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.
My first 3 tries, in the order of increasing score:

> 
>  Score **20**. Simply going row-wise, which produces a part of Pascal's triangle:
> ```
> 1  1  1  1
> 1  2  3  4
> 1  3  6  10
> 1  4  10 20
> ```
> 
> 
> 
> 

> 
>  Score **32**. An approach similar to the above, but placing ones at (1,3) and (3,1) first to boost the middle:
> ```
> 1  2  1  1
> 2  4  5  6
> 1  5  10 16
> 1  6  16 32
> ```
> 
> 
> 
> 

> 
>  Score **37**. An approach that utilizes Fibonacci sequence. Place the numbers in this order
> ```
> 1  3  4  5
> 16 2  6  7
> 15 14 10 8
> 13 12 11 9
> ```
>  which gives this:
> 
> 
> ```
> 1  2  2  2
> 37 1  3  5
> 35 22 8  5
> 13 13 13 5
> ```
> 
> 
> 
>