This is my response to Scojo's setting prompt for the week: Killers, Parity Lines and Negators.

The puzzle has been featured by Zetamath - many thanks to him for the feature! During his solve, he struggles to internalise upwards as being a "positive" vertical movement, as he is used to the Sudoku notation system in which row numbers increase as they go down. I commented that I would make a flipped version of the grid as we thought it wouldn't change the logic, only realising as the stream ended that it*would* in fact affect the vectors that need to be read from top to bottom.

For me, considering upwards to be positive feels less unnatural than reading a vector from bottom to top, so I haven't made the flipped version of the grid, but if I do use this ruleset again - which I'd like to because it gave rise to some fun deductions, and using a different line type may open up new possibilities - I will endeavour to make all vectors read from left to right, then release two versions according to your vertical movement preferences!

Place the digits -4 to 4 once each into every row, column and box. [For the answer check to work, write negative digits as regular positive digits]

Digits in a cage must sum to the indicated total, if given.

Draw a loop in the grid, which is a parity line, i.e. digits on the loop alternate between even and odd. The loop must not enter the same cell twice.

Nine cells in the grid are defined as a 'vertex': one in each row/column/box. The loop must pass through every vertex.

Every cage is a 'vector' in the form (x,y), read from left to right or top to bottom. A vector shows that, from the vertex in its box, the loop travels x cells rightwards, then y cells upwards, then arrives at another vertex.

The puzzle has been featured by Zetamath - many thanks to him for the feature! During his solve, he struggles to internalise upwards as being a "positive" vertical movement, as he is used to the Sudoku notation system in which row numbers increase as they go down. I commented that I would make a flipped version of the grid as we thought it wouldn't change the logic, only realising as the stream ended that it

For me, considering upwards to be positive feels less unnatural than reading a vector from bottom to top, so I haven't made the flipped version of the grid, but if I do use this ruleset again - which I'd like to because it gave rise to some fun deductions, and using a different line type may open up new possibilities - I will endeavour to make all vectors read from left to right, then release two versions according to your vertical movement preferences!

Place the digits -4 to 4 once each into every row, column and box. [For the answer check to work, write negative digits as regular positive digits]

Digits in a cage must sum to the indicated total, if given.

Draw a loop in the grid, which is a parity line, i.e. digits on the loop alternate between even and odd. The loop must not enter the same cell twice.

Nine cells in the grid are defined as a 'vertex': one in each row/column/box. The loop must pass through every vertex.

Every cage is a 'vector' in the form (x,y), read from left to right or top to bottom. A vector shows that, from the vertex in its box, the loop travels x cells rightwards, then y cells upwards, then arrives at another vertex.

Lösungscode: Row 9 (bottom row), left to right. (Negative digits appear as normal digits, i.e. "-4, 3, 0, -3" would be written as "4303" in this solution code)

Zuletzt geändert am 6. Februar 2024, 23:34 Uhr

Gelöst von appleponycmh, efnenu, agueybana, Kallor, yttrio, giladooshlon, Default, ViKingPrime, marty_sears, liushong, konklone, bernhard, Paletron, Uskok, bansalsaab, zhugelianglongming, csq527, OGRussHood, SKORP17, Uhu, Nell Gwyn, rich_27

Zuletzt geändert am 27. Februar 2024, 15:57 Uhr

**am 25. Februar 2024, 16:36 Uhr von rich_27**

Such a fantastic puzzle, I loved this so much. I hope you do more with this ruleset, it's great!

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TPP: Thank you for this and so many other lovely comments on my puzzles recently! Your solving spree is very impressive and I'm very flattered that you've attempted (with great success) so many of mine! :D

**am 28. Januar 2024, 14:01 Uhr von Uskok**

Nice puzzle but hard to mark everything. I used tens complement to write negative numbers: 6 for -4, 7 for -3, 8 for -2, 9 for -1.

**am 26. Januar 2024, 21:26 Uhr von marty_sears**

Another completely innovative and masterfully delivered bit of business here from the ivory-tinkling cyclist.

Uncharacteristically for me though, I was able to get my head round this logic rather quickly, which felt gooood! Once you've found the break-in, which is fairly well signposted, it all unfolds beautifully from there like a beautiful unfolding thing.

Deducing the nonsense in boxes 1 and 4 was my favourite part.

Etch-a-Sketch felt the perfect title for it too.

Thanks for another great puzzle and trip into your marvellous mind!

**am 26. Januar 2024, 17:22 Uhr von ViKingPrime**

This is remarkable. I'm speechless.

**am 25. Januar 2024, 11:32 Uhr von Default**

So much fun! Thanks

**am 25. Januar 2024, 00:04 Uhr von giladooshlon**

That is very original! It was hard to decide on a coloring strategy as there are so many different components to keep track of.

**am 24. Januar 2024, 20:49 Uhr von yttrio**

Excellent puzzle! It took me *way* too long to properly read the rule regarding how exactly the loop moves, but once I actually understood the rules it opened up some really nice logic!