# 9sums - a logical deduction puzzle

Each set of minuses contains an integer in the range [1,5]. The given numbers are the sum of the surrounding 4 numbers.

---    ---    ---    ---
13     14     11
---    ---    ---    ---
11     10     13
---    ---    ---    ---
11      9     14
---    ---    ---    ---


Can you solve the grid?

• Are there any rules on number of uses of each integer or using the same integer per row/column? – AHKieran Dec 11 '18 at 9:04
• @AHKieran; not yet! – JMP Dec 11 '18 at 9:07

 2      5      3      1
13     14     11
3      3      3      4
11     10     13
4      1      3      3
11      9     14
5      1      4      4


• Well done! I edited the answer to make it a little more readable. – Christoph Dec 11 '18 at 9:20

Not sure how to format this, but from left to right, then top to bottom, the numbers would be:
2, 5, 5, 1
5, 1, 3, 2
2, 3, 3, 5
5, 1, 2, 4

• from the looks of it there are multiple solutions, for some reason the other answer didn't appear until after I posted mine... – AHKieran Dec 11 '18 at 9:21

Its has many solutons, just need to start somewhere

3      3      3       1
13     14    11
2      5      3       4
11     10     13
2      1      1       5
11     9     14
5      3      4       4

Here is my solution in excel:

If you want to solve it by yourself, here is file 9sum.xlsx.

• Welcome to PuzzlingSE! Please include your answer in your post, and don't just link to it. This keeps it available, should you ever decide to delete the file and it does not force other people to download a file from an unknown person/source. As it stands this answer will most likely be deleted soon. – Christoph Dec 11 '18 at 11:24
• What you could do is take a screenshot of your excel and upload it as an answer – AHKieran Dec 11 '18 at 11:25

Also this grid is a solution :

2 4 5 2
3 4 1 3
3 1 4 5
4 3 1 4

• Welcome to Puzzling.SE. Take the tour. I have edited your answer to include it with a spoiler markup >!. – u_ndefined Dec 11 '18 at 10:14

This was fun!

For my solution, I started in the four corners and worked my way inwards, ignoring the 1-5 constraint. When I had completed the grid without constraints, I again started in the corners to adjust numbers to fit the constraints until all constraints were satisfied.

-3-    -4-    -4-    -1-
13     14     11
-1-    -5-    -1-    -5-
11     10     13
-3-    -2-    -2-    -5-
11      9     14
-5-    -1-    -4-    -3-


 2      5      3      2
13     14     11
3      3      3      3
11     10     13
3      2      2      5
11      9     14
4      2      3      4

Given the multiple possible answers, it is possible to add further restrictions, such as using integers from [2, 5] only (this solution qualifies!)