Analysis of 29 Jun 2012 6x6 Difficult with sums

In the short time I've been on this site, I've been thankful for clm's analysis and strategy posts. When I solved the puzzle in the title, it was in part due to the clear way in which clm has described how to use some of the less obvious strategies. I hope this post helps some of the beginners see how useful sums can be.

Here's the blank puzzle:

And here it is with some of the more obvious possibilities filled in:

I noticed two things when I got to this point: There are only two non-sum cages, the 3- and 2-, and there are two 10+ cages in the center which looked nicely symmetric and caught my eye. I started my sum analysis with the two subtraction cages; since everything else was an addition, I could add them all up and subtract from the total of all cells to find the sum of the two subtraction cages.

Total sum of all cells: 21 (sum of one column/row) x 6 = 126

Sum of addition cages: 21+15+18+16+9+10+3+5+10+6 = 113

Difference = sum of two subtraction cages = 13

Possibilities for subtraction cages, [x,y] with sum (S)

3- [1,4] (5), [2,5] (7), [3,6] (9)

2- [1,3] (4), [2,4] (6), [3,5] (8) ([4,6] is disallowed due to both the 10+ cage and the impossibility of making a sum of 13 with the 3- cage)

That turned out not to be immediately useful, so I set it aside to look at the large L shaped cages and the 10+ cages.

Due to the pattern of the puzzle, I could split it in half for analysis four different ways; I chose to start with the left side. Since d1 was 1, 2, 3, or 5, the other cells of the 21+ cage (a2,a1,b1,c1) had to sum up to 20, 19, 18, or 16. Cell c6 was under similar constraints as d1, and could only be 1, 2, 3, or 5. And from the first analysis, the 2- cage could only sum 4, 6, or 8. Finally, the sum of the addition cages in the three left hand columns was 16+9+10+6 = 41, so the remaining cells in those three columns had to sum to 22 to make the columns equal 63 (21 x 3).

So, below are the possibilities laid out a bit more clearly. Picking one from each has to sum 22.

a2,a1,b1,c1: 20 19 18 16

c6: 1 2 3 5

2- b4,c4: 4 6 8

Note the smallest possibilities for the partial 21+ cage and 2- cage sum to 20; that leads to the only possible solution being 16, 2, and 4. And from the earlier analysis, since the 2- sums to 4, the 3- must sum to 9. Let's see what that immediately gets us, without going one step further:

I put the above image up because as soon as we fill in the 2 and the [1,3] and [3,6], an unbelievably large number of cells become solved through the most basic calcudoku analysis. After doing said analysis and adding the remaining possibilities, here's what we have:

I think clm would call this puzzle "solved" at this point. :) And with good reason! Starting with 21+ again, since it has two solved cells, we see the remaining 3 cells need to sum 16. That forces [5,5,6]. Adding those cells and doing further basic analysis leads directly to the solution:

The moral of this story: sums can be quite useful with pattern puzzles! :)

Regards,

-Jake

ps If someone wants to tell me how to get the images to show up full size in the forum, I'd appreciate it!