McNugget Number
A McNugget number is a positive integer that can be obtained by adding together orders of McDonald's® Chicken McNuggetsTM (prior to consuming any), which originally came in boxes of 6, 9, and 20 (Vardi 1991, pp. 19-20 and 233-234; Wah and Picciotto 1994, p. 186). All integers are McNugget numbers except 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 28, 31, 34, 37, and 43. The value 43 therefore corresponds to the Frobenius number of ![<span style=]()
{6,9,20}" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline1.gif" style="height:15px; width:54px" />.
Since the Happy MealTM-sized nugget box (4 to a box) can now be purchased separately, the modern McNugget numbers are linear combinations of 4, 6, 9, and 20. These new-fangled numbers are much less interesting than before, with only 1, 2, 3, 5, 7, and 11 remaining as non-McNugget numbers. The value 11 therefore corresponds to the Frobenius number of ![<span style=]()
{4,6,9,20}" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline2.gif" style="height:15px; width:69px" />.
The greedy algorithm can be used to find a McNugget expansion of a given integer 
. This can also be done in the Wolfram Language using FrobeniusSolve[![<span style=]()
{" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline4.gif" style="height:15px; width:5px" />6, 9, 20![<span style=]()
}" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline5.gif" style="height:15px; width:5px" />, n]. The following table summarizes (classic) McNugget expansions for small integers.
 |
McNugget expansions |
| 6 |
![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline7.gif" style="height:15px; width:5px" />![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline8.gif" style="height:15px; width:5px" />1,0,0![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline9.gif" style="height:15px; width:5px" />![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline10.gif" style="height:15px; width:5px" /> |
| 9 |
![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline11.gif" style="height:15px; width:5px" />![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline12.gif" style="height:15px; width:5px" />0,1,0![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline13.gif" style="height:15px; width:5px" />![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline14.gif" style="height:15px; width:5px" /> |
| 12 |
![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline15.gif" style="height:15px; width:5px" />![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline16.gif" style="height:15px; width:5px" />2,0,0![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline17.gif" style="height:15px; width:5px" />![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline18.gif" style="height:15px; width:5px" /> |
| 15 |
![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline19.gif" style="height:15px; width:5px" />![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline20.gif" style="height:15px; width:5px" />1,1,0![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline21.gif" style="height:15px; width:5px" />![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline22.gif" style="height:15px; width:5px" /> |
| 18 |
![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline23.gif" style="height:15px; width:5px" />![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline24.gif" style="height:15px; width:5px" />0,2,0![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline25.gif" style="height:15px; width:5px" />,![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline26.gif" style="height:15px; width:5px" />3,0,0![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline27.gif" style="height:15px; width:5px" />![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline28.gif" style="height:15px; width:5px" /> |
| 20 |
![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline29.gif" style="height:15px; width:5px" />![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline30.gif" style="height:15px; width:5px" />0,0,1![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline31.gif" style="height:15px; width:5px" />![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline32.gif" style="height:15px; width:5px" /> |
| 21 |
![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline33.gif" style="height:15px; width:5px" />![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline34.gif" style="height:15px; width:5px" />2,1,0![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline35.gif" style="height:15px; width:5px" />![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline36.gif" style="height:15px; width:5px" /> |
| 24 |
![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline37.gif" style="height:15px; width:5px" />![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline38.gif" style="height:15px; width:5px" />1,2,0![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline39.gif" style="height:15px; width:5px" />,![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline40.gif" style="height:15px; width:5px" />4,0,0![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline41.gif" style="height:15px; width:5px" />![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline42.gif" style="height:15px; width:5px" /> |
| 26 |
![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline43.gif" style="height:15px; width:5px" />![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline44.gif" style="height:15px; width:5px" />1,0,1![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline45.gif" style="height:15px; width:5px" />![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline46.gif" style="height:15px; width:5px" /> |
| 27 |
![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline47.gif" style="height:15px; width:5px" />![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline48.gif" style="height:15px; width:5px" />0,3,0![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline49.gif" style="height:15px; width:5px" />,![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline50.gif" style="height:15px; width:5px" />3,1,0![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline51.gif" style="height:15px; width:5px" />![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline52.gif" style="height:15px; width:5px" /> |
| 29 |
![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline53.gif" style="height:15px; width:5px" />![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline54.gif" style="height:15px; width:5px" />0,1,1![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline55.gif" style="height:15px; width:5px" />![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline56.gif" style="height:15px; width:5px" /> |
| 30 |
![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline57.gif" style="height:15px; width:5px" />![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline58.gif" style="height:15px; width:5px" />2,2,0![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline59.gif" style="height:15px; width:5px" />,![<span style=]() {" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline60.gif" style="height:15px; width:5px" />5,0,0![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline61.gif" style="height:15px; width:5px" />![<span style=]() }" src="https://mathworld.wolfram.com/images/equations/McNuggetNumber/Inline62.gif" style="height:15px; width:5px" /> |
REFERENCES:
Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 19-20 and 233-234, 1991.
Wagon, S. "Greedy Coins." https://library.wolfram.com/infocenter/MathSource/5187/.
Wah, A. and Picciotto, H. Lesson 5.8, Problem 1 in Algebra Themes, Tools and Concepts. Mountain View, CA: Creative Publications, p. 186, 1994.
Wilson, D. rec.puzzles newsgroup posting, March 20, 1990.
