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: 
The best solution my computer found is

> 
>  41
> 
> 
> 

I suspect this is optimal.

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

I won't list the order, as you can simply choose the squares according to the increasing order of the numbers.
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)
> 
> 
>