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What goes at the far end of the line?
Level 3 · Lines
How to solve palindrome lines
A gray line reads the same from either end.
A palindrome line is a gray line that reads the same forward and backward. The first and last cells match, the second and second-to-last match, and so on toward the middle.
Each mirrored pair is one digit written twice. Anything you learn about one cell of the pair is true of the other, so the line carries information across the grid.
How setters write it
The same rule, worded the ways you’ll see it on puzzles.
- Digits on a gray line read the same forwards and backwards.
- Palindrome lines.
Worth memorizing
- Cells in mirrored positions hold the same digit.
- So mirrored cells can never share a row, column or box.
- An odd-length line has a free middle cell.
Solving tips
Pair up mirrored cells
Number the cells from both ends. On a 7-cell line, cells 1 and 7 match, 2 and 6 match, and 3 and 5 match. Cell 4 in the middle has no partner.
Copy candidates across
If one cell can only be 2, 5 or 8 and its mirror can only be 5, 8 or 9, both cells must be 5 or 8. Update both cells every time either one changes.
Borrow each end's limits
Mirrored cells hold the same digit, so they must sit in different rows, columns and boxes. If one end is in a box that already has a 7, the other end can't be 7 either.
Even lines turn at the middle
An even-length line has two middle cells that must match. They touch, so they can only be diagonal neighbors in two different boxes.
A worked example: a diagonal palindrome across two boxes
A 5-cell gray line runs diagonally from row 1, column 1 (r1c1) down to r5c5. So r1c1, r2c2 and r3c3 are in the top-left box, and r4c4 and r5c5 are in the center box. The top-left box already has 3, 6 and 9 off the line. The center box has 1, 4 and 7 off the line. Row 1 has a 2, row 4 has an 8, and column 5 has an 8.
- Pair the cells. r1c1 matches r5c5, and r2c2 matches r4c4. The middle cell, r3c3, has no partner.
- Each pair has one cell in each box, so its digit must be free in both boxes. The top-left box can still use 1, 2, 4, 5, 7 and 8. The center box can still use 2, 3, 5, 6, 8 and 9. The only digits in both lists are 2, 5 and 8, so all four paired cells come from 2, 5 and 8.
- Now the end pair. Row 1 has a 2, so r1c1 can't be 2. Column 5 has an 8, so r5c5 can't be 8. The pair shares one digit, so it can't be 2 or 8. Both ends are 5.
- Next the inner pair. r4c4 can't be 8 because row 4 has an 8. It can't be 5 because r5c5 is 5 and both cells are in the center box. So r4c4 is 2, and r2c2 is 2 as well.
- The middle cell is in the top-left box with 3, 6, 9, 5 and 2 already used. It can be 1, 4, 7 or 8, and the line itself gives it no extra limits.
- Carry the new digits outward. The two 5s sit in rows 1 and 5 and columns 1 and 5. The two 2s sit in rows 2 and 4 and columns 2 and 4. Remove 5 and 2 from every other cell in those rows and columns before you move on.
Practice
Reference table
| Cells | Matching pairs | Unpaired middle cell |
|---|---|---|
| 2 | 1 and 2 | none (cells 1 and 2 pair up) |
| 3 | 1 and 3 | 2 |
| 4 | 1 and 4, 2 and 3 | none (cells 2 and 3 pair up) |
| 5 | 1 and 5, 2 and 4 | 3 |
| 6 | 1 and 6, 2 and 5, 3 and 4 | none (cells 3 and 4 pair up) |
| 7 | 1 and 7, 2 and 6, 3 and 5 | 4 |
| 8 | 1 and 8, 2 and 7, 3 and 6, 4 and 5 | none (cells 4 and 5 pair up) |
| 9 | 1 and 9, 2 and 8, 3 and 7, 4 and 6 | 5 |
Going deeper
Why the line must bend
Mirrored cells hold the same digit, so they can't share a row, column or box. That rules out a lot of shapes. A palindrome of 2 or more cells can never run straight along one row or column, since its two ends would share it. It can't sit inside one box either.
So the shape of the line is itself a clue. For each pair, check that the two cells are in different rows, different columns and different boxes. If a pair fails any of these, you have misnumbered the cells. Count the cells along the line from each end, one cell at a time, and pair again.
Pairs carry digits between boxes
Each pair puts the same digit into two boxes, two rows and two columns. So a pair whose cells are in box 1 and box 5 means those two boxes share that digit in known places. If a digit is placed anywhere else in box 1, that digit is off the pair, and so it is off the pair's cell in box 5 too.
Keep the candidate lists of each pair identical at all times. Any digit removed from one cell comes off its partner. Over a long line this adds up. A single row clue at one end can finish a cell deep inside another box. It also helps to write each pair's candidates in both cells as center marks, even when one cell looks settled. Seeing the same list twice makes it easier to spot when the two cells drift apart.
Common mistakes
Thinking all digits must be different
A palindrome only says mirrored cells match. Two cells that are not a pair can hold the same digit, as long as sudoku rules allow it. Don't treat the line like a renban or a box.
Updating only one cell of a pair
Every time you remove a candidate from one cell, remove it from the partner too. Solvers often clean up one end, move on, and later find the other end still holds digits that are no longer possible.
Miscounting on a line that doubles back
Long palindromes often bend several times. Count cells, one step at a time, from both ends. Counting corners or straight segments gives the wrong pairs.
In mixed-rule puzzles
A palindrome can overlap other lines, and then each pair must satisfy both rules. On a German whispers line, a palindrome's even-length middle pair would have to differ by 5 while also being equal, so that can never happen. On a parity line, mirrored cells have the same parity, which means the two ends of an odd-length line are both odd or both even, and on an even-length line the middle pair would need two different parities while being equal, which is impossible. A palindrome that is also a thermo is impossible for 2 or more cells, since the ends can't be equal and rising. With killer cages, a pair that sits inside one cage can't happen, since cage digits don't repeat, but pairs in two different cages carry the same digit into both sums.
What the video shows
- A gray line reads the same from either end.
- Start with 7, then 2 in the middle...
- ...so the far end must be 7 as well.
Questions solvers ask
What is a palindrome line in sudoku?
It is a line whose digits read the same from either end, like 3, 8, 1, 8, 3.
Can a palindrome line have an odd number of cells?
Yes. The middle cell of an odd-length line has no partner, so it can be any digit the other rules allow.
Do all the digits on a palindrome line have to be different?
No. The line only requires mirrored cells to match. Other cells may repeat if sudoku rules allow.
What color is a palindrome line?
Usually gray. Check the rules text, since colors vary between puzzles.
Can a palindrome line be inside one box?
Only if it has a single cell. Any longer line would put two mirrored cells in the same box, and they would need to hold the same digit.
Can a palindrome line run straight along a row?
No. Its two end cells would both be in that row and would need the same digit.
How do you find the middle of a palindrome line?
Count the cells. On an odd-length line, the middle cell is the one with the same number of cells on each side. On an even-length line, the two middle cells form a pair.
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