McNuggets problem

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Task
McNuggets problem
You are encouraged to solve this task according to the task description, using any language you may know.
From Wikipedia: 
The McNuggets version of the coin problem was introduced by Henri Picciotto,
who included it in his algebra textbook co-authored with Anita Wah. Picciotto
thought of the application in the 1980s while dining with his son at
McDonald's, working the problem out on a napkin. A McNugget number is
the total number of McDonald's Chicken McNuggets in any number of boxes.
In the United Kingdom, the original boxes (prior to the introduction of
the Happy Meal-sized nugget boxes) were of 6, 9, and 20 nuggets.
Task

Calculate (from 0 up to a limit of 100) the largest non-McNuggets number (a number n which cannot be expressed with 6x + 9y + 20z = n where x, y and z are natural numbers).

11l[edit]

Translation of: Python
V nuggets = Set(0..100)
L(s, n, t) cart_product(0 .. 100 I/ 6,
                        0 .. 100 I/ 9,
                        0 .. 100 I/ 20)
   nuggets.discard(6*s + 9*n + 20*t)

print(max(nuggets))
Output:
43

Action![edit]

PROC Main()
  BYTE x,y,z,n
  BYTE ARRAY nuggets(101)

  FOR n=0 TO 100
  DO
    nuggets(n)=0
  OD

  FOR x=0 TO 100 STEP 6
  DO
    FOR y=0 TO 100 STEP 9
    DO
      FOR z=0 TO 100 STEP 20
      DO
        n=x+y+z
        IF n<=100 THEN
          nuggets(n)=1
        FI
      OD
    OD
  OD

  n=100
  DO
    IF nuggets(n)=0 THEN
      PrintF("The largest non McNugget number is %B%E",n)
      EXIT
    ELSEIF n=0 THEN
      PrintE("There is no result")
      EXIT
    ELSE
      n==-1
    FI
  OD
RETURN
Output:

Screenshot from Atari 8-bit computer 

The largest non McNugget number is 43

Ada[edit]

with Ada.Text_IO; use Ada.Text_IO;

procedure McNugget is
   Limit : constant                      := 100;
   List  : array (0 .. Limit) of Boolean := (others => False);
   N     : Integer;
begin
   for A in 0 .. Limit / 6 loop
      for B in 0 .. Limit / 9 loop
         for C in 0 .. Limit / 20 loop
            N := A * 6 + B * 9 + C * 20;
            if N <= 100 then
               List (N) := True;
            end if;
         end loop;
      end loop;
   end loop;
   for N in reverse 1 .. Limit loop
      if not List (N) then
         Put_Line ("The largest non McNugget number is:" & Integer'Image (N));
         exit;
      end if;
   end loop;
end McNugget;
Output:
The largest non McNugget number is: 43

ALGOL 68[edit]

BEGIN
    # Solve the McNuggets problem: find the largest n <= 100 for which there #
    # are no non-negative integers x, y, z such that 6x + 9y + 20z = n       #
    INT max nuggets = 100;
    [ 0 : max nuggets ]BOOL sum;
    FOR i FROM LWB sum TO UPB sum DO sum[ i ] := FALSE OD;
    FOR x FROM 0 BY 6 TO max nuggets DO
        FOR y FROM x BY 9 TO max nuggets DO
            FOR z FROM y BY 20 TO max nuggets DO
                sum[ z ] := TRUE
            OD # z #
        OD # y #
    OD # x # ;
    # show the highest number that cannot be formed                          #
    INT largest := -1;
    FOR i FROM UPB sum BY -1 TO LWB sum WHILE sum[ largest := i ] DO SKIP OD;
    print( ( "The largest non McNugget number is: "
           , whole( largest, 0 )
           , newline
           )
         )
END
Output:
The largest non McNugget number is: 43

APL[edit]

Works with: Dyalog APL
100 (⌈/(⍳⊣)~(⊂⊢)(+/×)¨(,⎕IO-⍨(⍳∘⌊÷))) 6 9 20
Output:
43

AppleScript[edit]

Generalised for other set sizes, and for other triples of natural numbers. Uses NSMutableSet, through the AppleScript ObjC interface: 

use AppleScript version "2.4"
use framework "Foundation"
use scripting additions


on run
    set setNuggets to mcNuggetSet(100, 6, 9, 20)
    
    script isMcNugget
        on |λ|(x)
            setMember(x, setNuggets)
        end |λ|
    end script
    set xs to dropWhile(isMcNugget, enumFromThenTo(100, 99, 1))
    
    set setNuggets to missing value -- Clear ObjC pointer value
    if 0 < length of xs then
        item 1 of xs
    else
        "No unreachable quantities in this range"
    end if
end run

-- mcNuggetSet :: Int -> Int -> Int -> Int -> ObjC Set
on mcNuggetSet(n, mcx, mcy, mcz)
    set upTo to enumFromTo(0)
    script fx
        on |λ|(x)
            script fy
                on |λ|(y)
                    script fz
                        on |λ|(z)
                            set v to sum({mcx * x, mcy * y, mcz * z})
                            if 101 > v then
                                {v}
                            else
                                {}
                            end if
                        end |λ|
                    end script
                    concatMap(fz, upTo's |λ|(n div mcz))
                end |λ|
            end script
            concatMap(fy, upTo's |λ|(n div mcy))
        end |λ|
    end script
    setFromList(concatMap(fx, upTo's |λ|(n div mcx)))
end mcNuggetSet


-- GENERIC FUNCTIONS ----------------------------------------------------

-- concatMap :: (a -> [b]) -> [a] -> [b]
on concatMap(f, xs)
    set lng to length of xs
    set acc to {}
    tell mReturn(f)
        repeat with i from 1 to lng
            set acc to acc & |λ|(item i of xs, i, xs)
        end repeat
    end tell
    return acc
end concatMap


-- drop :: Int -> [a] -> [a]
-- drop :: Int -> String -> String
on drop(n, xs)
    set c to class of xs
    if c is not script then
        if c is not string then
            if n < length of xs then
                items (1 + n) thru -1 of xs
            else
                {}
            end if
        else
            if n < length of xs then
                text (1 + n) thru -1 of xs
            else
                ""
            end if
        end if
    else
        take(n, xs) -- consumed
        return xs
    end if
end drop

-- dropWhile :: (a -> Bool) -> [a] -> [a]
-- dropWhile :: (Char -> Bool) -> String -> String
on dropWhile(p, xs)
    set lng to length of xs
    set i to 1
    tell mReturn(p)
        repeat while i ≤ lng and |λ|(item i of xs)
            set i to i + 1
        end repeat
    end tell
    drop(i - 1, xs)
end dropWhile

-- enumFromThenTo :: Int -> Int -> Int -> [Int]
on enumFromThenTo(x1, x2, y)
    set xs to {}
    repeat with i from x1 to y by (x2 - x1)
        set end of xs to i
    end repeat
    return xs
end enumFromThenTo

-- enumFromTo :: Int -> Int -> [Int]
on enumFromTo(m)
    script
        on |λ|(n)
            if m ≤ n then
                set lst to {}
                repeat with i from m to n
                    set end of lst to i
                end repeat
                return lst
            else
                return {}
            end if
        end |λ|
    end script
end enumFromTo

-- foldl :: (a -> b -> a) -> a -> [b] -> a
on foldl(f, startValue, xs)
    tell mReturn(f)
        set v to startValue
        set lng to length of xs
        repeat with i from 1 to lng
            set v to |λ|(v, item i of xs, i, xs)
        end repeat
        return v
    end tell
end foldl

-- Lift 2nd class handler function into 1st class script wrapper 
-- mReturn :: First-class m => (a -> b) -> m (a -> b)
on mReturn(f)
    if class of f is script then
        f
    else
        script
            property |λ| : f
        end script
    end if
end mReturn

-- sum :: [Num] -> Num
on sum(xs)
    script add
        on |λ|(a, b)
            a + b
        end |λ|
    end script
    
    foldl(add, 0, xs)
end sum

-- NB All names of NSMutableSets should be set to *missing value*
-- before the script exits.
-- ( scpt files can not be saved if they contain ObjC pointer values )
-- setFromList :: Ord a => [a] -> Set a
on setFromList(xs)
    set ca to current application
    ca's NSMutableSet's ¬
        setWithArray:(ca's NSArray's arrayWithArray:(xs))
end setFromList

-- setMember :: Ord a => a -> Set a -> Bool
on setMember(x, objcSet)
    missing value is not (objcSet's member:(x))
end setMember
Output:
43

Arturo[edit]

Translation of: Ruby
nonMcNuggets: function [lim][
    result: new 0..lim

    loop range.step:6 1 lim 'x [
        loop range.step:9 1 lim 'y [
            loop range.step:20 1 lim 'z 
                -> 'result -- sum @[x y z]
        ]
    ]

    return result
]

print max nonMcNuggets 100
Output:
46

AWK[edit]

# syntax: GAWK -f MCNUGGETS_PROBLEM.AWK
# converted from Go
BEGIN {
    limit = 100
    for (a=0; a<=limit; a+=6) {
      for (b=a; b<=limit; b+=9) {
        for (c=b; c<=limit; c+=20) {
          arr[c] = 1
        }
      }
    }
    for (i=limit; i>=0; i--) {
      if (!arr[i]+0) {
        printf("%d\n",i)
        break
      }
    }
    exit(0)
}
Output:
43

BASIC[edit]

10 DEFINT A-Z: DIM F(100)
20 FOR A=0 TO 100 STEP 6
30 FOR B=A TO 100 STEP 9
40 FOR C=B TO 100 STEP 20
50 F(C)=-1
60 NEXT C,B,A
70 FOR A=100 TO 0 STEP -1
80 IF NOT F(A) THEN PRINT A: END
90 NEXT A
Output:
 43

BCPL[edit]

get "libhdr"
manifest $( limit = 100 $)

let start() be
$(  let flags = vec limit
    for i = 0 to limit do flags!i := false
    
    for a = 0 to limit by 6
        for b = a to limit by 9
            for c = b to limit by 20
                do flags!c := true
                
    for i = limit to 0 by -1
        unless flags!i
        $(  writef("Maximum non-McNuggets number: %N.*N", i)
            finish
        $)
$)
Output:
Maximum non-McNuggets number: 43.

BQN[edit]

100 ((↕⊣)(⌈´⊣×⊣¬∘∊⥊∘⊢)(<⊢)(+´×)¨(↕⌊∘÷)) 6‿9‿20
Output:
43

C[edit]

#include <stdio.h>

int
main() {
    int max = 0, i = 0, sixes, nines, twenties;

loopstart: while (i < 100) {
        for (sixes = 0; sixes*6 < i; sixes++) {
            if (sixes*6 == i) {
                i++;
                goto loopstart;
            }

            for (nines = 0; nines*9 < i; nines++) {
                if (sixes*6 + nines*9 == i) {
                    i++;
                    goto loopstart;
                }

                for (twenties = 0; twenties*20 < i; twenties++) {
                    if (sixes*6 + nines*9 + twenties*20 == i) {
                        i++;
                        goto loopstart;
                    }
                }
            }
        }
        max = i;
        i++;
    }

    printf("Maximum non-McNuggets number is %d\n", max);

    return 0;
}
Output:
Maximum non-McNuggets number is 43

C#[edit]

using System;

public class McNuggets
{
   public static void Main()
   {
      bool[] isMcNuggetNumber = new bool[101];
      
      for (int x = 0; x <= 100/6; x++)
      {
         for (int y = 0; y <= 100/9; y++)
         {
            for (int z = 0; z <= 100/20; z++)
            {
               int mcNuggetNumber = x*6 + y*9 + z*20;
               if (mcNuggetNumber <= 100)
               {
                  isMcNuggetNumber[mcNuggetNumber] = true;
               }
            }
         } 
      }

      for (int mnnCheck = isMcNuggetNumber.Length-1; mnnCheck >= 0; mnnCheck--)
      {
         if (!isMcNuggetNumber[mnnCheck])
         {
            Console.WriteLine("Largest non-McNuggett Number less than 100: " + mnnCheck.ToString());
            break;
         }
      }
   }
}
Output:
Largest non-McNuggett Number less than 100: 43

Clojure[edit]

(defn cart [colls]
  (if (empty? colls)
    '(())
    (for [more (cart (rest colls))
          x (first colls)]
      (cons x more))))

(defn nuggets [[n6 n9 n20]] (+ (* 6 n6) (* 9 n9) (* 20 n20)))

(let [possible (distinct (map nuggets (cart (map range [18 13 6]))))
      mcmax (apply max (filter (fn [x] (not-any? #{x} possible)) (range 101)))]
  (printf "Maximum non-McNuggets number is %d\n" mcmax))
Output:
Maximum non-McNuggets number is 43

CLU[edit]

% Recursive nugget iterator.
% This yields all the nugget numbers of the given box sizes from start to max.
gen_nuggets = iter (start, max: int, sizes: sequence[int]) yields (int) 
    si = sequence[int]
    if si$empty(sizes) then 
        yield(start)
    else
        for i: int in int$from_to_by(start, max, si$bottom(sizes)) do
            for j: int in gen_nuggets(i, max, si$reml(sizes)) do
                yield(j)
            end
        end
    end
end gen_nuggets

start_up = proc ()
    max = 100
    ab = array[bool]
    po: stream := stream$primary_output()
    nuggets: ab := ab$fill(0,max+1,false)
    
    for nugget: int in gen_nuggets(0, max, sequence[int]$[6,9,20]) do
        nuggets[nugget] := true
    end
    
    maxn: int
    for i: int in ab$indexes(nuggets) do
        if ~nuggets[i] then maxn := i end
    end
    
    stream$putl(po, "Maximum non-McNuggets number: " || int$unparse(maxn))
end start_up
Output:
Maximum non-McNuggets number: 43

COBOL[edit]

       IDENTIFICATION DIVISION.
       PROGRAM-ID.  MCNUGGETS.

       DATA DIVISION.
       WORKING-STORAGE SECTION.
       01 NUGGETS.
          03 NUGGET-FLAGS     PIC X OCCURS 100 TIMES.
             88 IS-NUGGET     VALUE 'X'.

       01 A                   PIC 999.
       01 B                   PIC 999.
       01 C                   PIC 999.

       PROCEDURE DIVISION.
       BEGIN.
           MOVE SPACES TO NUGGETS.
           PERFORM A-LOOP VARYING A FROM 0 BY 6
               UNTIL A IS GREATER THAN 100.
           MOVE 100 TO A.

       FIND-LARGEST.
           IF IS-NUGGET(A), SUBTRACT 1 FROM A, GO TO FIND-LARGEST.
           DISPLAY 'Largest non-McNugget number: ', A.
           STOP RUN.
 
       A-LOOP.
           PERFORM B-LOOP VARYING B FROM A BY 9
               UNTIL B IS GREATER THAN 100.

       B-LOOP.
           PERFORM C-LOOP VARYING C FROM B BY 20
               UNTIL C IS GREATER THAN 100.
 
       C-LOOP.
           IF C IS NOT EQUAL TO ZERO, MOVE 'X' TO NUGGET-FLAGS(C).
Output:
Largest non-McNugget number: 043

Comal[edit]

0010 limit#:=100
0020 DIM nugget#(0:limit#)
0030 FOR a#:=0 TO limit# STEP 6 DO
0040   FOR b#:=a# TO limit# STEP 9 DO
0050     FOR c#:=b# TO limit# STEP 20 DO nugget#(c#):=TRUE
0060   ENDFOR b#
0070 ENDFOR a#
0080 FOR i#:=limit# TO 0 STEP -1 DO
0090   IF NOT nugget#(i#) THEN
0100     PRINT "Maximum non-McNuggets number: ",i#
0110     END
0120   ENDIF
0130 ENDFOR i#
Output:
Maximum non-McNuggets number: 43

Cowgol[edit]

include "cowgol.coh";
const LIMIT := 100;

var flags: uint8[LIMIT+1];
MemZero(&flags[0], @bytesof flags);

var a: @indexof flags;
var b: @indexof flags;
var c: @indexof flags;

a := 0;
while a <= LIMIT loop
    b := a;
    while b <= LIMIT loop
        c := b;
        while c <= LIMIT loop
            flags[c] := 1;
            c := c + 20;
        end loop;
        b := b + 9;
    end loop;
    a := a + 6;
end loop;

a := LIMIT;
loop
    if flags[a] == 0 then
        print("Maximum non-McNuggets number: ");
        print_i32(a as uint32);
        print_nl();
        break;
    end if;
    a := a - 1;
end loop;
Output:
Maximum non-McNuggets number: 43

Dart[edit]

import 'dart:math';
main() {
  var nuggets = List<int>.generate(101, (int index) => index);
  for (int small in List<int>.generate((100 ~/ (6 + 1)), (int index) => index)) {
    for (int medium in List<int>.generate((100 ~/ (9 + 1)), (int index) => index)) {
      for (int large in List<int>.generate((100 ~/ (20 + 1)), (int index) => index)) {
        nuggets.removeWhere((element) => element == 6 * small + 9 * medium + 20 * large);
      }
    }
  }
  print('Largest non-McNuggets number: ${nuggets.reduce(max).toString() ?? 'none'}.');
}
Output:
Largest non-McNuggets number: 43.

Draco[edit]

proc nonrec main() void:
    byte LIMIT = 100;
    [LIMIT+1] bool nugget;
    byte a, b, c;

    for a from 0 upto LIMIT do
        nugget[a] := false
    od;

    for a from 0 by 6 upto LIMIT do
        for b from a by 9 upto LIMIT do
            for c from b by 20 upto LIMIT do
                nugget[c] := true
            od
        od
    od;

    a := LIMIT;
    while nugget[a] do a := a - 1 od;
    writeln("Maximum non-McNuggets number: ", a)
corp
Output:
Maximum non-McNuggets number: 43

Dyalect[edit]

Translation of: Go
func mcnugget(limit) {
    var sv = Array.Empty(limit + 1, false)
    for s in 0^6..limit {
        for n in s^9..limit {
            for t in n^20..limit {
                sv[t] = true
            }
        }
    }
    for i in limit^-1..0 {
        if !sv[i] {
            print("Maximum non-McNuggets number is \(i)")
            return
        }
    }
}
 
mcnugget(100)
Output:
Maximum non-McNuggets number is 43

Elixir[edit]

Uses MapSet and Comprehension 

defmodule Mcnugget do
  def solve(limit) do
    0..limit
    |> MapSet.new()
    |> MapSet.difference(
      for(
        x <- 0..limit,
        y <- 0..limit,
        z <- 0..limit,
        Integer.mod(x, 6) == 0,
        Integer.mod(y, 9) == 0,
        Integer.mod(z, 20) == 0,
        x + y + z <= limit,
        into: MapSet.new(),
        do: x + y + z
      )
    )
    |> Enum.max()
  end
end

Mcnugget.solve(100) |> IO.puts
Output:
43

F#[edit]

// McNuggets. Nigel Galloway: October 28th., 2018
let fN n g = Seq.initInfinite(fun ng->ng*n+g)|>Seq.takeWhile(fun n->n<=100)
printfn "%d" (Set.maxElement(Set.difference (set[1..100]) (fN 20 0|>Seq.collect(fun n->fN 9 n)|>Seq.collect(fun n->fN 6 n)|>Set.ofSeq)))
Output:
43

Factor[edit]

USING: backtrack kernel math.ranges prettyprint sequences sets ;
101 <iota> [ 0 6 9 20 [ 100 swap <range> amb-lazy ] tri@ ] bag-of diff last .
Output:
43

FOCAL[edit]

01.10 F N=0,100;S T(N)=0
01.20 F A=0,6,100;F B=A,9,100;F C=B,20,100;S T(C)=-1
01.30 S N=101
01.40 S N=N-1
01.50 I (T(N))1.4
01.60 T %3,N,!
01.70 Q
Output:
=  43

FreeBASIC[edit]

Dim As Integer l(100), a, b, c, n
For a = 0 To 100/6
    For b =  0 To 100/9
        For c = 0 To 100/20
            n = a*6 + b*9 + c*20
            If n <= 100 Then l(n) = true
        Next c
    Next b
Next a
For n = 100 To 1 Step -1
    If l(n) = false Then Print "El mayor número que no sea McNugget es:"; n: Exit For
Next n
End
Output:
El mayor número que no sea McNugget es: 43

Frink[edit]

This is a nice demonstration for Frink's multifor loop which can perform arbitrarily-deeply-nested loops in a single statement. The "inner" (rightmost) loops can use values set by the "outer" (leftmost) as part of their bounds. 

a = toSet[0 to 100]

multifor [z,y,x] = [0 to 100 step 20, 0 to 100-z step 9, 0 to 100-z-y step 6]
   a.remove[x+y+z]

println[max[a]]
Output:
43

FutureBasic[edit]

local fn McNuggetsProblem
  BOOL l(100)
  long a, b, c, i, n
  
  for a = 0 to 100/6
    for b =  0 to 100/9
      for c = 0 to 100/20
        n = a * 6 + b * 9 + c * 20
        if n <= 100 then l(n) = YES
      next
    next
  next
  for i = 100 to 1 step -1
    if l(i) == NO then print "The maximum non-McNuggets number less than 100 is: "; i: exit for
  next
end fn

window 1,,( 0, 0, 450, 100 )
fn McNuggetsProblem

HandleEvents
Output:
The maximum non-McNuggets number less than 100 is: 43





Go[edit]

package main

import "fmt"

func mcnugget(limit int) {
    sv := make([]bool, limit+1) // all false by default
    for s := 0; s <= limit; s += 6 {
        for n := s; n <= limit; n += 9 {
            for t := n; t <= limit; t += 20 {
                sv[t] = true
            }
        }
    }
    for i := limit; i >= 0; i-- {
        if !sv[i] {
            fmt.Println("Maximum non-McNuggets number is", i)
            return
        }
    }
}

func main() {
    mcnugget(100)
}
Output:
Maximum non-McNuggets number is 43

Haskell[edit]

import Data.Set (Set, fromList, member)

------------------------ MCNUGGETS -----------------------

mcNuggets :: Set Int
mcNuggets =
  let size = enumFromTo 0 . quot 100
   in fromList $
        size 6
          >>= \x ->
            size 9
              >>= \y ->
                size 20
                  >>= \z ->
                    [ v
                      | let v = sum [6 * x, 9 * y, 20 * z],
                        101 > v
                    ]

--------------------------- TEST -------------------------
main :: IO ()
main =
  (putStrLn . go) $
    dropWhile (`member` mcNuggets) [100, 99 .. 1]
  where
    go (x : _) = show x
    go [] = "No unreachable quantities found ..."

Or equivalently, making use of the list comprehension notation: 

import Data.Set (Set, fromList, member)

gaps :: [Int]
gaps = dropWhile (`member` mcNuggets) [100,99 .. 1]

mcNuggets :: Set Int
mcNuggets =
  let size n = [0 .. quot 100 n]
  in fromList
       [ v
       | x <- size 6 
       , y <- size 9 
       , z <- size 20 
       , let v = sum [6 * x, 9 * y, 20 * z] 
       , 101 > v ]

main :: IO ()
main =
  print $
  case gaps of
    x:_ -> show x
    []  -> "No unreachable quantities found ..."

43

J[edit]

Brute force solution: calculate all pure (just one kind of box) McNugget numbers which do not exceed 100, then compute all possible sums, and then remove those from the list of numbers up to 100 (which is obviously a McNugget number), then find the largest number remaining: 

   >./(i.100)-.,+/&>{(* i.@>.@%~&101)&.>6 9 20 
43

Technically, we could have used 100 in place of 101 when we were finding how many pure McNugget numbers were in each series (because 100 is obviously a McNugget number), but it's not like that's a problem, either.

Java[edit]

public class McNuggets {

    public static void main(String... args) {
        int[] SIZES = new int[] { 6, 9, 20 };
        int MAX_TOTAL = 100;
        // Works like Sieve of Eratosthenes
        int numSizes = SIZES.length;
        int[] counts = new int[numSizes];
        int maxFound = MAX_TOTAL + 1;
        boolean[] found = new boolean[maxFound];
        int numFound = 0;
        int total = 0;
        boolean advancedState = false;
        do {
            if (!found[total]) {
                found[total] = true;
                numFound++;
            }
            
            // Advance state
            advancedState = false;
            for (int i = 0; i < numSizes; i++) {
                int curSize = SIZES[i];
                if ((total + curSize) > MAX_TOTAL) {
                    // Reset to zero and go to the next box size
                    total -= counts[i] * curSize;
                    counts[i] = 0;
                }
                else {
                    // Adding a box of this size still keeps the total at or below the maximum
                    counts[i]++;
                    total += curSize;
                    advancedState = true;
                    break;
                }
            }
            
        } while ((numFound < maxFound) && advancedState);
        
        if (numFound < maxFound) {
            // Did not find all counts within the search space
            for (int i = MAX_TOTAL; i >= 0; i--) {
                if (!found[i]) {
                    System.out.println("Largest non-McNugget number in the search space is " + i);
                    break;
                }
            }
        }
        else {
            System.out.println("All numbers in the search space are McNugget numbers");
        }
        
        return;
    }
}
Output:
Largest non-McNugget number in the search space is 43

JavaScript[edit]

(() => {
    'use strict';

    // main :: IO ()
    const main = () => {
        const
            size = n => enumFromTo(0)(
                quot(100, n)
            ),
            nuggets = new Set(
                size(6).flatMap(
                    x => size(9).flatMap(
                        y => size(20).flatMap(
                            z => {
                                const v = sum([6 * x, 9 * y, 20 * z]);
                                return 101 > v ? (
                                    [v]
                                ) : [];
                            }
                        ),
                    )
                )
            ),
            xs = dropWhile(
                x => nuggets.has(x),
                enumFromThenTo(100, 99, 1)
            );

        return 0 < xs.length ? (
            xs[0]
        ) : 'No unreachable quantities found in this range';
    };


    // GENERIC FUNCTIONS ----------------------------------

    // dropWhile :: (a -> Bool) -> [a] -> [a]
    const dropWhile = (p, xs) => {
        const lng = xs.length;
        return 0 < lng ? xs.slice(
            until(
                i => i === lng || !p(xs[i]),
                i => 1 + i,
                0
            )
        ) : [];
    };

    // enumFromThenTo :: Int -> Int -> Int -> [Int]
    const enumFromThenTo = (x1, x2, y) => {
        const d = x2 - x1;
        return Array.from({
            length: Math.floor(y - x2) / d + 2
        }, (_, i) => x1 + (d * i));
    };

    // ft :: Int -> Int -> [Int]
    const enumFromTo = m => n =>
        Array.from({
            length: 1 + n - m
        }, (_, i) => m + i);

    // quot :: Int -> Int -> Int
    const quot = (n, m) => Math.floor(n / m);

    // sum :: [Num] -> Num
    const sum = xs => xs.reduce((a, x) => a + x, 0);

    // until :: (a -> Bool) -> (a -> a) -> a -> a
    const until = (p, f, x) => {
        let v = x;
        while (!p(v)) v = f(v);
        return v;
    };

    // MAIN ---
    return console.log(
        main()
    );
})();
Output:
43

jq[edit]

Translation of: Clojure
[
 [range(18) as $n6  |
  range(13) as $n9  |
  range(6)  as $n20 |
  ($n6 * 6 + $n9 * 9 + $n20 * 20)] |
 unique |
 . as $possible |
 range(101) |
 . as $n |
 select($possible|contains([$n])|not)
] |
max
Output:
43

Julia[edit]

Simple brute force solution, though the BitSet would save memory considerably with larger max numbers. 

function mcnuggets(max)
    b = BitSet(1:max)
    for i in 0:6:max, j in 0:9:max, k in 0:20:max
        delete!(b, i + j + k)
    end
    maximum(b)
end

println(mcnuggets(100))
Output:


43

Kotlin[edit]

Translation of: Go
// Version 1.2.71

fun mcnugget(limit: Int) {
    val sv = BooleanArray(limit + 1)  // all false by default
    for (s in 0..limit step 6)
        for (n in s..limit step 9)
            for (t in n..limit step 20) sv[t] = true

    for (i in limit downTo 0) {
        if (!sv[i]) {
            println("Maximum non-McNuggets number is $i")
            return
        }
    }
}

fun main(args: Array<String>) {
    mcnugget(100)
}
Output:
Maximum non-McNuggets number is 43

Locomotive Basic[edit]

100 CLEAR
110 DIM a(100)
120 FOR a=0 TO 100/6
130   FOR b=0 TO 100/9
140     FOR c=0 TO 100/20
150       n=a*6+b*9+c*20
160       IF n<=100 THEN a(n)=1
170     NEXT c
180   NEXT b
190 NEXT a
200 FOR n=0 TO 100
210   IF a(n)=0 THEN l=n
220 NEXT n
230 PRINT"The Largest non McNugget number is:";l
240 END
Output:
The largest non McNugget number is: 43

Lua[edit]

function range(A,B)
    return function()
        return coroutine.wrap(function()
            for i = A, B do coroutine.yield(i) end
        end)
    end
end

function filter(stream, f)
    return function()
        return coroutine.wrap(function()
            for i in stream() do
                if f(i) then coroutine.yield(i) end
            end
        end)
    end
end

function triple(s1, s2, s3)
    return function()
        return coroutine.wrap(function()
            for x in s1() do
                for y in s2() do
                    for z in s3() do
                        coroutine.yield{x,y,z}
                    end
                end
            end
        end)
    end
end

function apply(f, stream)
    return function()
        return coroutine.wrap(function()
            for T in stream() do
                coroutine.yield(f(table.unpack(T)))
            end
        end)
    end
end

function exclude(s1, s2)
    local exlusions = {} for x in s1() do exlusions[x] = true end
    return function()
        return coroutine.wrap(function()
            for x in s2() do
                if not exlusions[x] then
                    coroutine.yield(x)
                end
            end
        end)
    end
end

function maximum(stream)
    local M = math.mininteger
    for x in stream() do
        M = math.max(M, x)
    end
    return M
end

local N = 100
local A, B, C = 6, 9, 20

local Xs = filter(range(0, N), function(x) return x  % A == 0 end)
local Ys = filter(range(0, N), function(x) return x  % B == 0 end)
local Zs = filter(range(0, N), function(x) return x  % C == 0 end)
local sum = filter(apply(function(x, y, z) return x + y + z end, triple(Xs, Ys, Zs)), function(x) return x <= N end)

print(maximum(exclude(sum, range(1, N))))
Output:
43

MAD[edit]

            NORMAL MODE IS INTEGER
            BOOLEAN NUGGET
            DIMENSION NUGGET(101)

            THROUGH CLEAR, FOR I=0, 1, I.G.100
CLEAR       NUGGET(I) = 0B

            THROUGH MARK, FOR A=0, 6, A.G.100
            THROUGH MARK, FOR B=A, 9, B.G.100
            THROUGH MARK, FOR C=B, 20, C.G.100
MARK        NUGGET(C) = 1B

SEARCH      THROUGH SEARCH, FOR I=100, -1, .NOT.NUGGET(I) 

            PRINT FORMAT F, I
            VECTOR VALUES F = $29HMAXIMUM NON-MCNUGGET NUMBER: ,I2*$
            END OF PROGRAM
Output:
MAXIMUM NON-MCNUGGET NUMBER: 43

Mathematica/Wolfram Language[edit]

FrobeniusNumber[{6, 9, 20}]
Output:
43

Modula-2[edit]

MODULE McNuggets;
FROM InOut IMPORT WriteCard, WriteString, WriteLn;

CONST Max = 100;
VAR a, b, c: CARDINAL;
    nugget: ARRAY [0..Max] OF BOOLEAN;

BEGIN
    FOR a := 0 TO Max DO
        nugget[a] := FALSE;
    END;

    FOR a := 0 TO Max BY 6 DO
        FOR b := a TO Max BY 9 DO
            FOR c := b TO Max BY 20 DO
                nugget[c] := TRUE;
            END;
        END;
    END;

    a := 100;
    REPEAT DEC(a); UNTIL NOT nugget[a];
    WriteString("Maximum non-McNuggets number: ");
    WriteCard(a, 2);
    WriteLn();
END McNuggets.
Output:
Maximum non-McNuggets number: 43

MiniZinc[edit]

%McNuggets. Nigel Galloway, August 27th., 2019
var 0..99: n;
constraint forall(x in 0..16,y in 0..11,z in 0..5)(6*x + 9*y + 20*z!=n);
solve maximize n;
output [show(n)]
Output:
43
----------
==========

Nim[edit]

const Limit = 100

var mcnuggets: array[0..Limit, bool]

for a in countup(0, Limit, 6):
  for b in countup(a, Limit, 9):
    for c in countup(b, Limit, 20):
      mcnuggets[c] = true

for n in countdown(Limit, 0):
  if not mcnuggets[n]:
    echo "The largest non-McNuggets number is: ", n
    break
Output:
The largest non-McNuggets number is: 43

Perl[edit]

Translation of: Raku
Library: ntheory
use ntheory qw/forperm gcd vecmin/;

sub Mcnugget_number {
    my $counts = shift;

    return 'No maximum' if 1 < gcd @$counts;

    my $min = vecmin @$counts;
    my @meals;
    my @min;

    my $a = -1;
    while (1) {
        $a++;
        for my $b (0..$a) {
            for my $c (0..$b) {
                my @s = ($a, $b, $c);
                forperm {
                    $meals[
                        $s[$_[0]] * $counts->[0]
                      + $s[$_[1]] * $counts->[1]
                      + $s[$_[2]] * $counts->[2]
                    ] = 1;
                } @s;
            }
        }
        for my $i (0..$#meals) {
            next unless $meals[$i];
            if ($min[-1] and $i == ($min[-1] + 1)) {
                push @min, $i;
                last if $min == @min
            } else {
                @min = $i;
            }
        }
        last if $min == @min
    }
    $min[0] ? $min[0] - 1 : 0
}

for my $counts ([6,9,20], [6,7,20], [1,3,20], [10,5,18], [5,17,44], [2,4,6], [3,6,15]) {
    print 'Maximum non-Mcnugget number using ' . join(', ', @$counts) . ' is: ' . Mcnugget_number($counts) . "\n"
}
Output:
Maximum non-Mcnugget number using 6, 9, 20 is: 43
Maximum non-Mcnugget number using 6, 7, 20 is: 29
Maximum non-Mcnugget number using 1, 3, 20 is: 0
Maximum non-Mcnugget number using 10, 5, 18 is: 67
Maximum non-Mcnugget number using 5, 17, 44 is: 131
Maximum non-Mcnugget number using 2, 4, 6 is: No maximum
Maximum non-Mcnugget number using 3, 6, 15 is: No maximum

Perl using Regex[edit]

use strict;
use warnings;

$_ = 1 . 0 x 100;
1 while s/ (?=1) (?:.{6}|.{9}|.{20}) \K 0 /1/x;
/01*$/ and print "Maximum non-Mcnugget number is: $-[0]\n";
Output:
Maximum non-Mcnugget number is: 43

Phix[edit]

Translation of: Go
with javascript_semantics
constant limit=100
sequence nuggets = repeat(false,limit+1)
for sixes=0 to limit by 6 do
    for nines=sixes to limit by 9 do
        for twenties=nines to limit by 20 do
            nuggets[twenties+1] = true
        end for
    end for
end for
printf(1,"Maximum non-McNuggets number is %d\n", rfind(false,nuggets)-1)
Output:
Maximum non-McNuggets number is 43

Also, since it is a bit more interesting, a

Translation of: Raku
with javascript_semantics
function Mcnugget_number(sequence counts)
 
    if gcd(counts)>1 then return "No maximum" end if
 
    atom cmin = min(counts)
    sequence meals = {}
    sequence smin = {}
 
    integer a = -1
    while true do
        a += 1
        for b=0 to a do
            for c=0 to b do
                sequence s = {a, b, c}
                for i=1 to factorial(3) do
                    sequence p = permute(i,s)
                    integer k = sum(sq_mul(p,counts))+1
--                  atom k = sum(sq_mul(p,counts))+1
                    if k>length(meals) then meals &= repeat(0,k-length(meals)) end if
                    meals[k] = 1
                end for
            end for
        end for
        for i=1 to length(meals) do
            if meals[i] then
                if length(smin) and smin[$]+1=i-1 then
                    smin = append(smin,i-1)
                    if length(smin)=cmin then exit end if
                else
                    smin = {i-1}
                end if
            end if
        end for
        if length(smin)=cmin then exit end if
    end while
    return sprintf("%d",iff(smin[1]?smin[1]-1:0))
end function
 
constant tests = {{6,9,20}, {6,7,20}, {1,3,20}, {10,5,18}, {5,17,44}, {2,4,6}, {3,6,15}}
for i=1 to length(tests) do
    sequence ti = tests[i]
    printf(1,"Maximum non-Mcnugget number using %V is: %s\n",{ti,Mcnugget_number(ti)})
end for
Output:
Maximum non-Mcnugget number using {6,9,20} is: 43
Maximum non-Mcnugget number using {6,7,20} is: 29
Maximum non-Mcnugget number using {1,3,20} is: 0
Maximum non-Mcnugget number using {10,5,18} is: 67
Maximum non-Mcnugget number using {5,17,44} is: 131
Maximum non-Mcnugget number using {2,4,6} is: No maximum
Maximum non-Mcnugget number using {3,6,15} is: No maximum

Picat[edit]

Using constraint modelling (cp solver). 

import cp.

go =>
  N :: 0..100,
  foreach(X in 0..16, Y in 0..11, Z in 0..5)
     6*X + 9*Y + 20*Z #!= N
  end,
  solve($[max(N)],N),
  println(n=N).
Output:
n = 43

PicoLisp[edit]

(de nuggets1 (M)
   (let Lst (range 0 M)
      (for A (range 0 M 6)
         (for B (range A M 9)
            (for C (range B M 20)
               (set (nth Lst (inc C))) ) ) )
      (apply max Lst) ) )

Generator from fiber: 

(de nugg (M)
   (co 'nugget
      (for A (range 0 M 6)
         (for B (range A M 9)
            (for C (range B M 20)
               (yield (inc C)) ) ) ) ) )
(de nuggets2 (M)
   (let Lst (range 0 M) 
      (while (nugg 100)
         (set (nth Lst @)) )
      (apply max Lst) ) )

Test versions against each other: 

(test
   T
   (=
      43
      (nuggets1 100)
      (nuggets2 100) ) )

PL/I[edit]

mcnugget: procedure options(main);
    declare nugget(0:100) bit, (a, b, c) fixed;    
    do a=0 to 100; nugget(a) = '0'b; end;
    
    do a=0 to 100 by 6;
        do b=a to 100 by 9;
            do c=b to 100 by 20;
                nugget(c) = '1'b;
            end;
        end;
    end;
    
    do a=100 to 0 by -1;
        if ^nugget(a) then do;
            put skip list('Maximum non-McNuggets number:', a);
            stop;
        end;
    end;
end mcnugget;
Output:
Maximum non-McNuggets number:        43

PL/M[edit]

100H:
BDOS: PROCEDURE (FN, ARG); DECLARE FN BYTE, ARG ADDRESS; GO TO 5; END BDOS;
EXIT: PROCEDURE; CALL BDOS(0,0); END EXIT;
PRINT: PROCEDURE (S); DECLARE S ADDRESS; CALL BDOS(9, S); END PRINT;

PRINT$NUMBER: PROCEDURE (N);
    DECLARE S (6) BYTE INITIAL ('...',13,10,'$');
    DECLARE P ADDRESS, (N, C BASED P) BYTE;
    P = .S(3);
DIGIT:
    P = P-1;
    C = N MOD 10 + '0';
    N = N/10;
    IF N>0 THEN GO TO DIGIT;
    CALL PRINT(P);
END PRINT$NUMBER;

DECLARE (A, B, C) BYTE;
DECLARE NUGGET (101) BYTE;

DO A=0 TO 100; NUGGET(A) = 0; END;
DO A=0 TO 100 BY 6;
    DO B=A TO 100 BY 9;
        DO C=B TO 100 BY 20;
            NUGGET(C) = -1;
        END;
    END;
END;

A = 100;
DO WHILE NUGGET(A); A = A-1; END;
CALL PRINT$NUMBER(A);
CALL EXIT;
EOF
Output:
43

PowerShell[edit]

Translation of: UNIX Shell
$possible = @{}
For ($i=0; $i -lt 18; $i++) {
  For ($j=0; $j -lt 13; $j++) {
    For ( $k=0; $k -lt 6; $k++ ) {
      $possible[ $i*6 + $j*9 + $k*20 ] = $true
    }
  }
}
 
For ( $n=100; $n -gt 0; $n-- ) {
  If ($possible[$n]) {
    Continue
  }
  Else {
    Break
  }
}
Write-Host "Maximum non-McNuggets number is $n"
Output:
Maximum non-McNuggets number is 43

Python[edit]

Python: REPL[edit]

It's a simple solution done on the command line: 

>>> from itertools import product
>>> nuggets = set(range(101))
>>> for s, n, t in product(range(100//6+1), range(100//9+1), range(100//20+1)):
	nuggets.discard(6*s + 9*n + 20*t)

	
>>> max(nuggets)
43
>>>

Single expression version (expect to be slower, however no noticeable difference on a Celeron B820 and haven't benchmarked): 

>>> from itertools import product
>>> max(x for x in range(100+1) if x not in
...   (6*s + 9*n + 20*t for s, n, t in
...     product(range(100//6+1), range(100//9+1), range(100//20+1))))
43
>>>

Using Set Comprehension[edit]

Translation of: FSharp
#Wherein I observe that Set Comprehension is not intrinsically dysfunctional. Nigel Galloway: October 28th., 2018
n = {n for x in range(0,101,20) for y in range(x,101,9) for n in range(y,101,6)}
g = {n for n in range(101)}
print(max(g.difference(n)))
Output:
43

List monad[edit]

A composition of pure functions, including dropwhile, which shows a more verbose and unwieldy (de-sugared) route to list comprehension, and reveals the underlying mechanics of what the (compact and elegant) built-in syntax expresses. May help to build intuition for confident use of the latter.

Note that the innermost function wraps its results in a (potentially empty) list. The resulting list of lists, some empty, is then flattened by the concatenation component of bind.

Works with: Python version 3.7
'''mcNuggets list monad'''

from itertools import (chain, dropwhile)


# mcNuggetsByListMonad :: Int -> Set Int
def mcNuggetsByListMonad(limit):
    '''McNugget numbers up to limit.'''

    box = size(limit)
    return set(
        bind(
            box(6)
        )(lambda x: bind(
            box(9)
        )(lambda y: bind(
            box(20)
        )(lambda z: (
            lambda v=sum([x, y, z]): (
                [] if v > limit else [v]
            )
        )())))
    )


# Which, for comparison, is equivalent to:

# mcNuggetsByComprehension :: Int -> Set Int
def mcNuggetsByComprehension(limit):
    '''McNuggets numbers up to limit'''
    box = size(limit)
    return {
        v for v in (
            sum([x, y, z])
            for x in box(6)
            for y in box(9)
            for z in box(20)
        ) if v <= limit
    }


# size :: Int -> Int -> [Int]
def size(limit):
    '''Multiples of n up to limit.'''
    return lambda n: enumFromThenTo(0)(n)(limit)


# -------------------------- TEST --------------------------
def main():
    '''List monad and set comprehension - parallel routes'''

    def test(limit):
        def go(nuggets):
            ys = list(dropwhile(
                lambda x: x in nuggets,
                enumFromThenTo(limit)(limit - 1)(1)
            ))
            return str(ys[0]) if ys else (
                'No unreachable targets in this range.'
            )
        return lambda nuggets: go(nuggets)

    def fName(f):
        return f.__name__

    limit = 100
    print(
        fTable(main.__doc__ + ':\n')(fName)(test(limit))(
            lambda f: f(limit)
        )([mcNuggetsByListMonad, mcNuggetsByComprehension])
    )


# ------------------------ GENERIC -------------------------

# bind (>>=) :: [a] -> (a -> [b]) -> [b]
def bind(xs):
    '''List monad injection operator.
       Two computations sequentially composed,
       with any value produced by the first
       passed as an argument to the second.
    '''
    return lambda f: chain.from_iterable(
        map(f, xs)
    )


# enumFromThenTo :: Int -> Int -> Int -> [Int]
def enumFromThenTo(m):
    '''Integer values enumerated from m to n
       with a step defined by nxt-m.
    '''
    def go(nxt, n):
        d = nxt - m
        return range(m, n - 1 if d < 0 else 1 + n, d)
    return lambda nxt: lambda n: go(nxt, n)


# ------------------------ DISPLAY -------------------------

# fTable :: String -> (a -> String) ->
# (b -> String) -> (a -> b) -> [a] -> String
def fTable(s):
    '''Heading -> x display function -> fx display function ->
       f -> xs -> tabular string.
    '''
    def gox(xShow):
        def gofx(fxShow):
            def gof(f):
                def goxs(xs):
                    ys = [xShow(x) for x in xs]
                    w = max(map(len, ys))

                    def arrowed(x, y):
                        return y.rjust(w, ' ') + ' -> ' + fxShow(f(x))
                    return s + '\n' + '\n'.join(
                        map(arrowed, xs, ys)
                    )
                return goxs
            return gof
        return gofx
    return gox


# MAIN ---
if __name__ == '__main__':
    main()
Output:
List monad and set comprehension - parallel routes:

    mcNuggetsByListMonad -> 43
mcNuggetsByComprehension -> 43

Quackery[edit]

0 temp put
100 6 / times
  [ i 6 * 
    100 9 / times
      [ dup i 9 * +
        100 20 / times
          [ dup i 20 * +
            dup 101 < if
              [ dup bit
                temp take | temp put ]
            drop ] 
        drop ] 
    drop ]
-1 temp take
101 times
  [ dup i bit & 0 =
    if 
      [ nip i swap
        conclude ] ]
drop dup 0 < iff
  [ drop 
    say "There are no non-McNugget numbers below 101" ]
else
  [ say "The largest non-McNugget number below 101 is "
    echo ]
char . emit

Output: 

The largest non-McNugget number below 101 is 43.

R[edit]

Assuming that the natural numbers start at 0.

There are two natural approaches. The first is to generate all valid x, y, and z and then apply the function: 

allInputs <- expand.grid(x = 0:(100 %/% 6), y = 0:(100 %/% 9), z = 0:(100 %/% 20))
mcNuggets <- do.call(function(x, y, z) 6 * x + 9 * y + 20 * z, allInputs)

The second is to find all of the valid 6x, 9y, and 20z, and then sum them: 

mcNuggets2 <- rowSums(expand.grid(seq(0, 100, 6), seq(0, 100, 9), seq(0, 100, 20)))

Either way, we get identical results, as checked by: 

all(mcNuggets == mcNuggets2)

For our final answer, note that our choice to remove values from the vector 0:100 means our outputs will already be sorted, unique, and no greater than 100. 

results <- setdiff(0:100, mcNuggets)
cat("The non-McNuggets numbers that are no greater than 100 are:", results, "\nThe largest is", max(results), "\n")

Ultimately, this can be done in one line: 

max(setdiff(0:100, rowSums(expand.grid(seq(0, 100, 6), seq(0, 100, 9), seq(0, 100, 20)))))

However, using seq without naming its arguments is considered bad practice. It works here, but breaking this code up is probably a better idea.

Output:
> all(mcNuggets == mcNuggets2)
[1] TRUE
The non-McNuggets numbers that are no greater than 100 are: 1 2 3 4 5 7 8 10 11 13 14 16 17 19 22 23 25 28 31 34 37 43 

The largest is 43 
> max(setdiff(0:100, rowSums(expand.grid(seq(0, 100, 6), seq(0, 100, 9), seq(0, 100, 20)))))
[1] 43

Racket[edit]

Translation of: Python
(one of them) 
#lang racket
(apply max (set->list (for*/fold ((s (list->set (range 1 101))))
                                 ((x (in-range 0 101 20))
                                  (y (in-range x 101 9))
                                  (n (in-range y 101 6)))
                        (set-remove s n))))

Raku[edit]

(formerly Perl 6)

Works with: Rakudo version 2018.09

No hard coded limits, no hard coded values. General purpose 3 value solver. Count values may be any 3 different positive integers, in any order, that are relatively prime.

Finds the smallest count value, then looks for the first run of consecutive count totals able to be generated, that is at least the length of the smallest count size. From then on, every number can be generated by simply adding multiples of the minimum count to each of the totals in that run. 

sub Mcnugget-number (*@counts) {

    return '∞' if 1 < [gcd] @counts;

    my $min = min @counts;
    my @meals;
    my @min;

    for ^Inf -> $a {
        for 0..$a -> $b {
            for 0..$b -> $c {
                ($a, $b, $c).permutations.map: { @meals[ sum $_ Z* @counts ] = True }
            }
        }
        for @meals.grep: so *, :k {
            if @min.tail and @min.tail + 1 == $_ {
                @min.push: $_;
                last if $min == +@min
            } else {
                @min = $_;
            }
        }
        last if $min == +@min
    }
    @min[0] ?? @min[0] - 1 !! 0
}

for (6,9,20), (6,7,20), (1,3,20), (10,5,18), (5,17,44), (2,4,6), (3,6,15) -> $counts {
    put "Maximum non-Mcnugget number using {$counts.join: ', '} is: ",
        Mcnugget-number(|$counts)
}
Output:
Maximum non-Mcnugget number using 6, 9, 20 is: 43
Maximum non-Mcnugget number using 6, 7, 20 is: 29
Maximum non-Mcnugget number using 1, 3, 20 is: 0
Maximum non-Mcnugget number using 10, 5, 18 is: 67
Maximum non-Mcnugget number using 5, 17, 44 is: 131
Maximum non-Mcnugget number using 2, 4, 6 is: ∞
Maximum non-Mcnugget number using 3, 6, 15 is: ∞

REXX[edit]

This REXX version generalizes the problem (does not depend on fixed meal sizes),   and also checks for:

  •   a meal that doesn't include McNuggets   (in other words, zero nuggets)
  •   a meal size that includes a double order of nuggets
  •   a meal size that includes a single nugget   (which means, no largest McNugget number)
  •   excludes meals that have a multiple order of nuggets
  •   automatically computes the high value algebraically instead of using   100.
/*REXX pgm solves the  McNuggets problem:  the largest McNugget number for given meals. */
parse arg y                                      /*obtain optional arguments from the CL*/
if y='' | y=","  then y= 6 9 20                  /*Not specified?  Then use the defaults*/
say 'The number of McNuggets in the serving sizes of: '    space(y)
$=
#= 0                                             /*the Y list must be in ascending order*/
z=.
       do j=1  for words(y);      _= word(y, j)  /*examine  Y  list for dups, neg, zeros*/
       if _==1               then signal done    /*Value ≡ 1?  Then all values possible.*/
       if _<1                then iterate        /*ignore zero and negative # of nuggets*/
       if wordpos(_, $)\==0  then iterate        /*search for duplicate values.         */
            do k=1  for #                        /*   "    "  multiple     "            */
            if _//word($,k)==0  then iterate j   /*a multiple of a previous value, skip.*/
            end   /*k*/
       $= $ _;      #= # + 1;     $.#= _         /*add─►list; bump counter; assign value*/
       end        /*j*/
if #<2                     then signal done      /*not possible, go and tell bad news.  */
_= gcd($)        if _\==1  then signal done      /* "     "       "  "   "    "    "    */
if #==2   then z= $.1 * $.2  -  $.1  -  $.2      /*special case, construct the result.  */
if z\==.  then signal done
h= 0                                             /*construct a theoretical high limit H.*/
       do j=2  for #-1;  _= j-1;       _= $._;       h= max(h, _ * $.j  -  _  -  $.j)
       end   /*j*/
@.=0
       do j=1  for #;    _= $.j                  /*populate the  Jth + Kth   summand.   */
         do a=_  by _  to h;           @.a= 1    /*populate every multiple as possible. */
         end   /*s*/

         do k=1  for h;  if \@.k  then iterate
         s= k + _;       @.s= 1                  /*add two #s;   mark as being possible.*/
         end   /*k*/
       end     /*j*/

       do z=h  by -1  for h  until \@.z          /*find largest integer not summed.     */
       end     /*z*/
say
done:  if z==.  then say 'The largest McNuggets number not possible.'
                else say 'The largest McNuggets number is: '          z
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
gcd: procedure; $=;    do j=1  for arg();  $=$ arg(j);  end;  $= space($)
     parse var $ x $;     x= abs(x);
       do  while $\=='';  parse var $ y $;  y= abs(y);  if y==0  then iterate
         do  until y==0;  parse value  x//y  y   with   y  x;  end
       end;              return x
output   when using the default inputs:
The number of McNuggets in the serving sizes of:  6 9 20

The largest McNuggets number is:  43

Ring[edit]

Nuggets = list(100)

for six = 0 To 100/6
    for nine =  0 To 100/9
        for twenty = 0 To 100/20
            n = six*6 + nine*9 + twenty*20
            If n <= 100 and not (six = 0 and nine = 0 and twenty = 0)
               Nuggets[n] = true 
            ok
        next
    next
next

for n = 100 to 1 step -1
    if Nuggets[n] = false
       ? "Maximum non-McNuggets number is: " + n
       exit
    ok
next
Output:
Maximum non-McNuggets number is: 43

Ruby[edit]

Translation of: Go
def mcnugget(limit)
  sv = (0..limit).to_a

  (0..limit).step(6) do |s|
    (0..limit).step(9) do |n|
      (0..limit).step(20) do |t|
        sv.delete(s + n + t)
      end
    end
  end

  sv.max
end

puts(mcnugget 100)
Output:
43

Generic solution, allowing for more or less then 3 portion-sizes: 

limit = 100
nugget_portions = [6, 9, 20]

arrs = nugget_portions.map{|n| 0.step(limit, n).to_a }
hits = arrs.pop.product(*arrs).map(&:sum)
p ((0..limit).to_a - hits).max # => 43

Rust[edit]

No hard limits. Generalization of Rødseth’s Algorithm explained in post. Working code: Rust playground. 

fn main() {
    let test_cases = vec![
        [6, 9, 20],
        [12, 14, 17],
        [12, 13, 34],
        [5, 9, 21],
        [10, 18, 21],
        [71, 98, 99],
        [7_074_047, 8_214_596, 9_098_139],
        [582_795_988, 1_753_241_221, 6_814_151_015],
        [4, 30, 16],
        [12, 12, 13],
        [6, 15, 1],
    ];
    for case in &test_cases {
        print!("g({}, {}, {}) = ", case[0], case[1], case[2]);
        println!(
            "{}",
            match frobenius(case.to_vec()) {
                Ok(g) => format!("{}", g),
                Err(e) => e,
            }
        );
    }
}

fn frobenius(unsorted_a: Vec<i64>) -> Result<i64, String> {
    let mut a = unsorted_a;
    a.sort();
    assert!(a[0] >= 1);
    if gcd(gcd(a[0], a[1]), a[2]) > 1 {
        return Err("Undefined".to_string());
    }
    let d12 = gcd(a[0], a[1]);
    let d13 = gcd(a[0] / d12, a[2]);
    let d23 = gcd(a[1] / d12, a[2] / d13);
    let mut a_prime = vec![a[0] / d12 / d13, a[1] / d12 / d23, a[2] / d13 / d23];
    a_prime.sort();
    let rod = if a_prime[0] == 1 {
        -1
    } else {
        // Rødseth’s Algorithm
        let mut a1 = a_prime[0];
        let mut s0 = congruence(a_prime[1], a_prime[2], a_prime[0]);
        let mut s = vec![a1];
        let mut q: Vec<i64> = vec![];
        while s0 != 0 {
            s.push(s0);
            let s1 = if s0 == 1 { 0 } else { s0 - (a1 % s0) };
            let q1 = (a1 + s1) / s0;
            q.push(q1);
            a1 = s0;
            s0 = s1;
        }
        let mut p = vec![0, 1];
        let mut r = (s[1] * a_prime[1] - p[1] * a_prime[2]) / a_prime[0];
        let mut i = 1;
        while r > 0 {
            let p_next = q[i - 1] * p[i] - p[i - 1];
            p.push(p_next);
            r = (s[i + 1] * a_prime[1] - p_next * a_prime[2]) / a_prime[0];
            i += 1;
        }
        let v = i - 1;
        -a_prime[0] + a_prime[1] * (s[v] - 1) + a_prime[2] * (p[v + 1] - 1)
            - (a_prime[1] * s[v + 1]).min(a_prime[2] * p[v])
    };
    Ok(rod * d12 * d13 * d23 + a[0] * (d23 - 1) + a[1] * (d13 - 1) + a[2] * (d12 - 1))
}

fn gcd(a: i64, b: i64) -> i64 {
    if b == 0 {
        a
    } else {
        gcd(b, a % b)
    }
}

fn congruence(a: i64, c: i64, m: i64) -> i64 {
    // Solves ax ≡ c mod m
    let aa = a % m;
    let cc = (c + a * m) % m;
    if aa == 1 {
        cc
    } else {
        let y = congruence(m, -cc, aa);
        (m * y + cc) / aa
    }
}
Output:
g(6, 9, 20) = 43
g(12, 14, 17) = 61
g(12, 13, 34) = 79
g(5, 9, 21) = 22
g(10, 18, 21) = 65
g(71, 98, 99) = 1307
g(7074047, 8214596, 9098139) = 48494282357
g(582795988, 1753241221, 6814151015) = 173685179295403
g(4, 30, 16) = Undefined
g(12, 12, 13) = 131
g(6, 15, 1) = -1

Swift[edit]

func maxNugget(limit: Int) -> Int {
  var (max, sixes, nines, twenties, i) = (0, 0, 0, 0, 0)

  mainLoop: while i < limit {
    sixes = 0

    while sixes * 6 < i {
      if sixes * 6 == i {
        i += 1
        continue mainLoop
      }

      nines = 0

      while nines * 9 < i {
        if sixes * 6 + nines * 9 == i {
          i += 1
          continue mainLoop
        }

        twenties = 0

        while twenties * 20 < i {
          if sixes * 6 + nines * 9 + twenties * 20 == i {
            i += 1
            continue mainLoop
          }

          twenties += 1
        }

        nines += 1
      }

      sixes += 1
    }

    max = i
    i += 1
  }

  return max
}

print(maxNugget(limit: 100))
Output:
43

Tailspin[edit]

templates largestNonMcNuggetNumber
  @: { largest: 0, mcNuggetNumbers: [1..$+20 -> 0] };
  @.mcNuggetNumbers([6,9,20]): 1..3 -> 1;
  1..$ -> #
  $@.largest !
  when <?($@.mcNuggetNumbers($) <=0>)> do @.largest: $;
  otherwise @.mcNuggetNumbers([$ + 6, $ + 9, $ + 20]): 1..3 -> 1;
end largestNonMcNuggetNumber

100 -> largestNonMcNuggetNumber -> !OUT::write
Output:
43

UNIX Shell[edit]

Translation of: Clojure
Works with: bash
Works with: ksh
Works with: zsh
possible=()
for (( i=0; i<18; ++i )); do
  for (( j=0; j<13; ++j )); do
    for (( k=0; k<6; ++k )); do
      (( n = i*6 + j*9 + k*20 ))
      if (( n )); then
        possible[n]=1
      fi
    done
  done
done

for (( n=100; n; n-- )); do
  if [[ -n ${possible[n]} ]; then
    continue
  fi
  break
done

printf 'Maximum non-McNuggets number is %d\n' $n
Output:
Maximum non-McNuggets number is 43
Works with: sh
possible=
i=0
while [ $i -lt 18 ]; do
  j=0
  while [ $j -lt 13 ]; do
    k=0
    while [ $k -lt 6 ]; do
      possible="${possible+$possible }"`expr $i \* 6 + $j \* 9 + $k \* 20`
      k=`expr $k + 1`
    done
    j=`expr $j + 1`
  done
  i=`expr $i + 1`
done

n=100
while [ $n -gt 0 ]; do
  if echo "$possible" | tr ' ' '\n' | fgrep -qx $n; then
    n=`expr $n - 1`
    continue
  fi
  break
done
echo "Maximum non-McNuggets number is $n"
Output:
Maximum non-McNuggets number is 43

V (Vlang)[edit]

Translation of: Go
fn mcnugget(limit int) {
    mut sv := []bool{len: limit+1} // all false by default
    for s := 0; s <= limit; s += 6 {
        for n := s; n <= limit; n += 9 {
            for t := n; t <= limit; t += 20 {
                sv[t] = true
            }
        }
    }
    for i := limit; i >= 0; i-- {
        if !sv[i] {
            println("Maximum non-McNuggets number is $i")
            return
        }
    }
}
 
fn main() {
   mcnugget(100)
}
Output:
Maximum non-McNuggets number is 43

VTL-2[edit]

10 N=0
20 :N+1)=0
30 N=N+1
40 #=100>N*20
50 A=0
60 B=A
70 C=B
80 :C+1)=160
90 C=C+20
100 #=100>C*80
110 B=B+9
120 #=100>B*70
130 A=A+6
140 #=100>A*60
150 N=101
160 N=N-1
170 #=:N+1)
180 ?="Largest non-McNuggets number: ";
190 ?=N
Output:
Largest non-McNuggets number: 43

Wren[edit]

Translation of: Go
var mcnugget = Fn.new { |limit|
    var sv = List.filled(limit+1, false)
    var s = 0
    while (s <= limit) {
        var n = s
        while (n <= limit) {
            var t = n
            while (t <= limit) {
                sv[t] = true
                t = t + 20
            }
            n = n + 9
        }
        s = s + 6
    }
    for (i in limit..0) {
        if (!sv[i]) {
            System.print("Maximum non-McNuggets number is %(i)")
            return
        }
    }
}

mcnugget.call(100)
Output:
Maximum non-McNuggets number is 43

XPL0[edit]

int N, A(101), X, Y, Z;
[for N:= 0 to 100 do A(N):= false;
for X:= 0 to 100/6 do
    for Y:= 0 to 100/9 do
        for Z:= 0 to 100/20 do
            [N:= 6*X + 9*Y + 20*Z;
            if N <= 100 then A(N):= true;
            ];
for N:= 100 downto 0 do
    if A(N) = false then
        [IntOut(0, N);
        exit;
        ];
]
Output:
43

zkl[edit]

Translation of: Python
nuggets:=[0..101].pump(List());	// (0,1,2,3..101), mutable
foreach s,n,t in ([0..100/6],[0..100/9],[0..100/20])
   { nuggets[(6*s + 9*n + 20*t).min(101)]=0 }
println((0).max(nuggets));
Output:
43
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