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Tap every digit that could go next to the 3.
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Level 3 · Lines
How to solve parity lines
Along the line, digits alternate between odd and even.
Along a parity line, digits alternate between odd and even. An odd digit always sits next to an even one on the line, and an even digit next to an odd one.
The line splits into two sets of cells, and every other cell shares a parity. Work out one cell and you know the parity of every cell on the line.
How setters write it
The same rule, worded the ways you’ll see it on puzzles.
- Adjacent digits on a parity line have different parity.
- Digits alternate between odd and even along the line.
Worth memorizing
- Every other cell shares a parity. Mark one cell and the whole line follows.
- There are five odd digits and four even ones, so long lines in one box are tight.
Solving tips
Every other cell matches
Cells 1, 3 and 5 on the line share a parity, and so do cells 2 and 4. If cell 1 is odd, cells 3 and 5 are odd too.
Evens run out first
There are five odd digits (1, 3, 5, 7, 9) and only four even ones (2, 4, 6, 8). A line with all nine cells in one box must start and end on odd digits.
Count odds and evens per box
Each box has five odd and four even digits. If a line already puts four even digits in a box, the other five cells of that box are all odd.
Neighbor sums are odd
An odd digit plus an even digit always makes an odd number. Two touching cells on a parity line can never add to 10, so they can never share an X or a two-cell cage of 10.
A worked example: a 5-cell parity line in one box
A parity line winds through five cells of one box. Number them 1 to 5 along the line. Cell 2 is a given 4. The box also has a 1 and a 9 in cells off the line, and the column of cell 4 already has a 2 and a 6.
- Cell 2 is even, so cells 1 and 3 are odd. Cell 3 is odd, so cell 4 is even, and then cell 5 is odd. The line reads odd, even, odd, even, odd.
- The box has five odd digits: 1, 3, 5, 7 and 9. The 1 and the 9 are off the line. The line needs three odd digits, so cells 1, 3 and 5 hold 3, 5 and 7 in some order.
- Cell 4 is even and can't be 4, which is already at cell 2. So it is 2, 6 or 8. Its column has a 2 and a 6, so cell 4 is 8.
- Now count the rest of the box. It has nine cells. Five are on the line and two hold the 1 and the 9. The last two cells must hold the even digits the line didn't use, which are 2 and 6.
- Nothing so far orders the 3, 5 and 7 on the line. Mark them as center marks in all three odd cells, and look to the rows and columns for the next step.
Practice
Reference table
| Cells | First cell | Odd cells | Even cells | Possible |
|---|---|---|---|---|
| 2 | odd | 1 | 1 | yes |
| 2 | even | 1 | 1 | yes |
| 3 | odd | 2 | 1 | yes |
| 3 | even | 1 | 2 | yes |
| 4 | odd | 2 | 2 | yes |
| 4 | even | 2 | 2 | yes |
| 5 | odd | 3 | 2 | yes |
| 5 | even | 2 | 3 | yes |
| 6 | odd | 3 | 3 | yes |
| 6 | even | 3 | 3 | yes |
| 7 | odd | 4 | 3 | yes |
| 7 | even | 3 | 4 | yes |
| 8 | odd | 4 | 4 | yes |
| 8 | even | 4 | 4 | yes |
| 9 | odd | 5 | 4 | yes |
| 9 | even | 4 | 5 | no (only 4 even digits) |
Going deeper
Five odd, four even
Every row, column and box holds five odd digits and four even ones. That imbalance is the main tool for parity lines. One set of line cells (every other cell) can hold at most four cells in a row, column or box if it is even, and at most five if it is odd.
So if every other cell of a line puts five cells in one box, that set must be odd. It then uses all five odd digits in the box, and every other cell of that box, on or off the line, is even. The table on this page shows which lengths allow each starting parity.
Parity and sums
Two neighbors on the line are one odd and one even, so they always add to an odd number. Three cells in a row along the line add to an even number if they go odd, even, odd, and an odd number if they go even, odd, even.
Use this with cages, arrows and sandwich clues. A 2-cell stretch of line inside a cage of 10 is impossible, since 10 is even. If a 3-cell stretch starts on an odd digit and sits in a cage of 15, something is wrong with your coloring, since that total should be even.
Lines that fill most of a box
When a parity line covers 8 cells of one box, it has four odd and four even cells. The box needs five odd digits, so the ninth cell, off the line, must be odd.
When a line covers all 9 cells of a box, or all 9 cells of a row or column, it must start and end on odd digits. The odd cells are cells 1, 3, 5, 7 and 9 along the line, and they hold the five odd digits. The even cells are cells 2, 4, 6 and 8, and they hold 2, 4, 6 and 8.
Common mistakes
Thinking the line is all odd or all even
A parity line alternates. It is odd, even, odd or even, odd, even. Solvers who mix it up with odd and even cell markers sometimes fill a whole line with one parity.
Forgetting 5 is odd
5 is easy to skip when you list odd digits in your head. There are five odd digits (1, 3, 5, 7, 9), and each box needs all of them.
Assuming no repeats on the line
Cells two steps apart share a parity, and they can hold the same digit if they are in different rows, columns and boxes. The line only controls odd and even.
In mixed-rule puzzles
Parity lines combine cleanly with other rules. With Kropki, a black dot on a parity pair must be 1 and 2 or 3 and 6, since 2 and 4 and 4 and 8 are both even. A white dot always fits, since consecutive digits always have different parity. With X and V, no pair on a parity line can sit on an X, since X pairs add to 10, which is even. A V pair adds to 5, which is odd, so 1 and 4 or 2 and 3 both fit. A 3-cell renban that is also a parity line must have its middle digit in the middle cell, since that digit is the odd one out for parity. With odd and even cell markers, one circle or square on the line fixes the parity of every cell.
What the video shows
- Along the line, digits alternate between odd and even.
- 8 then 2 are both even.
- Odd, even, odd, even.
Questions solvers ask
What is a parity line in sudoku?
A line where neighboring digits alternate between odd and even.
What does parity mean in sudoku?
Parity means whether a digit is odd or even. The odd digits are 1, 3, 5, 7 and 9, and the even digits are 2, 4, 6 and 8.
Can digits repeat on a parity line?
Yes, if the cells don't share a row, column or box. The line only controls odd and even.
What is the difference between parity lines and odd-even sudoku?
Parity lines control neighbors along a line. Odd-even sudoku marks single cells with circles for odd digits and squares for even digits.
Can a parity line start with an even digit?
Yes. Either end can be odd or even. The line only needs neighbors to switch.
How many even digits can go on a parity line in one box?
At most four, since each box only has 2, 4, 6 and 8. A 9-cell line in one box must start and end on odd digits.
Do parity lines alternate high and low?
No. They alternate odd and even. 1, 2, 3, 4 is a valid parity line, and so is 9, 2, 7, 4.
RAT RUN 5: Disparity
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