Packets of Buttons

The function $buttons$ has $8$ base cases which are the minimum number of packets to purchase for $n$ buttons where $0\leqslant{n}\leqslant{7}$.

$L(0) = 0$

$L(1) = 1$

$L(2) = 2$

$L(3) = 3$

$L(4) = 1$

$L(5) = 2$

$L(6) = 3$

$L(7) = 1$

For a number of buttons $n>7$, the vector $L$ is appended with the minimum value of the base cases $L$ after having subtracted the packet sizes $1,\ 4,\&\ 7$ from $n$, incrementing the total number of packets $i$ by $1$ in each iteration with the condition that $i\leqslant{n}$.

It then returns the final value of the vector $L$.

The Greedy Method

Alternatively, a greedy algorithm does actually works for $n=20$ buttons although it will not work for all values of $n$. The function $greedyButtons$ calculates the minimum number of packets required to purchase for $n$ buttons with $3$ base cases:

$n>=7$

$n>=4$

$n>=1$

The function returns a vector with the required number of each sized packet. The sum of this vector is the total minimum number of packets required to purchase to have $20$ buttons.