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Look at the row where you need to make 41 from 8 numbers:
That's the breakthrough you need. Now just follow the deductions like normal and you will eventually end up with the following final solved grid (SPOILER WARNING! Only reveal if you really want to see it!):
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Solution:
Took 1 hour and 35 minutes according to the timer!
(For all images, click/open in new tab for a larger, clearer version)
Let's start off by ignoring the sawtooth rule for now, and focusing on the XV conditions:
One last key image before we start filling in numbers:
Now we can start adding some numbers:
A few more numbers:
Now let's look at ...
11
Solution is not unique.
8
Solution:
Progress:
7
Solution:
Step-by-step:
7
Step-by-step solution:
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Step 6:
Step 7/solution:
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The finished grid:
Progress Screenshots:
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My word, this was a tricky puzzle! I've been pipped to the post by @user39583 mid-writeup (fair play) but I have visualised the solve differently, using 3 separate grids for each of the 3 sub-puzzles, and it took me all evening anyway, so I shall follow through with my explanation nonetheless in case it is of use to others with a similar way of thinking...
...
5
23 hours and no takers? Here is my solution.
How to start:
And some more:
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Well, I solved it, but now that I go back and look at my notes I am a little confused at some of my deductions. I did keep a lot of the clues in memory, as the program provided doesn't allow for much note-taking, so the process may not be obvious from the pictures. I will provide my process at the 3 places I got stuck. Step1:
Step 2:
Step 3:
Finally, ...
4
Fibonacci was already answered correctly by trolley813, so just the other two:
Odds:
The placement of the 1s for this one is pretty straightforward. Starting from the bottom row, every time there is just one column where the 1 can go.
After placing the 1s, everything basically falls into place without any difficult deductions.
Pi:
This one is slightly more ...
4
Here's the finished grid:
I only took one screenshot of my progress, but here it is:
Hopefully it's correct...
3
The sawtooth rule gives us some super powerful tools. The other answer uses them, but doesn't outright abuse them, so here's another approach that does.
Analyzing the constraints
By sudoku, we know there will be a row in the grid that has a nine in the second column. That row is severely restricted, it's basically just all the digits in descending order, ...
2
That is a very satisfying slitherlink puzzle to solve.
The way I started is to look at a 3 with two adjacent 1s.
Suppose we cross out two of the edges of one of the 1s as shown here:
This very quickly leads to a contradiction due to the adjacent 3 and the other 1:
This means that the edge for that 1 is actually one of those crossed out sides, so I should ...
2
If my comment is correct, and an Arrows Puzzle is one in which you are given a grid of numbers and asked to arrange arrows around the border such that each cell is seen by [cell's number] arrows, then the following grid has at least two solutions:
Many more grids of similar construction exist.
1
Another way without trial and error:
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