Sudoku 10×10 – Hard

Sudoku 10×10

Sudoku 10×10 transfers the well-known Sudoku principle to a grid with 10 rows and 10 columns. The grid is additionally divided into 10 clearly outlined regions, each consisting of 2×5 fields. Some characters are given; all others are completed through logical exclusion.

The puzzle type was originally published under the name Number Place. Despite the digits and letters, no calculations are performed: each character is merely a distinguishable symbol. It is only important that each appears exactly once in every row, column, and region.

In this size, the digits 1 to 9 and the letter A are used.

Basic Rules

• Each empty cell receives exactly one of the 10 allowed characters.
• Each row contains each allowed character exactly once.
• Each column contains each allowed character exactly once.
• Each 2×5 region contains each allowed character exactly once.
• Pre-filled entries must not be changed.
• A candidate is only allowed if the character does not already appear in the corresponding row, column, or region.

Regions, Symbols, and Units

Row, column, and region are collectively called units. Each unit contains 10 cells and, at the end, the same complete set of symbols. Two cells influence each other if they share at least one unit.

The regions are 2×5 fields large in this size. Their shape is part of the rule: they are rectangular and cover 2 rows and 5 columns. Always follow the thick boundary lines when scanning; an adjacent cell visually can already belong to a different region.

Strategic Overview

Start with almost complete units and cells with few candidates. Then look for a specific character across an entire unit. If direct placements do not occur, examine the interfaces between regions and lines and small groups of candidates.

The illustrations use circled candidates for confirmed or reserved possibilities. Candidates marked in red are eliminated by the respective deduction. A red result is not yet a placement as long as more than one candidate remains.

Each technique answers a different question: For the last missing character, a unit is completed; for a naked single, a cell is checked; for a hidden single, a character is tracked. Pointing and Claiming use the overlap of region and line; pairs reserve two characters for two cells.

1. The last missing character in a row

In the first intermediate state, row 1 is almost complete except for the cell in row 1, column 9. All allowed characters except 9 are already visible there. Therefore, the only empty cell must be 9.

The conclusion does not require a lengthy candidate analysis: a complete row must contain each character exactly once. Since only 9 is missing, any other entry already violates the row rule.

The same pattern can occur in a column or region. Immediately check the intersecting column and the corresponding 2×5 region after placing the character, as the new character might eliminate other candidates there.

2. Recognizing a naked single

Now consider row 1, column 8. In the visible row, column, and region, all allowed characters except 6 already occur. Although there are still several empty cells in the three units, only one candidate remains for this particular cell.

Thus, 6 is placed. A cell with exactly one permissible possibility is called a naked single. The candidate is 'naked' because it is directly visible as the only note in the cell.

Always verify naked singles against all three units. Considering only the row and column might overlook a digit eliminated by the region, leading to an incorrect candidate list.

3. Finding a hidden single in a unit

In row 1, the character 2 is missing. The cell in row 1, column 2 has candidates {1, 2, 6} and is thus not a naked single. However, in all other empty cells of this unit, 2 is blocked by crossing units.

Since row 1 must contain the character 2, only position in row 1, column 2 remains. This is a hidden single: the certainty only becomes visible when tracking a specific character across the entire unit.

Scan for missing characters one after the other. An extended candidate list in the target cell is not an argument against; what matters is whether the considered character remains possible somewhere else in the same unit.

4. Pointing: A Region points to a line

In region 1, 9 can only be in row 2, column 1, row 2, column 3, and row 2, column 4. All these possibilities lie collectively in row 2. Which one is correct is still unknown, but it is certain that the region places its 9 on this line.

Therefore, no other cell in row 2 can keep the candidate 9 outside this region. It is removed from row 2, column 6. Afterwards, only {3, 5} remain as allowed candidates.

This block-line interaction is called Pointing or Locked Candidates Type 1. The observation starts in a region and affects exactly one line or column outward.

5. Claiming: A line claims a region

The reverse also works. In row 2, 6 can only be in row 2, column 1; row 2, column 2; row 2, column 3; and row 2, column 4. All possible positions of this line are within region 1.

Row 2 thus claims the character 6 within this region. Other cells in the region that do not belong to row 2 lose this candidate; in the example, it affects row 1, column 2. The candidate list is thereby shortened to {1, 2}.

This technique is called Claiming, line-region interaction or Locked Candidates Type 2. Remember the viewing direction: Pointing starts in a region and acts outward on exactly one line or column; claiming starts in a line or column and affects a region.

6. Discovering a naked pair

In row 10, row 10, column 3, and row 10, column 4 each have exactly the two candidates {2, 4}. No other possibilities are available in these two cells.

The two characters must therefore be distributed precisely between these two cells. It has not yet been decided which cell gets which character; but it is already proven that both characters are bound in these cells for the unit.

A naked pair consists of two cells in the same unit, whose common candidate set includes exactly two characters. Only this precise restriction allows eliminations in the other cells of the unit.

7. Applying the naked pair

Now consider row 10, column 2 within the same unit. Before the pair deduction, this cell had candidates {2, 3, 5, A}. Candidates 2 are already reserved entirely for the pair cells.

The reserved candidates are removed from the target cell. Remaining are {3, 5, A}. The cell remains open but is clearly restricted. The removal applies only to units containing both pair cells and the target cell.

This second look separates discovery from consequence cleanly: the pair itself does not solve its two cells but can remove candidates outside the pair cells and thereby generate a new single or reduce the candidate set.

8. Revealing a hidden pair

In column 10, 1 and 5 can only be in row 7, column 10, and row 10, column 10. At least one of these cells contains additional notes; therefore, the pair is initially hidden in larger candidate lists.

The two pair characters must be distributed across these two cells. Consequently, 9 is removed from row 7, column 10, and 2 from row 10, column 10. The order of the pair characters remains open, but all other possibilities in the pair cells are eliminated.

In a hidden pair, you do not look for two identical candidate lists. Instead, you track two characters that only occur together in the same two cells within a unit. Removing the other notes makes it a visible naked pair.

Proven solving sequence

1. Scan rows, columns, and regions with few gaps for the last missing character.
2. Calculate candidates for highly restricted cells and place naked singles.
3. Follow each missing character within a unit and look for hidden singles.
4. Check regions for candidates confined to a single line or column and apply Pointing.
5. Check lines and columns for candidates entirely within a region and apply Claiming.
6. Compare candidate lists for naked pairs; additionally, look for hidden pairs in larger grids.
7. After each placement or elimination, return to singles before searching for more challenging patterns.

Common mistakes

• Forget the region: A character may be missing in row and column but still forbidden in the 2×5 region.
• Misread region boundaries: Especially rectangular regions are easily confused with adjacent cells. The thick lines are authoritative.
• Incomplete candidates: A supposed pair is invalid if a third candidate is still possible in one of the pair cells.
• Mix up Pointing and Claiming: Pointing eliminates outside a region along the same line; Claiming does so inside the region but outside the considered line.
• Misinterpret a reservation as order: A pair determines which two characters occupy the cells but not which character goes where.
• Retain outdated notes: After each new clue, candidate sets in the cell’s row, column, and region change.
• Guessing: If no step is visible, a systematic scan is more reliable than an unfounded trial entry.

Tips for Beginners

• Use a fixed order, such as regions, rows, columns, then individual symbols.
• Mark candidates small but complete. Note not only a favorite but every possibility allowed by the three units.
• Formulate a brief proof before each entry: Which unit enforces the character, or what pattern allows removal?
• After a result, first check for new singles; a pair or interface elimination often prepares a simpler deduction.
• Check completed units against the full symbol set and look for duplicates immediately.
• In letter Sudoku, use the given symbol order without assigning numerical values to letters.

Summary

Sudoku 10×10 relies entirely on the interplay of rows, columns, and 2×5 regions. Last missing characters and singles provide direct entries; Pointing, Claiming, and pairs create secure candidate eliminations.

The grid size changes the number of symbols and the shape of regions but not the logic. By updating candidates carefully, trusting every deduction on a visible unit, and restarting with simple techniques after each result, you can solve the puzzle step by step without guessing.