## McNugget Numbers

### December 9, 2011

Since 6, 9, and 20 are coprime and their least common multiple is 180, we know that any number larger than 180 is a McNugget number. To find the list of numbers less than 180 that are McNugget numbers we calculate as follows:

`(define non-mcnuggets`

(sort <

(list-minus

(range 181)

(unique =

(list-of m

(x range (+ (/ 180 6) 1))

(y range (+ (/ 180 9) 1))

(z range (+ (/ 180 20) 1))

(m is (+ (* 6 x) (* 9 y) (* 20 z)))

(<= m 180))))))

The list comprehension computes *m*=6*x*+9*y*+20*z* for all combinations of *x*, *y* and *z* that could be less than 180, then `list-minus`

figures those numbers that are not on the list. Here is `list-minus`

:

`(define (list-minus xs ys)`

(let loop ((xs xs) (zs (list)))

(cond ((null? xs) zs)

((member (car xs) ys) (loop (cdr xs) zs))

(else (loop (cdr xs) (cons (car xs) zs))))))

The list of non-McNugget numbers is 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 28, 31, 34, 37, and 43. If you’re interested in the math, the Frobenius number of the set {6 9 20} is 43, which is the largest non-McNugget number; Google will point you to more.

We used list comprehensions, unique and range from the Standard Prelude. You can run the program at http://programmingpraxis.codepad.org/01h2PiHK.

[…] today’s Programming Praxis exercise, our goal is to determine all the numbers that are not McNugget […]

My Haskell solution (see http://bonsaicode.wordpress.com/2011/12/09/programming-praxis-mcnugget-numbers/ for a version with comments):

[…] response to this challenge McNugget numbers I wrote C code based on a different approach than exhibited in the LISP solution and obviously […]

My independent C solution: http://ctopy.wordpress.com/2011/12/09/mcnugget-numbers-c-solution/

[…] I heard from my friend about the new programming praxis problem I started thinking how to implement the solution. I wrote down the numbers from 1 to 20 and […]

Python solution along the same lines as the C one, but self-documenting,

My thinking is described here: http://ctopy.wordpress.com/2011/12/09/mcnugget-numbers-python-solution/

I’ve tried to make a Haskell solution using somewhat less mathematical knowlegde (for example the one that we only have to check the numbers smaller than 180).

The ugly function mc calculates the infinite list of McNugget numbers in increasing order. For example the first 50:

And notmc gives the solution /it stops if 6 consecutive McNugget numbers are found, because each larger number can be generated easily by adding 6 to them repeatedly/:

Go version here: http://play.golang.org/p/WwY0X0PE3V

And a haskell copy of the popular solution of Tomasz Kwiatkowskis:

[/sourcecode]

notmc = helper [0] [1..]

where helper _ [] = []

helper is (x:xs)

| length is > 5 && head is – 5 == is!!5 = []

| otherwise = if any (\a-> elem a is) [x-6,x-9,x-20]

then helper (x:is) xs

else x:helper is xs

[/sourcecode]

Probably not the most efficient solution, but in relatively short Python:

The use of the procedures from the

`itertools`

module ensures no intermediate lists are constructed.I still don’t get why 181 or 182 is not a mcnugget number. You it can not summed using 6,9,20

22*6 + 1*9 + 2*20 = 181

24*6 + 2*9 + 1*20 = 182

Java approach:

Please I have a question … How it is determined that all numbers after LCM “180” are MCNuggets number?