Is there a largest number that is impossible to achieve by combining 6s, 9s and 20s? You might combine 6 and 9 to get 15, but it's impossible to get 16, or 17 and so on. Is there a largest number you can't achieve by adding some combination of only these 3 numbers? If there's not how do you prove it?
These are the "non-McNugget numbers", of which the largest is 43.
Consider the following cases:


$A_{44}=\{6,6,6,6,20\}$
$A_{45}=\{6,6,6,6,6,6,9\}$
$A_{46}=\{6,20,20\}$
$A_{47}=\{6,6,6,9,20\}$
$A_{48}=\{6,6,6,6,6,6,6,6\}$
$A_{49}=\{9,20,20\}$
$A_{n+6}=A_{n}\cup\{6\}$


So every $n\geq44$ can be represented as a sum of $6$s, $9$s and $20$s.

Now you just need to show that $43$ cannot, and you're done.

Here is a short Python script for asserting this:

def func(val,arr):
    if val &gt; 0:
        for k in arr:
            if val &gt;= k and func(val-k,arr):
                return True
    return val == 0

print func(43,[6,9,20])