McNuggets Problem

From Rosetta Code
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).

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 0 BY 9 TO max nuggets DO
FOR z FROM 0 BY 20 TO max nuggets DO
INT nuggets = x + y + z;
IF nuggets <= max nuggets THEN sum[ nuggets ] := TRUE FI
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 largest := i; sum[ i ] DO SKIP OD;
print( ( "The largest non McNugget number is: "
, whole( largest, 0 )
, newline
)
)
END
Output:
The largest non McNugget number is: 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

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

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

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.

Dyalect[edit]

Translation of: Go
func mcnugget(limit) {
var sv = Array.empty(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 += 20
}
n += 9
}
s += 6
}
for i in limit..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 ] [email protected] ] bag-of diff last .
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

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)
 
gaps :: [Int]
gaps = dropWhile (`member` mcNuggets) [100,99 .. 1]
 
mcNuggets :: Set Int
mcNuggets =
let size = enumFromTo 0 . quot 100
in fromList $
size 6 >>=
\x ->
size 9 >>=
\y ->
size 20 >>=
\z ->
let v = sum [6 * x, 9 * y, 20 * z]
in [ v
| 101 > v ]
 
main :: IO ()
main =
print $
case gaps of
x:_ -> show x
[] -> "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

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

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)-.,+/&>{(* [email protected]>[email protected]%~&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.

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

Mathematica[edit]

Complement[Range[100], 
Select[6 #[[1]] + 9 #[[2]] + 20 #[[3]] & /@
Tuples[Range[0, 17], 3], # < 101 &]][[-1]]
Output:
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
----------
==========

Perl[edit]

Translation of: Perl 6
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

Perl 6[edit]

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 == [email protected]min
} else {
@min = $_;
}
}
last if $min == [email protected]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: ∞

Phix[edit]

Translation of: Go
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: Perl_6
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
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 %s is: %s\n",{sprint(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

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

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) ) )

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 ABSTRACTIONS ------------------------------------
 
# 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: list(
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: list(go(nxt, n))
 
 
# FORMATTING ----------------------------------------------
 
# 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 go(xShow, fxShow, f, xs):
ys = [xShow(x) for x in xs]
w = max(map(len, ys))
return s + '\n' + '\n'.join(map(
lambda x, y: y.rjust(w, ' ') + ' -> ' + fxShow(f(x)),
xs, ys
))
return lambda xShow: lambda fxShow: lambda f: lambda xs: go(
xShow, fxShow, f, xs
)
 
 
# MAIN ---
if __name__ == '__main__':
main()
Output:
List monad and set comprehension - parallel routes:

    mcNuggetsByListMonad -> 43
mcNuggetsByComprehension -> 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))))

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

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..$+20 -> 0] };
@.mcNuggetNumbers([6,9,20]): 1..3 -> 1;
1..$ -> #
[email protected] !
<?([email protected]($) <0>)> @.largest: $;
<> @.mcNuggetNumbers([$ + 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

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