we'll use a 3-element @racket[struct]
+ to model the player and boss.
+
+@chunk[<day22-player>
+       (require racket rackunit)
+       (provide (all-defined-out))
+
+       (define BASE-HP 50)
+       (define BASE-MANA 500)
+
+       (struct player (hp attack mana) #:transparent)
+
+       (define (make-player)
+         (player BASE-HP 0 BASE-MANA))
+
+       (define (make-boss hp attack)
+         (player hp attack 0))
+
+       (define (hp+ p val)
+         (match-define (player hp attack mana) p)
+         (player (+ hp val) attack mana))
+
+       (define (hp- p val)
+         (hp+ p (- val)))
+
+       (define (mana+ p val)
+         (match-define (player hp attack mana) p)
+         (player hp attack (+ mana val)))
+
+       (define (mana- p val)
+         (mana+ p (- val)))
+
+]
+
+Next, a few constants and helper functions to model spells.
+The key functions are @racket[apply-spell], which represents the actions a player
+ can take during their turn, and @racket[apply-effect], which implements the
+ delayed action of the @racket[shield], @racket[poison], and @racket[recharge] spells.
+Both functions are parameterized over the player, boss, and current active
+ spells.
+These three values represent the game state at any moment.
+
+@chunk[<day22-spells>
+       (define MAGIC-MISSILE-DAMAGE 4)
+       (define DRAIN-DAMAGE 2)
+       (define SHIELD-ARMOR 7)
+       (define SHIELD-DURATION 6)
+       (define POISON-DAMAGE 3)
+       (define POISON-DURATION 6)
+       (define RECHARGE-MANA 101)
+       (define RECHARGE-DURATION 5)
+
+       (define ALL-SPELLS
+         '(magic-missile drain shield poison recharge))
+
+       (define (unknown-spell spell)
+         (raise-user-error 'day22 "Unknown spell '~a'" spell))
+
+       (define (spell-mana s)
+         (case s
+          [(magic-missile) 53]
+          [(drain)         73]
+          [(shield)        113]
+          [(poison)        173]
+          [(recharge)      229]
+          [else            (unknown-spell s)]))
+
+       (define (apply-spell spell player0 boss active-spells)
+         (define player (mana- player0 (spell-mana spell)))
+         (case spell
+          [(magic-missile)
+           (values player (hp- boss MAGIC-MISSILE-DAMAGE) active-spells)]
+          [(drain)
+           (values (hp+ player DRAIN-DAMAGE) (hp- boss DRAIN-DAMAGE) active-spells)]
+          [(shield)
+           (values player boss (cons (cons SHIELD-DURATION 'shield) active-spells))]
+          [(poison)
+           (values player boss (cons (cons POISON-DURATION 'poison) active-spells))]
+          [(recharge)
+           (values player boss (cons (cons RECHARGE-DURATION 'recharge) active-spells))]
+          [else
+           (unknown-spell spell)]))
+
+       (define (apply-effects player boss active-spells)
+         (for/fold ([p+ player]
+                    [b+ boss]
+                    [a+ '()])
+                   ([ctr+spell (in-list active-spells)])
+           (match-define (cons ctr spell) ctr+spell)
+           (define a++ (if (= 1 ctr) a+ (cons (cons (- ctr 1) spell) a+)))
+           (case spell
+            [(poison)
+             (values p+ (hp- b+ POISON-DAMAGE) a++)]
+            [(recharge)
+             (values (mana+ p+ RECHARGE-MANA) b+ a++)]
+            [(shield)
+             (values p+ b+ a++)]
+            [else
+             (unknown-spell spell)])))
+]
+
+@section{What's the least (mana) we can spend and win?}
+
+Starting with 50 health, 500 mana and a boss with @math{N} hit points and @math{A}
+ attack points, the least mana we can spend and win is either:
+@itemize[
+  @item{
+    The cost of one @emph{Magic Missile}, plus the least-mana solution for
+     a player with @math{50 - M} hit points, @math{500 - 53} mana,
+     and a boss with @math{N - 4} hit points.
+  }
+  @item{
+    The cost of one @emph{Drain} spell, plus the least-mana solution for
+     a player with @math{50 + 2} hit points, @math{500 - 73} mana, and
+     a boss with @math{N - 2} hit points.
+  }
+  @item{
+    The cost of one @emph{Shield} spell, plus the least-mana solution for:
+    @itemize[
+      @item{
+        a player with @math{50 - max(1, (M - 7))} hit points and @math{500 - 113} mana,
+      }
+      @item{
+        a boss with @math{N} hit points,
+      }
+      @item{
+        the @emph{Shield} spell active for 4 more turns.
+      }
+    ]
+  }
+  @item{
+    The cost of one @emph{Poison} spell, plus the least-mana solution for
+     a player with @math{50 - 7} hit points and @math{500 - 173} mana and
+     a boss with @math{N - 6} hit points, given that the @emph{Poison} spell
+     is active for 4 more turns.
+  }
+  @item{
+    The cost of one @emph{Recharge} spell, plus the least-mana solution for
+    a player with @math{50 - 7} hit points and @math{500 - 229 + (101 * 2)} mana
+    against a boss with @math{N} hit points, given that the @emph{Recharge} spell
+    is active for 3 more turns.
+  }
+]
+The correct alternative is the one that happens to be cheapest.
+Since we've outlined the alternatives and know the end conditions for the game,
+ we can write an algorithm that tries every alternative and returns the best
+ sequence of spells.
+
+Never mind the @racket[hard-mode?] parameter for now.
+
+@chunk[<day22-dp>
+       (define (win/least-mana player boss #:hard-mode? [hard-mode? #f])
+         (or
+           (let maybe-win/least-mana ([player player]
+                                      [boss boss]
+                                      [active-spells '()]
+                                      [current-turn 0])
+             (cond
+              [(lose? player boss)
+               #f]
+              [(win? player boss)
+               '()]
+              [else
+               (define-values (p+ b+ a+) (apply-effects player boss active-spells))
+               (define next-turn (+ 1 current-turn))
+               (if (even? current-turn)
+                 (let ([p+ (if hard-mode? (hp- p+ 1) p+)])
+                   (minimize-mana
+                     (for/list ([spell (in-list ALL-SPELLS)]
+                                #:when (and (not (active? spell a+))
+                                            (has-enough-mana? p+ spell)))
+                       (define-values (p++ b++ a++) (apply-spell spell p+ b+ a+))
+                       (define future-spells (maybe-win/least-mana p++ b++ a++ next-turn))
+                       (and future-spells (cons spell future-spells)))))
+                 (maybe-win/least-mana (boss-attack p+ b+ (active? 'shield a+)) b+ a+ next-turn))]))
+             (raise-user-error 'day22 "Impossible for ~a to beat ~a. Sorry.\n" player boss)))
+
+       (define (win? player boss)
+         (dead? boss))
+
+       (define (dead? player)
+         (<= (player-hp player) 0))
+
+       (define lose?
+         (let ([MIN-MANA (apply min (map spell-mana ALL-SPELLS))])
+           (λ (player boss)
+             (or (dead? player)
+                 (< (player-mana player) MIN-MANA)))))
+
+       (define (boss-attack player boss shield?)
+         (define boss-damage (max 1 (- (player-attack boss) (if shield? SHIELD-ARMOR 0))))
+         (hp- player boss-damage))
+
+       (define (active? spell active-spells)
+         (for/or ([ctr+spell (in-list active-spells)])
+           (eq? spell (cdr ctr+spell))))
+
+       (define (has-enough-mana? player spell)
+         (<= (spell-mana spell) (player-mana player)))
+
+       (define (spell*-mana spell*)
+         (for/sum ([spell (in-list spell*)])
+           (spell-mana spell)))
+
+       (define (minimize-mana spell**)
+         (for/fold ([best #f])
+                   ([other (in-list spell**)])
+           (if (or (not best)
+                   (and best other (< (spell*-mana other) (spell*-mana best))))
+             other
+             best)))
+]
+
+@chunk[<day22-q1>
+       (define (q1 input-str)
+         (define player (make-player))
+         (define boss (apply make-boss (filter integer? (map string->number (string-split input-str)))))
+         (spell*-mana (win/least-mana player boss)))
+]
+
+@section{Hard Mode}
+
+On hard mode, the player loses one hit point on their turn.
+Thanks to the optional keyword argument to @racket[win/least-mana], the answer
+ is easy find.
 
-@section{What's the least mana we can spend and win?}
+@chunk[<day22-q2>
+       (define (q2 input-str)
+         (define player (make-player))
+         (define boss (apply make-boss (filter integer? (map string->number (string-split input-str)))))
+         (spell*-mana (win/least-mana player boss #:hard-mode? #t)))]
 
-Did I mention I hate RPGs? Well, this question is much more RPG-ish than @secref{Day_21}. It involves mana and spellscasting. So I'm taking a pass. If someone devises a concise solution, I'll be happy to add it to this page.
+@section{Testing Day 22}
 
-Otherwise, onward to @secref{Day_23}.
+The algorithm as written is very slow.
+Unfortunately, there's no better way to solve this problem than trying everything,
+ but while the tests are running, think about what kind of caching strategy could
+ improve performance.
 
+@chunk[<day22-test>
+       (module+ test
+         (define input-str (file->string "day22-input.txt"))
+         (check-equal? (q1 input-str) 1269)
+         (check-equal? (q2 input-str) 1309))]