Logic Masters Deutschland e.V.

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.

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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

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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.

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Play Online:

     f-puzzles: Sick and Twisted Sandwiches

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     penpa: Sick and Twisted Sandwiches