Sick and Twisted Sandwiches
(Published on 10. January 2023, 04:15 by Andrewsarchus)
About:
- This puzzle was created as part of a Secret Satan puzzle exchange (like "Secret Santa", but with more diabolical puzzles). This puzzle was my gift to Piatato, and he has encouraged me to share it here as well.
Rules:
- Normal Sudoku rules apply.
- Normal Kropki pair rules apply. (no negative constraint)
- Clues outside of the grid are Satanic Sandwich clues.
- A satanic sandwich consists of a path formed by connecting a sequence of ordinary sandwiches by their crusts (digits 1 and 9), until the sandwich sequence contains exactly three copies of the digit 6 between the crusts in their segments.
- The 6s as well as the crusts do not contribute to the sandwich sum. All other digits between the crusts contribute their value multiplied by their segment number.
- For example, in the first segment, all sandwiched digits (except 6s) are counted at face value. In the 2nd segment, sandwiched digits are counted double their value, and in the 3rd segment, they are counted triple, and so on.
- If a 6 appears in the clued row/column before the first crust, then that 6 doesn't count towards the quota of three 6s since it is not actually in the sandwich.
A Simple Example
In the example above, the sandwich corresponding to the 108 clue starts at the first crust in Row 5 as we move away from the clue (the 1 in R5C5).
We do not count the 6 in R5C2 towards our quota of three 6s because it is not between the crusts in the row.
When we reach the second crust in row 5 (the 9 in R5C8), we turn towards the other crust in that column.
We continue the process of turning at each second crust until we encounter our 3rd 6 inside of the sandwich at R8C4, which completes our quota of three sixes in the sandwich.
The sandwich terminates at the next crust after we encounter our 3rd 6.
The first segment is scored 1x(2+3) for a total of 5
The second segment is scored 2x(2+3) for a total of 10
The third segment is scored 3x(2+4+3) for a total of 27 (recall that 6s score 0 points)
The fourth segment is scored 4x(3+5+2+4) for a total of 56
The fifth segment is scored 5x(2) for a total of 10
Adding up each segment’s total, we get 5+10+27+56+10 = 108
A Diabolical Example
In this example, it takes quite a few more twists and turns to reach the three 6 quota. In fact there are only two sandwiched-sixes reachable by the clue, so the sandwich loops back on itself and doesn’t end until the first 6 is visited a second time.
We start at the clue and move along the row. The red line segment shows our path before encountering the first crust. The 6 at R5C2 is ignored, because we have not yet reached the first crust.
Once we reach R5C4, we encounter the first crust, and the sandwich begins. Our path is now shown in blue. Eligible digits (2,3,4,5,7,8) in the first segment are scored 1x their value.
When we reach the second crust, we must turn and move towards the other crust in the column. Unfortunately, the 6 in the column is in the other direction, so we will not encounter it. Eligible digits in the second segment are scored 2x their value. At the second crust in the column, we turn again, and go towards the other crust in our new row, scoring eligible digits at 3x their value.
Segment number 4 in this example is empty, as there are no cells between the crusts in column 2.
Segment number 5 goes from R1C2 to R1C6, scoring eligible digits at 5x their value.
When we turn down in column 6 along the 6th sandwich segment, we end up intersecting our previous path. Segment 6 intersects our path through segment 1 at R5C6, so this cell gets counted multiple times. We’ve already counted it 1x for segment 1, and now we will count it 6x for segment 6, so this cell has now contributed a total of 7x its value to the sandwich sum. As we continue, we will intersect ourselves a few more times.
We do not reach our first sandwiched-six until the 8th segment. Although it scores 0 points for the sandwich sum, it does move us one more step to meeting the quota of three 6s. The eligible digits in this segment (the 2 and the 8) are scored at 8x their value.
We encounter our second sandwiched-six in the 12th segment, but we will encounter no more 6s this pass through the loop, so after completing the 14th segment, we arrive back at the first crust, and segment 15 is overlaid on top of segment 1. This time through the loop, we score the eligible digits in Row 5 at 15x their value.
We continue in this manner, scoring the 16th segment’s eligible digits at 16x their value, and the 17th segment’s eligible digits at 17x their value, etc. until we reach the 6 again at R7C3 while traversing the 22nd segment. We have now met the quota of three sandwiched-sizes, but we are not done until we get to the end of this segment (the crust at R4C3), scoring the eligible digits (the 2 and the 8 again) in the 22nd segment at 22x their value.
It’s left as an exercise for the solver to verify that the clue of 3024 is accurate for the given example.
Play Online:
f-puzzles: Sick and Twisted Sandwiches
ctc: Sick and Twisted Sandwiches
penpa: Sick and Twisted Sandwiches
Solution code: Row 6 and Column 9
(left to right, top to bottom, no spaces)
Comments
on 8. October 2024, 18:36 by DiMono
What an elegant and engaging solve path!
on 22. February 2023, 21:01 by grkles
where's my exploding head emoji when i need it
on 31. January 2023, 06:17 by AstralSky
Surprisingly smooth and not that calculation heavy as I first thought it might be. Fun solve and very nice puzzle!
on 16. January 2023, 16:49 by JayForty
Nice and smooth and not too heavy on the mental arithmetics. I appreciated the duality of the segments, it made for a flexible solve path. Thank you!
on 15. January 2023, 07:55 by henrypijames
Only 3¼ difficulty for me, and I'm astonished to have solved this puzzle using less than half of the outside clues. Didn't need a calculator either or to even write anything down - all arithmetics doable in my head.
It's a pity no enormous diabolical sandwich like the second example is actually featured. Has it been put aside for the follow-up puzzle (and then the third puzzle featuring an infinitely evil clue: ∞)? ;D
on 15. January 2023, 03:48 by Samish
The idea of satanic sandwich clues is really amazing in this context, and turns out to make a great puzzle, it was both intimidating and a joy to solve once on the right tracks!
on 13. January 2023, 04:00 by Andrewsarchus
changed second example title
on 12. January 2023, 06:37 by Ekkojensen
That was a very fun ride! Thanks for setting!
Agree with the others here. The ruleset is, at first glance, devilishly chaotic and daunting (perfect for what you were aiming at), but once you work out its implications, surprisingly smooth and fun... If you like math. ;)
on 11. January 2023, 16:49 by Vebby
Wonderful puzzle! Fairly smooth solve path after grasping the break-in. Thanks Andrew :)
on 10. January 2023, 11:27 by Piatato
Thanks a lot for this great gift! The ruleset is hilariously thematic and evil-looking, but the puzzle itself is wonderfully smooth and fun. Every step is telegraphed beautifully. Just awesome!
on 10. January 2023, 09:02 by filuta
there is quite a contrast between the shock you get after reading the rules and how well the puzzle eventually solves after you understand how to approach it, there is no way around doing some serious arithmetic though. but if you don't mind that it really is brilliant -both the concept and the puzzle
