We set this puzzle to highlight a particular mathematical property and stunning pattern that emerges from a simple constraint. It was featured on Cracking the Cryptic as *A Modular Miracle* on 31 August 2022. It also appears in our Indigestion Collection 2 and our Showcase Puzzles.

Estimated difficulty: 3.5/5

*Rules*

Normal sudoku rules apply: every row, column, and 3x3 box must include a complete set of the digits 1-9.

Killer cages: digits within cages marked by dashed lines must sum to the number shown in the upper left corner of the cage.

Every diagonal of length 7, 8, or 9 is either modular or unimodular. On modular lines, every set of three sequential digits contains one digit from {147}, one from {258}, and one from {369}. On unimodular lines, all digits are from the same class (all from {147} or all from {258} or all from {369}).

A harder version of Box Twist, Starburst first appeared in the Caging Constraints puzzle pack by BremSter. Starburst also appears in our Indigestion Collection 2.

Estimated difficulty: 3/5

*Rules*

Normal sudoku rules apply: every row, column, and box must contain a complete set of the digits 1-9.

Normal killer sudoku rules apply: digits inside a cage marked by dashed lines cannot repeat and must sum to the number shown in the upper left corner.

Modular lines: every set of three sequential digits along a line must contain a complete set of residues modulo 3, i.e. one digit from {1,4,7}, one from {2,5,8}, and one from {3,6,9}. Digits may repeat on a line if allowed by other rules.

This puzzle serves as a gentle introduction to how the modular lines and killer cage logic interact. It originally appeared as "Full o'Beans, Missing Broccoli" in the Caging Constraints puzzle pack by BremSter. This puzzle also appears in the GAS Collection.

Estimated difficulty: 1.5/5

*Rules*

Normal sudoku rules apply: every row, column, and box must contain a complete set of the digits 1-9.

Normal killer sudoku rules apply: digits inside a cage marked by dashed lines cannot repeat and must sum to the number shown in the upper left corner.

Modular lines: every set of three sequential digits along a line must contain a complete set of residues modulo 3, i.e. one digit from {1,4,7}, one from {2,5,8}, and one from {3,6,9}. Digits may repeat on a line if allowed by other rules.

This puzzle was set for, and initially appeared in, the Caging Constraints puzzle pack curated by BremSter. You can find the entire puzzle pack here.

Estimated difficulty: 2.5/5

*Rules*

Normal sudoku rules apply: every row, column, and box must contain a complete set of the digits 1-9.

Normal killer sudoku rules apply: digits inside a cage marked by dashed lines cannot repeat and must sum to the number shown in the upper left corner.

Normal disjoint groups rules apply: digits in the same relative position within different 3x3 boxes must be different.

Our first puzzle of August 2022! We are very happy to have had this puzzle featured on two different channels: by BremSter on 7 October 2022 and by Sudoku Sauce on 9 October 2022.

Estimated difficulty: 1.5/5

*Rules*

Normal sudoku rules apply: every row, column, and box must contain a complete set of the digits 1-9.

Anti-knight constraint: Cells a chess knight's move apart cannot contain the same digit.

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