This is not an eating competition, instead you have to think.

## The Challenge

Chicken McNuggets can be ordered in the sizes of 6, 9 and 20 pieces. Being allowed to use a combination of the sizes, what is the highest number that you can NOT order?

Example: 98 pieces can be ordered using a combination (4×20 pieces + 2×9 pieces). 14 pieces cannot be ordered (no combination of 6, 9 and 20 equals 14)

## Solution

To solve this we created a small optimisation model that for a proposed number verifies if it is possible to order that quantity.

- The variables are the number of 6-packs, 9-packs and 20-packs respectively.
- The objective is to maximise the total number of nuggets ordered.
- The total number of nuggets ordered is constrained to not be larger than a quantity currently investigated
- In a loop the program tests order quantities in a decreasing order until the maximum order is not equal the constraint/proposed quantity.

## Example Code

## Console Output

class TheMcNuggetsChallenge ojAlgo 2019-05-09 OPTIMAL 100.0 @ { 0, 0, 5 } OPTIMAL 99.0 @ { 5, 1, 3 } OPTIMAL 98.0 @ { 0, 2, 4 } OPTIMAL 97.0 @ { 8, 1, 2 } OPTIMAL 96.0 @ { 0, 4, 3 } OPTIMAL 95.0 @ { 1, 1, 4 } OPTIMAL 94.0 @ { 0, 6, 2 } OPTIMAL 93.0 @ { 4, 1, 3 } OPTIMAL 92.0 @ { 2, 0, 4 } OPTIMAL 91.0 @ { 4, 3, 2 } OPTIMAL 90.0 @ { 2, 2, 3 } OPTIMAL 89.0 @ { 0, 1, 4 } OPTIMAL 88.0 @ { 8, 0, 2 } OPTIMAL 87.0 @ { 0, 3, 3 } OPTIMAL 86.0 @ { 1, 0, 4 } OPTIMAL 85.0 @ { 0, 5, 2 } OPTIMAL 84.0 @ { 4, 0, 3 } OPTIMAL 83.0 @ { 0, 7, 1 } OPTIMAL 82.0 @ { 7, 0, 2 } OPTIMAL 81.0 @ { 2, 1, 3 } OPTIMAL 80.0 @ { 0, 0, 4 } OPTIMAL 79.0 @ { 2, 3, 2 } OPTIMAL 78.0 @ { 0, 2, 3 } OPTIMAL 77.0 @ { 8, 1, 1 } OPTIMAL 76.0 @ { 0, 4, 2 } OPTIMAL 75.0 @ { 1, 1, 3 } OPTIMAL 74.0 @ { 0, 6, 1 } OPTIMAL 73.0 @ { 4, 1, 2 } OPTIMAL 72.0 @ { 2, 0, 3 } OPTIMAL 71.0 @ { 4, 3, 1 } OPTIMAL 70.0 @ { 5, 0, 2 } OPTIMAL 69.0 @ { 0, 1, 3 } OPTIMAL 68.0 @ { 5, 2, 1 } OPTIMAL 67.0 @ { 0, 3, 2 } OPTIMAL 66.0 @ { 1, 0, 3 } OPTIMAL 65.0 @ { 0, 5, 1 } OPTIMAL 64.0 @ { 4, 0, 2 } OPTIMAL 63.0 @ { 0, 7, 0 } OPTIMAL 62.0 @ { 7, 0, 1 } OPTIMAL 61.0 @ { 2, 1, 2 } OPTIMAL 60.0 @ { 0, 0, 3 } OPTIMAL 59.0 @ { 2, 3, 1 } OPTIMAL 58.0 @ { 0, 2, 2 } OPTIMAL 57.0 @ { 5, 3, 0 } OPTIMAL 56.0 @ { 0, 4, 1 } OPTIMAL 55.0 @ { 1, 1, 2 } OPTIMAL 54.0 @ { 0, 6, 0 } OPTIMAL 53.0 @ { 4, 1, 1 } OPTIMAL 52.0 @ { 2, 0, 2 } OPTIMAL 51.0 @ { 4, 3, 0 } OPTIMAL 50.0 @ { 5, 0, 1 } OPTIMAL 49.0 @ { 0, 1, 2 } OPTIMAL 48.0 @ { 8, 0, 0 } OPTIMAL 47.0 @ { 0, 3, 1 } OPTIMAL 46.0 @ { 1, 0, 2 } OPTIMAL 45.0 @ { 0, 5, 0 } OPTIMAL 44.0 @ { 4, 0, 1 } Not possible to order 43 nuggets

If you believe there is a number greater than 100 that cannot be ordered, then you need to modify and run the code yourself.