A restaurant (KFX) serves chicken nuggets in 6, 9 or 20 packs. Show that you can buy $n$ nuggets if $n>43$.
I solved this problem in the following elementary way: $$44=6+9+9+20,$$ $$45=9+9+9+9+9,$$ $$46=6+20+20,$$ $$47=9+9+9+20,$$ $$48=6+6+9+9+9+9,$$ $$49=9+20+20$$ But then, by adding $6$ packs to these, I can order any $n>49$ too.
What is the significance of 43 or 44 here? Can you solve this question in more theoretical way?
Thanks in advance.
The generating function which calculates the number of ways for ordering nuggets is $$g(x)=\frac{1}{(1-x^6)(1-x^9)(1-x^{20})}$$ Upto 300th order its series expansion at $x=0$:
43 is the largest power term whose coefficient is zero. After 43 there is no other skipping. $+2x^{44}$ term means there are 2 ways to order 44 nuggets: 6+9+9+20 and 6+6+6+6+20.