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Level 3 · Lines
How to solve entropic lines
Any three cells in a row on the line hold one low (1–3), one middle (4–6) and one high (7–9) digit.
Entropic lines split the digits into three groups: low (1 to 3), middle (4 to 6) and high (7 to 9). Any three cells in a row along the line must hold one digit from each group.
The rule turns the line into a repeating pattern. Once you know the groups of two neighboring cells, you know the group of every cell on the line.
How setters write it
The same rule, worded the ways you’ll see it on puzzles.
- Any set of three sequential cells along an entropic line contains a low, a middle and a high digit.
- Entropic lines.
Worth memorizing
- The pattern repeats every three cells. Cells 1, 4 and 7 on the line share a group.
- Once you know two cells’ groups, the third is forced.
Solving tips
The pattern repeats every three
Cells 1, 4 and 7 on the line are in the same group, as are cells 2, 5 and 8. If cell 1 is a 2, cells 4 and 7 are also low.
Two groups force the third
If one cell is 1 and the next is 8, the third cell must be middle, so it is 4, 5 or 6.
Neighbors never share a group
Two cells next to each other on the line are always in different groups. A 5 on the line can't sit next to a 4 or a 6.
Count groups in a box
A box holds exactly three digits from each group. If six cells in a row along the line sit in one box, they use two lows, two middles and two highs, so the box's other three cells hold one of each.
A worked example: a 6-cell line in one box
An entropic line snakes through six cells of one box. Number them 1 to 6 along the line. Cell 1 is a given 2 and cell 3 is a given 8. The box has a 3 in a cell off the line, and the column of cell 6 already has a 9.
- Cells 1 and 3 are low and high, so cell 2 must be middle: 4, 5 or 6.
- The pattern repeats every three cells. Cell 4 matches cell 1, so it is low. Cell 5 matches cell 2, so it is middle. Cell 6 matches cell 3, so it is high.
- Cell 4 is low and must differ from the 2 in the same box. The box's 3 is off the line, so cell 4 is 1.
- Cell 6 is high and must differ from the 8, so it is 7 or 9. Its column has a 9, so cell 6 is 7.
- Now count the box. The line holds two lows (2, 1), two highs (8, 7) and two middles (cells 2 and 5). The three cells off the line hold one of each: the 3, the 9, and the middle digit the line doesn't use.
Practice
Reference table
| Cells on line | Same group as cell 1 | Same group as cell 2 | Same group as cell 3 |
|---|---|---|---|
| 3 | 1 (1) | 1 (2) | 1 (3) |
| 4 | 2 (1, 4) | 1 (2) | 1 (3) |
| 5 | 2 (1, 4) | 2 (2, 5) | 1 (3) |
| 6 | 2 (1, 4) | 2 (2, 5) | 2 (3, 6) |
| 7 | 3 (1, 4, 7) | 2 (2, 5) | 2 (3, 6) |
| 8 | 3 (1, 4, 7) | 3 (2, 5, 8) | 2 (3, 6) |
| 9 | 3 (1, 4, 7) | 3 (2, 5, 8) | 3 (3, 6, 9) |
Going deeper
Three sets of cells in any order
An entropic line splits into three sets: cells 1, 4, 7 and so on, cells 2, 5, 8 and so on, and cells 3, 6, 9 and so on. Each set is all low, all middle or all high. The rule doesn't say which set gets which group, and it doesn't require low, middle, high in that order. Any order works, as long as it repeats.
Mark each set with its own color. Once one cell in a set is known, the group of that whole set is known. Once two sets are known, the third follows. It helps to watch rows and boxes where the groups are nearly used up. If a row already has all three low digits placed off the line, no line cell in that row can be low, and the set that cell belongs to can't be the low set.
Counting groups in rows and boxes
Each row, column and box holds exactly three low, three middle and three high digits. So one set of line cells can have at most three cells in any row, column or box. If a set has three cells in a box, those cells hold all three digits of their group.
This works outside the line too. Suppose the line puts two lows in a row. The row's third low must be in a cell off the line, and none of the row's other cells can be low. That can place a digit far from the line.
The same count works for a box that the line fills. If the line covers all nine cells of a box, each set of line cells has three cells there, and each set uses all three digits of its group. A 3-cell line inside one box holds one digit from each group, so the box's other six cells hold two of each.
Sums of three cells
Any three cells in a row along the line hold one low, one middle and one high. Their smallest possible total is 1 + 4 + 7 = 12 and their largest is 3 + 6 + 9 = 18.
This helps when the line runs through cages or arrows. A 3-cell entropic stretch inside a cage of 12 must be 1, 4, 7. In a cage of 18 it must be 3, 6, 9.
Common mistakes
Forcing a low, middle, high order
The groups can come in any order along the line. 8, 2, 5 is fine. The only rule is that any three cells in a row along the line have one of each.
Checking only one set of three
The rule applies to every three cells in a row along the line, and those sets overlap. On a 4-cell line you must check cells 1, 2, 3 and also cells 2, 3, 4.
Losing the pattern at a box border
The line doesn't reset when it crosses into a new box. Keep counting cells along the whole line, so the repeat of every three cells stays correct.
In mixed-rule puzzles
Entropic lines give a group for every cell, so they work well with anything that limits size. A white Kropki dot on an entropic pair can only be 3 and 4 or 6 and 7, since those are the only consecutive pairs from different groups. On a thermo, a 3-cell overlap must rise low, middle, high, so it is one digit from each group in that order. A 3-cell renban can never lie along three cells in a row of an entropic line, since three consecutive digits come from at most two groups. Parity lines add odd and even on top. For example, a low cell that must also be even is 2.
What the video shows
- Any three cells in a row on the line hold one low (1–3), one middle (4–6) and one high (7–9) digit.
- 1 is low, 8 is high, 4 is middle.
- 8, 4, 7 has two highs.
- So the next digit must be low again: 2 works.
Questions solvers ask
What is an entropic line in sudoku?
A line where every run of three neighboring cells has one digit from 1 to 3, one from 4 to 6 and one from 7 to 9.
Can two middle digits be next to each other on an entropic line?
No. Neighboring cells on the line always come from different groups.
What are the low, middle and high digits on an entropic line?
Low is 1, 2 and 3. Middle is 4, 5 and 6. High is 7, 8 and 9.
How are entropic lines different from whispers lines?
Whispers lines set a minimum gap between neighbors. Entropic lines sort digits into three groups and cycle through them every three cells.
Does an entropic line have to go low, middle, high in order?
No. Any order works. The line just repeats the same order every three cells.
What is the smallest sum of three cells on an entropic line?
12, from 1 + 4 + 7. The largest is 18, from 3 + 6 + 9.
Can a digit repeat on an entropic line?
Yes. Cells three apart on the line share a group, and they can hold the same digit if they don't share a row, column or box.
Record Highs
RAT RUN 17: Trapdoors
Nautilus (Entropic lines, XV, Kropki pairs)
Copycat Collaboration
Entropic Tree
Picking Blackcurrants