Question: Here is an interesting game. You start with an empty 4x4 grid. At each turn you can choose an empty cell and place a value in it. The placed value is given by the following rules:

* If the chosen cell has no neighboring (horizontal or vertical) values then the placed value is 1.
* Otherwise the placed value is the sum of all neighboring (horizontal or vertical) values.

What is the largest value that you can achieve in this game?

Answer: 
Here is my answer:

> 
>  38
> 
> 
> 

as here

> 
>  [![enter image description here](https://i.stack.imgur.com/zoihu.jpg)](https://i.stack.imgur.com/zoihu.jpg)
> 
> 
> 

the order is as below

> 
>  [![enter image description here](https://i.stack.imgur.com/NTZeg.jpg)](https://i.stack.imgur.com/NTZeg.jpg)
> 
> 
>
I found three solutions that all achieve the following score, with *more or less* the same chain but finishing in different positions:

> 
>  41
> 
> 
> 

corner:

> 
> 
> ```
>  41 27  1  1
>  14 26  3  2
>  14  9  4  1
>   5  5  5  1
> ```
> 
> 
> 
> 

edge:

> 
> 
> ```
>  14 41  1  1
>  14 26  3  2
>  14  9  4  1
>   5  5  5  1
> ```
> 
> 
> 
> 

interior:

> 
> 
> ```
>  14 15  1  1
>  14 41  3  2
>  14  9  4  1
>   5  5  5  1
> ```
> 
> 
> 
> 

A computer search determined that this score is optimal.