Radio Astronomy Systems Research Group

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===== Answer =====

The largest number of McNuggets that you cannot purchase is 43.

There is a simple way to find this number. Write down a list of number starting at 1.

|1|2|3|4|5|6|7|8|9|10|
|11|12|13|14|15|16|17|18|19|20|
|21|22|23|24|25|26|27|28|29|30|
|31|32|33|34|35|36|37|38|39|40|
|41|42|43|44|45|46|47|48|49|50|

Mark off the numbers you know you can make with only one of each box

|1|2|3|4|5|X|7|8|X|10|
|11|12|13|14|15|16|17|18|19|X|
|21|22|23|24|25|26|27|28|29|30|
|31|32|33|34|35|36|37|38|39|40|
|41|42|43|44|45|46|47|48|49|50|

Now move to the smallest number you know you can make (6), add one box of each 6, 9, and 20 and mark off these values.

|1|2|3|4|5|X|7|8|X|10|
|11|X|13|14|X|16|17|18|19|X|
|21|22|23|24|25|X|27|28|29|30|
|31|32|33|34|35|36|37|38|39|40|
|41|42|43|44|45|46|47|48|49|50|

Now go to the next lowest number you know you can make (9) and add one of each box again.

|1|2|3|4|5|X|7|8|X|10|
|11|X|13|14|X|16|17|X|19|X|
|21|22|23|24|25|X|27|28|X|30|
|31|32|33|34|35|36|37|38|39|40|
|41|42|43|44|45|46|47|48|49|50|

Continue this process for a while...

|1|2|3|4|5|X|7|8|X|10|
|11|X|13|14|X|16|17|X|19|X|
|X|22|23|X|25|X|X|28|X|X|
|31|X|X|34|X|X|37|X|X|X|
|X|X|43|X|X|X|X|X|X|X|
|X|X|X|X|X|X|X|X|X|X|

And you'll end up with all the number you know you can make marked.

Notice that if you ever get six consecutive numbers that you know you can make, you can add one box of 6 to each of these and you will be able to make any other larger number. Since we have the values 44-49, we can make any number larger than 43.

Radio Astronomy Systems Research Group

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