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#lang scribble/lp2
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@(require scribble/manual aoc-racket/helper)
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@aoc-title[22]
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@defmodule[aoc-racket/day22]
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@italic{This solution contributed by guest star @link["https://github.com/bennn"]{Benjamin Greenman}, because I had no patience for this puzzle. — MB}
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@link["http://adventofcode.com/day/22"]{The puzzle}. Once again, our @link-rp["day22-input.txt"]{input} tells us the boss's stats.
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@chunk[<day22>
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<day22-player>
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<day22-spells>
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<day22-dp>
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<day22-q1>
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<day22-q2>
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<day22-test>]
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@section{You're a Wizard, Henry}
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The rules of the game are different this time.
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Instead of items, attack power, and defense power, we have mana and spells.
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The boss still has a fixed attack power, so we'll use a 3-element @racket[struct]
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to model the player and boss.
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@chunk[<day22-player>
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(require racket rackunit)
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(provide (all-defined-out))
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(define BASE-HP 50)
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(define BASE-MANA 500)
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(struct player (hp attack mana) #:transparent)
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(define (make-player)
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(player BASE-HP 0 BASE-MANA))
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(define (make-boss hp attack)
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(player hp attack 0))
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(define (hp+ p val)
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(match-define (player hp attack mana) p)
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(player (+ hp val) attack mana))
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(define (hp- p val)
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(hp+ p (- val)))
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(define (mana+ p val)
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(match-define (player hp attack mana) p)
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(player hp attack (+ mana val)))
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(define (mana- p val)
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(mana+ p (- val)))
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]
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Next, a few constants and helper functions to model spells.
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The key functions are @racket[apply-spell], which represents the actions a player
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can take during their turn, and @racket[apply-effect], which implements the
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delayed action of the @racket[shield], @racket[poison], and @racket[recharge] spells.
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Both functions are parameterized over the player, boss, and current active
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spells.
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These three values represent the game state at any moment.
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@chunk[<day22-spells>
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(define MAGIC-MISSILE-DAMAGE 4)
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(define DRAIN-DAMAGE 2)
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(define SHIELD-ARMOR 7)
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(define SHIELD-DURATION 6)
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(define POISON-DAMAGE 3)
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(define POISON-DURATION 6)
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(define RECHARGE-MANA 101)
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(define RECHARGE-DURATION 5)
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(define ALL-SPELLS
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'(magic-missile drain shield poison recharge))
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(define (unknown-spell spell)
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(raise-user-error 'day22 "Unknown spell '~a'" spell))
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(define (spell-mana s)
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(case s
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[(magic-missile) 53]
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[(drain) 73]
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[(shield) 113]
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[(poison) 173]
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[(recharge) 229]
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[else (unknown-spell s)]))
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(define (apply-spell spell player0 boss active-spells)
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(define player (mana- player0 (spell-mana spell)))
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(case spell
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[(magic-missile)
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(values player (hp- boss MAGIC-MISSILE-DAMAGE) active-spells)]
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[(drain)
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(values (hp+ player DRAIN-DAMAGE) (hp- boss DRAIN-DAMAGE) active-spells)]
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[(shield)
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(values player boss (cons (cons SHIELD-DURATION 'shield) active-spells))]
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[(poison)
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(values player boss (cons (cons POISON-DURATION 'poison) active-spells))]
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[(recharge)
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(values player boss (cons (cons RECHARGE-DURATION 'recharge) active-spells))]
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[else
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(unknown-spell spell)]))
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(define (apply-effects player boss active-spells)
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(for/fold ([p+ player]
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[b+ boss]
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[a+ '()])
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([ctr+spell (in-list active-spells)])
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(match-define (cons ctr spell) ctr+spell)
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(define a++ (if (= 1 ctr) a+ (cons (cons (- ctr 1) spell) a+)))
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(case spell
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[(poison)
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(values p+ (hp- b+ POISON-DAMAGE) a++)]
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[(recharge)
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(values (mana+ p+ RECHARGE-MANA) b+ a++)]
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[(shield)
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(values p+ b+ a++)]
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[else
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(unknown-spell spell)])))
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]
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@section{What's the least (mana) we can spend and win?}
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Starting with 50 health, 500 mana and a boss with @math{N} hit points and @math{A}
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attack points, the least mana we can spend and win is either:
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@itemize[
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@item{
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The cost of one @emph{Magic Missile}, plus the least-mana solution for
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a player with @math{50 - M} hit points, @math{500 - 53} mana,
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and a boss with @math{N - 4} hit points.
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}
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@item{
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The cost of one @emph{Drain} spell, plus the least-mana solution for
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a player with @math{50 + 2} hit points, @math{500 - 73} mana, and
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a boss with @math{N - 2} hit points.
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}
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@item{
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The cost of one @emph{Shield} spell, plus the least-mana solution for:
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@itemize[
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@item{
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a player with @math{50 - max(1, (M - 7))} hit points and @math{500 - 113} mana,
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}
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@item{
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a boss with @math{N} hit points,
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}
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@item{
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the @emph{Shield} spell active for 4 more turns.
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}
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]
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}
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@item{
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The cost of one @emph{Poison} spell, plus the least-mana solution for
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a player with @math{50 - 7} hit points and @math{500 - 173} mana and
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a boss with @math{N - 6} hit points, given that the @emph{Poison} spell
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is active for 4 more turns.
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}
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@item{
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The cost of one @emph{Recharge} spell, plus the least-mana solution for
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a player with @math{50 - 7} hit points and @math{500 - 229 + (101 * 2)} mana
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against a boss with @math{N} hit points, given that the @emph{Recharge} spell
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is active for 3 more turns.
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}
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]
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The correct alternative is the one that happens to be cheapest.
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Since we've outlined the alternatives and know the end conditions for the game,
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we can write an algorithm that tries every alternative and returns the best
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sequence of spells.
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Never mind the @racket[hard-mode?] parameter for now.
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@chunk[<day22-dp>
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(define (win/least-mana player boss #:hard-mode? [hard-mode? #f])
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(or
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(let maybe-win/least-mana ([player player]
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[boss boss]
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[active-spells '()]
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[current-turn 0])
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(cond
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[(lose? player boss)
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#f]
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[(win? player boss)
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'()]
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[else
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(define-values (p+ b+ a+) (apply-effects player boss active-spells))
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(define next-turn (+ 1 current-turn))
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(if (even? current-turn)
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(let ([p+ (if hard-mode? (hp- p+ 1) p+)])
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(minimize-mana
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(for/list ([spell (in-list ALL-SPELLS)]
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#:when (and (not (active? spell a+))
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(has-enough-mana? p+ spell)))
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(define-values (p++ b++ a++) (apply-spell spell p+ b+ a+))
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(define future-spells (maybe-win/least-mana p++ b++ a++ next-turn))
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(and future-spells (cons spell future-spells)))))
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(maybe-win/least-mana (boss-attack p+ b+ (active? 'shield a+)) b+ a+ next-turn))]))
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(raise-user-error 'day22 "Impossible for ~a to beat ~a. Sorry.\n" player boss)))
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(define (win? player boss)
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(dead? boss))
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(define (dead? player)
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(<= (player-hp player) 0))
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(define lose?
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(let ([MIN-MANA (apply min (map spell-mana ALL-SPELLS))])
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(λ (player boss)
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(or (dead? player)
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(< (player-mana player) MIN-MANA)))))
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(define (boss-attack player boss shield?)
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(define boss-damage (max 1 (- (player-attack boss) (if shield? SHIELD-ARMOR 0))))
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(hp- player boss-damage))
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(define (active? spell active-spells)
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(for/or ([ctr+spell (in-list active-spells)])
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(eq? spell (cdr ctr+spell))))
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(define (has-enough-mana? player spell)
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(<= (spell-mana spell) (player-mana player)))
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(define (spell*-mana spell*)
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(for/sum ([spell (in-list spell*)])
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(spell-mana spell)))
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(define (minimize-mana spell**)
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(for/fold ([best #f])
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([other (in-list spell**)])
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(if (or (not best)
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(and best other (< (spell*-mana other) (spell*-mana best))))
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other
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best)))
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]
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@chunk[<day22-q1>
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(define (q1 input-str)
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(define player (make-player))
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(define boss (apply make-boss (filter integer? (map string->number (string-split input-str)))))
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(spell*-mana (win/least-mana player boss)))
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]
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@section{Hard Mode}
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On hard mode, the player loses one hit point on their turn.
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Thanks to the optional keyword argument to @racket[win/least-mana], the answer
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is easy find.
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@chunk[<day22-q2>
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(define (q2 input-str)
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(define player (make-player))
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(define boss (apply make-boss (filter integer? (map string->number (string-split input-str)))))
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(spell*-mana (win/least-mana player boss #:hard-mode? #t)))]
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@section{Testing Day 22}
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The algorithm as written is very slow.
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Unfortunately, there's no better way to solve this problem than trying everything,
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but while the tests are running, think about what kind of caching strategy could
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improve performance.
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@chunk[<day22-test>
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(module+ test
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(define input-str (file->string "day22-input.txt"))
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(check-equal? (q1 input-str) 1269)
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(check-equal? (q2 input-str) 1309))]
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