Rossini Sudoku – Hard

Rossini Sudoku

Rossini Sudoku is a variant of the classic 9x9 Sudoku with arrows outside the grid. Each arrow relates to the three cells nearest to it. The numbers in these three cells must strictly increase in the direction of the arrow. The arrowhead points to the largest of the three numbers.

The three numbers do not have to be consecutive. A sequence like 1, 3, 7 is just as allowed as 2, 3, 5. The only requirement is that each value in arrow direction is greater than the previous one.

If a side of a row or column lacks an arrow, the three nearest numbers from that side may not be strictly increasing or decreasing. This reverse rule is an important additional hint.

Basic Rules

• The playing field consists of 9 rows and 9 columns.
• In each row, the numbers 1 to 9 must appear exactly once.
• In each column, the numbers 1 to 9 must appear exactly once.
• In each 3x3 block, the numbers 1 to 9 must appear exactly once.
• An arrow outside the grid relates to the three cells in the same row or column nearest to it.
• The three numbers must be strictly increasing in the direction of the arrow.
• The arrowhead points to the largest number in the sequence.
• The numbers do not have to follow each other consecutively. Sequences like 1, 3, 7 or 2, 3, 5 are allowed.
• If an arrow is missing, the three border cells from that side do not form a strictly increasing or decreasing sequence.
• The given numbers cannot be changed.
• The puzzle is solved when all regular Sudoku rules and all Rossini hints are fulfilled.

Strategies for Solving

1. Arrows do not require directly consecutive numbers

A Rossini arrow only describes the order. There can be gaps between the numbers. This is also seen in the following example puzzle.

Above the fourth column, an arrow points down. The first three numbers in this column are 1, 6, and 9. They increase strictly in arrow direction:

1 < 6 < 9

The numbers are not consecutive, but still meet the hint requirements.

Next to the sixth row on the left, an arrow points to the right. The first three numbers of this row are 2, 3, and 5:

2 < 3 < 5

This sequence also contains a gap between 3 and 5. For Rossini Sudoku, this is fully correct.

2. A Rossini arrow decides between two visible candidates

In the following grid, we look at the third cell of the sixth row. To the left of this row, an arrow points to the right. Therefore, the first three numbers of this row must increase strictly from left to right.

For the empty third cell of the sixth row, only 1 and 5 remain possible according to normal Sudoku rules:

• Already in the third column are 9, 4, 7, 6, 2, and 3.
• In the sixth row, are already 2, 3, 8, 7, and 4.
• In the left middle 3x3 block, are already 8, 7, 9, 4, 6, 2, and 3.
• Thus, only 1 and 5 are possible for the cell in question.

The arrow requires for the first three cells of the row:

2 < 3 < ?

But 1 is not greater than 3 and thus is excluded. Only 5 satisfies both Sudoku rules and the Rossini rule.

Hence, 5 is clearly derived. It is visible in the grid why initially only 1 and 5 are possible and why the arrow then excludes 1.

3. An arrow can reduce candidates without immediately setting a number

Not every Rossini hint leads directly to a placement. Often, it initially only removes individual candidates.

In the following grid, the arrow above the fourth column points down. The first two cells of this column contain 1 and 6. The third cell from above is still empty.

According to normal Sudoku rules, the empty third cell of the fourth column can still be 2, 7, or 9. But the arrow requires:

1 < 6 < ?

2 is less than 6 and is excluded. Candidates 7 and 9 remain possible.

The arrow does not assign a number here yet. However, it provides a safe progression by removing candidate 2.

4. Correct reading of the arrow direction

An arrow next to a row can point to the left. Then, the three right border cells are considered from right to left.

In the first row of the solution, the arrow on the right side of the grid points to the left:

The three rightmost numbers from left to right are 6, 4, and 3. Following the arrow from right to left, one reads:

3 < 4 < 6

The numbers increase in arrow direction. Again, the sequence is not gapless, as 5 is missing between 4 and 6. This is allowed.

5. Using missing arrows as exclusions

In our example puzzle, all possible arrows are provided. Therefore, the absence of an arrow also carries meaning.

Next to the fourth row on the left, there is no arrow. The first three numbers of this row in the solution are:

1 8 7

This sequence is not strictly increasing because 7 is smaller than 8. It also isn't strictly decreasing because beginning with 1, it increases to 8. The sequence is neither increasing nor decreasing and thus satisfies the condition for a page without an arrow.

If the three numbers were, for example, 1, 3, and 7, an arrow on that side would be required. Without an arrow, such a strictly increasing sequence would be disallowed.

6. Deriving bounds from position within the sequence

An increasing triplet requires three different magnitudes. This yields simple boundaries:

• The first number in arrow direction can be at most 7, because two larger numbers must still follow.
• The middle number cannot be 1 or 9.
• The last number in arrow direction can be at least 3, because two smaller numbers must precede.

These limits rarely lead directly to a number but often help eliminate candidates at the grid's edges.

Typical solving process

1. Check arrows with two known affected numbers.
2. Always read the three cells in the actual arrow direction.
3. Note only the magnitudes, and gaps between numbers are allowed.
4. Determine standard Sudoku candidates for the affected cells.
5. Eliminate candidates that do not fit into a strictly increasing triplet.
6. Check pages without arrows and exclude strictly increasing or decreasing sequences there.
7. Use each new entry for rows, columns, blocks, and Rossini hints again.

Common Mistakes

• Assuming the three numbers must be directly consecutive.
• Falsely excluding a valid sequence like 1, 3, 7.
• Considering the wrong three cells.
• Reversing the arrow direction.
• Only checking the Rossini rule and forgetting the row, column, or 3x3 block.
• Placing a number when the arrow only reduces candidates; multiple options remain.
• Overlooking the significance of a page without an arrow.

Tips for Beginners

• Mentally mark exactly the three cells directly next to an arrow.
• Read the values in arrow direction and formulate the condition as a less-than chain.
• Don't think in consecutive numbers, but only in "less than" or "greater than".
• Check arrows with two known values first.
• Use Rossini hints even if they only exclude one candidate.
• Verify that pages without arrows do not allow for strictly increasing or decreasing sequences among the first three numbers.

Rossini Sudoku extends classic Sudoku with clear ordering logic. The arrows require strictly increasing sequences without the necessity of immediate consecutive numbers. Properly reading the arrow direction, considering visible Sudoku candidates, and utilizing missing arrows enable deducing additional clues without guesswork.