# Objects in Mirror

This puzzle is part of the Monthly Topic Challenge #1: Restricted Title: xkcd 1xxx and is based on https://xkcd.com/1125/.

[Penpa link at the bottom, if you want to skip the story]

Objects in Mirror ... are different than they appear

Sue had found herself in another cursed temple. You'd think she'd learn, but this time it was really her friend Doe's fault. They'd been exploring for Doe's 9th birthday, stepped under an overhang, and fallen down a hole. Worse, a rock moved to block the entrance they'd just fallen down.

At least this temple seemed to be pretty empty. Except for a table in the middle, that is. It didn't look like Doe was moving any time soon (you couldn't blame them for hiding in a corner, not really - it was their first temple) so Sue went to inspect the table. On the top was a design with some strange arrows.

The design looked like an Arrow Sudoku. But after ten minutes of fruitless thinking, Sue was pretty sure it was impossible. There had to be something more! She cracked her knuckles and carefully looked around the table some more. Without touching anything, of course. (Doing that willy-nilly was a sure sign of a cursed-temple amateur.)

By this time Doe had worked up the courage to come nearer. They reached out and grabbed at the mirror Sue had been looking at before so much as a warning could be said. Strangely enough, Doe's fingers went through the mirror, as if it was only a frame, despite the fact it reflected light as well. Sue chalked it up to cursed-temple magic and took the opportunity to read the writing along the edge while Doe just stood there hyperventilating. Amateur.

Through the mirror is the answer you seek

Answers! Answers were great, maybe that would get them out. Sue gently pried the mirror out of Doe's hands and placed it in the suspiciously convenient grooves on opposite sides of the Arrow Sudoku. Looking through it now, nothing interesting happened - still just a mirror which your hand could go through. Wait, through?

Had the middle arrow always been poking out, just a bit? Sue wasn't sure. She gently touched it, and when nothing happened, grasped it more firmly. It seemed to - rotate? And it took the table with it. Ah-hah! Sue threw all her weight into rotating the table around and through the frame of this magical mirror. Once the arrows were settled on the other side (this table must be transparent, Sue mused) new shapes shimmered into being, each perfectly reflecting the corresponding arrow on the other side of the mirror. Except... different.

Now this, this Sue could solve. She'd show Doe the ropes of cursed-temple puzzles. It'd be the greatest 9th birthday ever.

Standard Sudoku, Thermo, and Arrow rules apply. Briefly:

• Place the numbers 1-9
• Numbers do not repeat in a row, column, or heavy-outlined box
• Numbers strictly increase from the bulb of a thermometer (dark background) to the other end
• Numbers along an arrow add up to the number in that arrow's circle
• Cells within both a thermometer and an arrow must abide by both special rules

This is a logic puzzle, and is intended to be solved logically. Please include description in your answer as to how the puzzle was solved. This could be as little as explaining a few key deductions, or as much detail as you would like. Answers with just the final solution will not be accepted.

Click on images for an uncompressed version. Or, here is a word-based description of the puzzle:

• No numbers are given
• There are five thermometers:
• Starting at R1C2, then going R2C3-R2C4-R1C4
• Starting at R1C7, then going R2C7-R3C7
• Starting at R5C4, then going R5C3-R4C3-R4C2
• Starting at R7C8, then going R6C8-R5C8
• Starting at R9C1, then going R8C2-R7C3-R6C4-R5C5
• There are six arrows:
• Circle at R1C3, then arrow R2C3-R3C3
• Circle at R1C8, then arrow R2C7-R2C6-R1C6
• Circle at R4C5, then arrow R5C5-R6C5
• Circle at R5C6, then arrow R5C7-R4C7-R4C8
• Circle at R7C2, then arrow R6C2-R5C2
• Circle at R9C9, then arrow R8C8-R7C7-R6C6-R5C5

Finally, here is a link to a Penpa version of the puzzle.

P.S. - if you want a slightly bigger challenge, try to solve without the R7C2-R5C2 arrow. It's only there for antisymmetry. This isn't worth anything extra except imaginary bobble points.

• I'm extremely excited to see this, and also to eventually see it solved! Commented Jul 18, 2022 at 22:20

## Final grid:

Step 1: First steps

The obvious place is to start in the middle cell, as that's where we have the most information. From the thermometer, we know it can't be less than 5, and from the long arrow, as the total can be a maximum of 9, the remaining 3 cells must be a 1, 1 and 2 - making the middle cell a 5, and hence the bottom right a 9. We have our first entries, and a few other candidates can be filled.

Step 2: Progression

Now focusing on what we have, the thermometer must be 5-4-3-2-1, which also fixes the 1-2 pair bottom right. The middle arrow is now restricted and must be 5+2/3 = 7/8. A few other candidates can be filled.

Step 3: Intersections

A lot of candidates can be filled in around the grid now, but the interesting bit is the top row. The top right intersection must be a 3 or 4, as there is a 1 in column 6 restricting the arrow, creating a 2/3 pair and the arrow must be an 8/9. The thermometer then is also restricted, and notice that it cannot start with a 1, as that would mean the arrow further down is not possible. This places several numbers and restricts many other candidates down further.

There is only one place for a 9 in the middle right square, and the middle thermometer can't start with a 6 or there is no place for an 8 and 9 in the middle square. The major breakthrough is that this creates a 6/7/8/9 quadruplet in the 4th row, placing a 5 in the thermometer, and in turn completely solving the right arrow. 1 is now isolated in the middle left square, partially solving the thermometer top left.

Step 5: Falling like dominoes

Major breakthrough found, the puzzle becomes much easier; the top left arrow can be resolved, an isolated 1 bottom middle square, and 4 middle right leads to a 3 bottom right - placing a 4 at the start of the thermometer. A 6 can be placed top right, and then top middle. A 3 can also be placed top left, and a 1 top middle. At this point, the grid is nearly solvable by pure sudoku logic.

Step 6: Finishing touches

A 1 can be placed top right, and a 3 and 2 middle left. Column 3 can be solved, solving the arrow in column 2. A 9 in the middle left places an 8 higher up, in turn solving a lot of the top. A 3 middle right solves a thermometer and an arrow, and from here the puzzle can be solved (very satisfyingly!) using normal sudoku logic.

• I feel like perhaps my major breakthrough was not the intended one, although it worked out rather nicely, but nevertheless, very fun puzzle! Had me backtracking several times and some of the intersection logic was making me go in circles! Commented Jul 19, 2022 at 0:47
• Indeed, it wasn't entirely intended, but instead a product of the middle arrow (the very last clue I added in order to force a single solution). However I noticed it and thought it was fine because it was still fun/ah-hah!/difficult. My intended path places the 1 in the middle-left box very early, and then resolves the middle-right arrow earlier than yours. But both work :) Commented Jul 19, 2022 at 0:49
• @bobble ah how did I miss that 1!! Very well built, has a couple nice gotcha moments, and finally got me my sudoku answer badge after a year of being on 19/20 answers :P Commented Jul 19, 2022 at 0:56