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).

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

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

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

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)
import Data.List (uncons)
 
gaps :: [Int]
gaps = dropWhile (`member` mcNuggets) [100,99 .. 1]
 
mcNuggets :: Set Int
mcNuggets =
fromList $
[0 .. quot 100 6] >>=
\x ->
[0 .. quot 100 9] >>=
\y ->
[0 .. quot 100 20] >>=
\z ->
let v = sum [6 * x, 9 * y, 20 * z]
in [ v
| 101 > v ]
 
main :: IO ()
main =
print $
case uncons gaps of
Just (x, _) -> show x
Nothing -> "No unreachable quantities found ..."
43

JavaScript[edit]

(() => {
'use strict';
 
// main :: IO ()
const main = () => {
 
const
upTo = enumFromTo(0),
nuggets = new Set(
bindList(
upTo(quot(100, 6)),
x => bindList(
upTo(quot(100, 9)),
y => bindList(
upTo(quot(100, 20)),
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 ----------------------------------
 
// bindList (>>=) :: [a] -> (a -> [b]) -> [b]
const bindList = (xs, mf) => [].concat.apply([], xs.map(mf));
 
// 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.

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

Perl[edit]

Translation of: Perl 6
Library: ntheory
use ntheory qw/forperm vecall vecmin/;
 
sub Mcnugget_number {
my $counts = shift;
 
return 'No maximum' if vecall { 0 == $_%2 } @$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 $counts ([6,9,20], [6,7,20], [1,3,20], [10,5,18], [5,17,44], [2,4,6]) {
print 'Maximum non-Mcnugget number using ' . join(', ', @$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

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, though if they are all even, no maximum non-McNugget number is possible (can't form odd numbers).

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 'No maximum' if so all @counts »%%» 2;
 
my $min = [min] @counts;
my @meals;
my @min;
 
for ^Inf -> $a {
for 0..$a -> $b {
for 0..$b -> $c {
($a, $b, $c).permutations.map: {
for flat $_ Z* @counts {
@meals[sum $^first, $^second, $^third] = 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) -> $counts {
put "Maximum non-Mcnugget number using {$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: No maximum

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


Equivalently, a composition of pure functions, including dropwhile, which shows a more verbose and unwieldy (de-sugared) route to set 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 concatMap.

from itertools import (chain, dropwhile)
 
 
def main():
upTo = enumFromTo(0)
mcNuggets = set(
concatMap(
lambda x:
concatMap(
lambda y:
concatMap(
lambda z: (
lambda v=sum([6 * x, 9 * y, 20 * z]): (
[v] if 101 > v else []
)
)()
)(upTo(100 // 20))
)(upTo(100 // 9))
)(upTo(100 // 6))
)
xs = list(dropwhile(
lambda x: x in mcNuggets,
enumFromThenTo(100)(99)(1))
)
print(
xs[0] if xs else 'No unreachable quantities found in this range.'
)
 
 
# GENERIC ABSTRACTIONS ------------------------------------
 
# concatMap :: (a -> [b]) -> [a] -> [b]
def concatMap(f):
return lambda xs: list(chain.from_iterable(map(f, xs)))
 
 
# enumFromThenTo :: Int -> Int -> Int -> [Int]
def enumFromThenTo(m):
return lambda next: lambda n: (
list(range(m, 1 + n, next - m))
)
 
 
# enumFromTo :: Int -> Int -> [Int]
def enumFromTo(m):
return lambda n: list(range(m, 1 + n))
 
 
if __name__ == '__main__':
main()
Output:
43

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

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