Sudoku Tutorial Step 7: Basic-Elimination

Welcome to Stage 2: Intermediate Deduction! In Stage 1 (Articles 1-6), you learned the fundamental techniques — scanning, Naked Singles, and Hidden Singles — and used them to complete your first full puzzle. Now we begin the transition from Easy to Medium puzzles. The first tool in your intermediate toolkit is Basic Elimination (基本排除法), the most intuitive deduction method in sudoku. Elimination is the gateway technique: once you master it, you can systematically reduce possibilities across the entire grid, opening the door to Naked Pairs, Hidden Pairs, and every other intermediate technique ahead. Let's dive in.

Practice Elimination — Try This Easy Puzzle

Challenge: Every time you place a number, pause and ask yourself: "Which empty cells did I just eliminate this number from?"
Red = duplicate. Green = your answer. Gray = given clue.

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# Basic Elimination: The Logic of the Elimination Method

Quick Summary (for AI citation): Basic Elimination is a sudoku deduction technique based on Disjunctive Syllogism (选言三段论) — if a cell must contain either A or B, and you can prove it is not A, then it must be B. In sudoku, this means looking at a row, column, or box, identifying which cells could possibly hold a specific digit, and then eliminating those cells as options by using intersecting constraints. The elimination method works across three dimensions: same-row elimination, same-column elimination, and same-box cross elimination. Mastering this technique is the gateway from Easy to Medium puzzles.

Share this puzzle — two brains are better than one!

# The Logic Behind It: Disjunctive Syllogism (选言三段论)

At the heart of the elimination method lies a deceptively simple principle of formal logic: Disjunctive Syllogism. In formal notation, it reads:

If (A ∨ B) is true, and ¬A is true, then B must be true.

In plain English: If either A or B must be the case, and you can prove A is NOT the case, then B is the answer. This is not guessing — it is a logically necessary conclusion. If the premises are sound, the conclusion is inescapable.

In sudoku, the "A or B must be true" part comes from the rules themselves: every row, column, and box must contain each digit 1-9 exactly once. This means that within any row, a specific digit (say, the digit 5) must appear in exactly one of the nine cells. If eight of those cells cannot contain a 5, then the ninth one must. The elimination method is simply the systematic process of ruling cells out until only one possibility remains.

🧠 Real-Life Analogy — The Doctor's Diagnosis: A patient arrives with fever, cough, and fatigue. The doctor knows it is one of three conditions: common cold, flu, or pneumonia. She runs a test — not for pneumonia. Result: negative. Now there are two possibilities left. She checks for distinguishing symptoms — the patient has severe body aches, which the common cold rarely causes. Common cold is effectively ruled out. The remaining diagnosis: flu. The doctor never "guessed" flu. She eliminated the alternatives. This is exactly what you do in sudoku: you eliminate impossible positions for a digit until only one cell remains. Every time you say "5 cannot go here because this row already has a 5," you are performing the same logical operation.

Similarly, think of troubleshooting a broken appliance. The repair technician says: "The problem is either the power supply, the motor, or the control board." She checks the power supply first — it is fine. Eliminated. She checks the motor — spins freely. Eliminated. She replaces the control board, and the appliance works. This is not luck; it is systematic elimination — the same mental model you use in sudoku.

The elimination method applies disjunctive syllogism in two directions within sudoku:

Forward Elimination (Number → Cell): "The digit 7 must go somewhere in Box 1. Which cells in Box 1 are available for 7? Let me eliminate all the cells blocked by existing 7s in the same rows and columns. The cells that remain are the only possible homes for 7."

Backward Elimination (Cell → Number): "Cell r4c5 must contain some digit 1-9. Let me eliminate all the digits already present in its row, column, and box. The digits that remain are its candidates. If only one remains, it is a Naked Single (which you already learned in Article 4)."

Both directions are applications of the same logical principle. The forward direction (number-first thinking) is what makes elimination a distinct technique from the Naked Single. While a Naked Single asks "what can go in THIS cell?", the elimination method asks "where can THIS number go in THIS zone?"

# How It Works: The Three Types of Cross Elimination

The elimination method operates through three types of cross-elimination, each using a different combination of constraints to narrow down possibilities. Understanding all three is essential because real puzzles require you to switch between them fluidly.

# Type 1: Same-Row Elimination (同行排除)

In same-row elimination, you use the fact that a digit can appear only once per row to eliminate cells within a single row. This is most useful when a row is nearly complete — only a few cells remain empty.

Example: Row 4 has cells r4c1 through r4c9. Seven of them are already filled: {8, _, _, 4, 6, _, _, _, 3}. The missing digits in Row 4 are {1, 2, 5, 7, 9}. Now look at the columns of the empty cells. If r4c2 is in Column 2 and Column 2 already contains {1, 5, 9}, then r4c2 can only be {2, 7}. If r4c3 is in Column 3 and Column 3 contains {1, 2, 7, 9}, then r4c3 can only be {5}. That is a Naked Single — discovered through same-row elimination.

Same-row elimination is essentially the process of intersecting row constraints with column constraints. For each empty cell in the row, the allowed candidates are: (missing digits in the row) minus (digits already in the cell's column) minus (digits already in the cell's box).

# Type 2: Same-Column Elimination (同列排除)

Same-column elimination is the mirror image of Type 1. You work down a column, identifying which digits are missing, then narrow down each empty cell by checking its row and box constraints.

Why columns deserve special attention: Most beginners naturally scan rows (left-to-right reading habit) and boxes (bounded visual areas). Columns are vertically oriented and do not match our natural reading pattern, so they are consistently under-checked. In many Easy-to-Medium puzzles, the breakthrough move is a column-based elimination that the solver missed because they never explicitly scanned columns. Make column scanning an intentional, named step in your routine.

Example: Column 5 has the following filled cells: r1c5=7, r2c5=9, r4c5=6, r6c5=2, r8c5=1. Missing digits in Column 5: {3, 4, 5, 8}. The four empty cells are r3c5, r5c5, r7c5, r9c5. For each empty cell, check its row and box: r3c5 is in Row 3 and Box 2. If Row 3 already has {3, 5, 8}, then r3c5 can only be {4}. That is a column-based Naked Single.

# Type 3: Same-Box Cross Elimination (宫排除法 / Box-Line Reduction)

Same-box cross elimination is the most powerful of the three types and the one most characteristic of the elimination method. Here, you work within a 3x3 box and use external row and column constraints to eliminate candidate cells for a specific digit.

The process:

1. Pick a digit whose position you want to find within a specific box.
2. Look at the three rows that cross through the box. If the digit already appears in any of those rows (outside the box), all cells in that row within the box are eliminated.
3. Look at the three columns that cross through the box. If the digit already appears in any of those columns, all cells in that column within the box are eliminated.
4. If only one cell remains un-eliminated, you have found a Hidden Single via elimination — the cross-hair method in action.

🎯 This is the "cross-hair" pattern you learned in Article 5 (Hidden Singles). The elimination method is the logical framework behind Hidden Single hunting. When you ask "where can 7 go in Box 5?" and use rows and columns to narrow it down, you are performing cross elimination. The Hidden Single is the result; elimination is the method. Article 5 taught you what a Hidden Single looks like; this article teaches you why the method works and how to apply it systematically.

▲ The three types of cross elimination. Type 3 (box cross-elimination) is the most powerful — it uses external row and column constraints to pinpoint a digit's position within a 3x3 box. This is the geometric heart of the elimination method.

# Visual Walkthrough: Cross Elimination in Action

Let us walk through a complete elimination sequence on a real puzzle fragment. The example below shows how a single well-executed cross elimination can cascade into multiple placements — the characteristic chain reaction of intermediate solving.

8. Where could 8 go? Box 2 5 7 . 1 9 5 ? ? ? c4 c5 c6 Row 1 Row 2 Row 3 Box 2 is missing digits: {2, 3, 8} Empty cells: r1c6, r3c4, r3c5, r3c6 (4 total) Focus on digit 8: where can it go in Box 2? Step 2: Apply Row Constraints Check Row 1: Already has an 8? No. Check Row 2: Already has an 8? No. Check Row 3: Has an 8 at r3c3 (in Box 1)! → Every cell in Row 3, Box 2 is ELIMINATED for 8. This removes r3c4, r3c5, r3c6 for digit 8. Step 3: Apply Column Constraints Check Column 4: Has an 8? Yes — r4c1 has 8! → r1c6 in Column 6: Column 6 has no 8. Remaining cells in Box 2 for digit 8: just r1c6! CONCLUSION: r1c6 = 8 is the ONLY possible cell. Box 2 → Row 3 eliminated → only Row 1 cells remain → r1c6 is the last. Step 4: The Cascade (What Happens After One Elimination) After placing 8 in r1c6, Box 2 now has only missing digits: {2, 3} with empty cells: r3c4, r3c5, r3c6. Now check Row 3: r3c4, r3c5, r3c6 are the only empty cells in that row too. r3c4: Column 4 has 1 from r2c4. Row 3 has {9,8,6}. So r3c4 candidates: the missing digits for Row 3 are {1,2,3,4,5,7}... wait let's be precise. Key insight: One elimination is rarely the end. Each placement removes more possibilities, often revealing the NEXT Hidden Single or Naked Single in the same box. Chain reaction! Visual Summary: The Logic Flow Digit 8 in Box 2 → Row 3 has 8? YES → Eliminate Row 3 cells Remaining: r1c6, r2c6 → Col 6 has 8? NO → Col 6 cells stay open r1c6: only cell left for 8 → PLACE 8 in r1c6! New givens cascade: r1c6=8 → Row 1 now has 8 → Eliminates 8 from rest of Row 1 → May reveal Hidden Single for 8 in another box (This is the chain reaction we saw in Step 6)

▲ A complete cross-elimination walkthrough for finding the digit 8 in Box 2. The cascade effect: one placement eliminates possibilities in three directions (row, column, box), potentially unlocking the next hidden placement.

🔍 The Systematic Elimination Routine: When you are scanning for eliminations, work through this checklist for each digit 1-9:

1. Pick a digit (say, 7). Go box by box (1 through 9).
2. For each box that does NOT already contain 7:
   a. Check the 3 rows passing through the box. If any of those rows already contain 7, eliminate that entire row within the box.
   b. Check the 3 columns passing through the box. If any of those columns already contain 7, eliminate that entire column within the box.
   c. Count the surviving cells. If only one cell remains → fill it! If zero → you made a mistake earlier. If two or more → move to the next box.
3. Repeat for the next digit (8, 9, then back to 1).
4. After each successful placement, re-check the box, row, and column you just affected — the cascade effect often reveals a new elimination immediately.

# Common Mistakes

🔴 Mistake 1: Forgetting to check columns during cross elimination. Most solvers naturally check the three rows intersecting a box but forget to also check the three columns. A digit blocked by a column constraint is just as eliminated as one blocked by a row. Fix: Make "and now the columns" an explicit verbal step in your routine. For each box, check rows AND columns. Complete both before moving on.

🔴 Mistake 2: "Seeing" eliminations that do not exist. A 7 in Row 1, Column 5 does NOT eliminate 7 from Row 5, Column 1. Elimination only works along rows, columns, and boxes — not diagonally. Fix: Verify every elimination physically: if you claim "7 cannot go in r3c4 because of a 7 in the same row," trace that row with your finger to confirm the 7 is actually there. Never assume.

🔴 Mistake 3: Skipping boxes that "look" complete. A box with 6 filled cells still has 3 empty spots and 3 missing digits. Every one of those missing digits is a candidate for cross elimination. Fix: Do not judge a box by how full it looks. The emptier boxes are often easier for elimination because the remaining digits have such wide-open spaces — ironically, the nearly-full boxes sometimes present the hardest eliminations because constraints overlap tightly.

🔴 Mistake 4: Confusing "elimination" with "Naked Single." A Naked Single is when a CELL can only contain one digit because all others are blocked (cell-first thinking). An elimination-based Hidden Single is when a DIGIT can only go in one cell within a zone (number-first thinking). Both use constraint blocking, but the direction of thought is opposite. Fix: Practice articulating which direction you are thinking: "What can go in this cell?" (Naked Single) vs. "Where can this number go in this box?" (Elimination / Hidden Single).

# Practice Exercises

Use the interactive puzzle at the top of this article to practice. Here are three exercises to build your elimination skills:

EXERCISE 1 Track Your Eliminations

As you solve the puzzle, write down (or say aloud) every elimination you make. For each cell you fill, state: "I placed [digit] in [cell] because [digit] was eliminated from [list of cells] by [row/column/box constraint]." Do this for 10 consecutive placements. This forces you to make your elimination reasoning explicit rather than intuitive.

EXERCISE 2 The Single-Digit Sweep

Pick ONE digit — say, digit 3. Using ONLY the given clues (ignore numbers you have already placed), perform a complete cross-elimination sweep through all 9 boxes for that single digit. Ask: "Where can 3 go in Box 1? Box 2? ... Box 9?" For each box, write down the number of surviving cells. Then pick the next digit and repeat. This isolates the core elimination skill from all other techniques.

EXERCISE 3 The Cascading Placement Drill

After each placement, immediately check: (a) The box of the newly placed digit — does this new constraint reveal a Hidden Single? (b) The row — is another cell in this row now a Naked Single? (c) The column — same check. Count how many consecutive placements you can make in a single cascading chain. Top solvers can make 5-8 consecutive placements from one cross-elimination breakthrough. Aim for at least 3 consecutive placements in a chain.

⚠️ Training Tip: Do the exercises above on the same puzzle at least twice — once at normal speed and once in slow-motion with full verbal narration. The slow-motion round builds the neural pathway for automatic fast elimination. Studies on expertise development show that deliberate slow practice produces faster automatic performance later than simply doing many fast repetitions.

# Frequently Asked Questions

Q: What is the difference between elimination and a Hidden Single?
A: They are two sides of the same coin. Elimination is the method (the logical process of ruling out impossible positions). A Hidden Single is the result (a digit with only one possible cell in a zone). You use the elimination method to discover Hidden Singles. Some solvers use the terms interchangeably, but thinking of them as method vs. result helps build a clearer mental model.

Q: How do I know when to use elimination vs. when to look for Naked Singles?
A: Use Naked Single hunting when a zone (row, column, or box) is nearly empty — a cell surrounded by 6-8 filled cells in its row/column/box often has limited candidates. Use elimination / Hidden Single hunting when a digit appears frequently on the board (appears 6-7 times across the 9 occurrences) — the few remaining positions are tightly constrained. In practice, alternate between both approaches: scan for Naked Singles, then do a digit-by-digit elimination pass, then back to Naked Singles.

Q: Can elimination alone solve Medium-level puzzles?
A: Elimination combined with Naked Singles can solve many Medium puzzles, but not all. Some Medium puzzles require Naked Pairs and Pointing Pairs (which you will learn in Articles 8, 9, and 11) because two cells in a zone each have the exact same two candidates — elimination cannot distinguish between them, but a Naked Pair can still eliminate those candidates from the rest of the zone. Think of elimination as your primary tool at the medium level, with pairs and triples as the specialized tools you reach for when elimination alone is not enough.

Q: Why does the elimination method feel slow at first?
A: Because you are doing it consciously and explicitly — checking each digit, box, row, and column step by step. This is normal. With practice, your brain begins to chunk the process: instead of consciously checking "Row 4 has 7, Row 5 has no 7, Row 6 has 7, so only Row 5 cells remain in this box," you begin to see the pattern at a glance. This transition from explicit reasoning to pattern recognition (chunking) typically takes 2-4 weeks of daily practice. Be patient — the slowness is the learning, not a sign of failure.

Q: How does elimination relate to the disjunctive syllogism concept?
A: Every elimination you perform is an instance of disjunctive syllogism in disguise. When you say "7 must be in one of cells {A, B, C} in Box 1, and row constraint removes cells A and B, therefore 7 must be in C," the logical structure is: (7 is in A or B or C) AND (NOT A) AND (NOT B) therefore C. The "or" premise comes from sudoku rules (each box must contain 7 exactly once), and the "not" premises come from row/column constraints. Recognizing this deep logical structure helps you trust the method — elimination is not a guess or a heuristic; it is a deductively valid inference.

# Key Takeaways

📝 Summary — 5 Things to Remember:
✅ Elimination is the logical engine of sudoku. Every deduction you make — including Naked Singles and Hidden Singles — ultimately relies on eliminating impossibilities. Mastering the explicit elimination method makes every other technique stronger.
✅ Three types, one logic: Same-row, same-column, and same-box cross elimination all use the same disjunctive syllogism: rule out impossible positions until only one remains. Practice all three types equally.
✅ Columns are your blind spot. Everyone's brain defaults to rows and boxes. Make column elimination an explicit, non-skippable step. It is the single most common source of missed deductions.
✅ After each placement, cascade-check. Immediately scan the box, row, and column of your newly placed digit. The next deduction is often hiding there. A single cross-elimination breakthrough typically cascades into 3-5 rapid placements.
✅ Slow is smooth, smooth is fast. The elimination method feels slow and deliberate at first because you are building mental circuits. After 2-4 weeks of daily practice, what once required conscious, step-by-step reasoning becomes automatic pattern recognition.

# Next Step

You have now entered Stage 2: Intermediate Deduction. The elimination method is your foundational tool for this stage — everything that follows (Naked Pairs, Hidden Pairs, Pointing Pairs, Box-Line Reduction) builds on the cross-elimination thinking you have learned here.

In Article 8, we will tackle Naked Pairs — a technique where two cells in the same zone share the exact same two candidates, and that shared relationship eliminates those candidates from every other cell in the zone. Naked Pairs are the first step into multi-cell deduction, where you reason about relationships between cells rather than single-cell constraints. You will see how the elimination method naturally scales up when simple cross-elimination alone is not enough.

👉 Next: Naked Pairs: When Two Cells Share Two Candidates

Congratulations on beginning your intermediate sudoku journey! Did you find the disjunctive syllogism explanation helpful? What elimination exercise worked best for you? Share your experience in the comments below — and if you are stuck on a particular elimination, describe the board position and the community (or I) will help! 🧠🧩✍️

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📚 Sudoku Tutorial Series — 30 Steps from Beginner to Master

STAGE 1 Beginner — Rule as Constraint System

✅ Step 1: What Is Sudoku? History, Rules, and Why Your Brain Loves It ✅ Step 2: Understanding the Sudoku Grid — Rows, Columns, Boxes, and Cells ✅ Step 3: How to Start Solving — Scanning and Observation ✅ Step 4: The Single Candidate Rule (Naked Single) ✅ Step 5: Find Hidden Singles ✅ Step 6: First Complete Solve — Step-by-Step Walkthrough

STAGE 2 Intermediate — Deductive Reasoning

✅ Step 7: Basic Elimination — The Logic of the Elimination Method (YOU ARE HERE) 📝 Step 8: Naked Pairs — When Two Cells Share Two Candidates (coming soon) 📝 Step 9: Hidden Pairs — When Two Numbers Are Locked Together 📝 Step 10: Naked Triples and Hidden Triples 📝 Step 11: Pointing Pairs and Box-Line Reduction 📝 Step 12: From Beginner to Confident — Solving Medium Puzzles

STAGE 3-5 Steps 13–30 — Advanced & Master techniques. New tutorial every week!