The case notes
When a diagonal is modular or unimodular
Every diagonal with at least seven cells, in either direction, is either modular or unimodular. modular means every three consecutive cells contain one digit from each of one-four-seven, two-five-eight, and three-six-nine. unimodular means the entire diagonal stays within one of those groups. Watch a worked example, then try the idea in a puzzle.
Put it into practice
Diagonals of 7+ cells: modular or unimodular
Every diagonal with at least seven cells has to choose one pattern: modular or unimodular. That includes diagonals in both directions, whether drawn or not.
1/4/7 · 2/5/8 · 3/6/9
The three groups are one-four-seven, two-five-eight, and three-six-nine. A modular diagonal visits all three in every run of three cells. A unimodular diagonal stays in just one group.
8 · 1 · 9 → two-five-eight
8, 1 and 9 belong to three different groups. This diagonal cannot be unimodular; it must be modular. The next cell returns to two-five-eight.
Row 4 leaves 5
2 and 8 already use the other digits from that group in row four. So the next diagonal cell is 5.
4 · 7 · 1 → one group throughout
Now take the crossing diagonal. 4, 7 and 1 are all in one-four-seven. That rules out modular, so the entire diagonal has to stay in this one group.
Row 4 leaves 4
1 and 7 are already in row four. The next cell on this diagonal must therefore be 4. Each qualifying diagonal makes its own choice.
Read the transcript
Every diagonal with at least seven cells has to choose one pattern: modular or unimodular. That includes diagonals in both directions, whether drawn or not. The three groups are one-four-seven, two-five-eight, and three-six-nine. A modular diagonal visits all three in every run of three cells. A unimodular diagonal stays in just one group. 8, 1 and 9 belong to three different groups. This diagonal cannot be unimodular; it must be modular. The next cell returns to two-five-eight. 2 and 8 already use the other digits from that group in row four. So the next diagonal cell is 5. Now take the crossing diagonal. 4, 7 and 1 are all in one-four-seven. That rules out modular, so the entire diagonal has to stay in this one group. 1 and 7 are already in row four. The next cell on this diagonal must therefore be 4. Each qualifying diagonal makes its own choice.
