# Induction of the N-Block Game!

The following problem is about a game played with wooden blocks stacked vertically.

The way the game is played is as so:

1. There is a stack of n blocks available on a table in front of you;
2. With every move you make you can split the stack into two piles which, under your choice, can be equal or not;
3. After every move, your score is increased based on the product of the sizes of the two new stacks;
4. The game is played until all stacks are of size 1;
5. The sum of all the products is the total score.

The following is a simulation of 8 blocks:

$$\boxed{8\quad \longrightarrow \quad { 4 }_{ 1 }\quad and\quad { 4 }_{ 2 }\quad =\quad (4\quad \cdot \quad 4)\quad =\quad 16\\ { 4 }_{ 1 }\quad \longrightarrow \quad 3\quad and\quad 1\quad =\quad (3\quad \cdot \quad 1)\quad =\quad 3\\ { 4 }_{ 2 }\quad \longrightarrow \quad { 2 }_{ 1 }\quad and\quad { 2 }_{ 2 }\quad =\quad (2\quad \cdot \quad 2)\quad =\quad 4\\ 3\quad \longrightarrow \quad { 2 }_{ 3 }\quad and\quad 1\quad =\quad (2\quad \cdot \quad 1)\quad =\quad 2\\ { 2 }_{ 1 }\quad \longrightarrow \quad 1\quad and\quad 1\quad =\quad (1\quad \cdot \quad 1)\quad =\quad 1\\ { 2 }_{ 2 }\quad \longrightarrow \quad 1\quad and\quad 1\quad =\quad (1\quad \cdot \quad 1)\quad =\quad 1\\ { 2 }_{ 3 }\quad \longrightarrow \quad 1\quad and\quad 1\quad =\quad (1\quad \cdot \quad 1)\quad =\quad 1\\ \\ Total\quad Score\quad =\quad 16+3+4+2+1+1+1\quad =\quad 28}$$

Let S(n) represent the maximum score one could possibly achieve playing this game with n blocks.

What is S(2014)?

×