Thomas Snyder (aka Dr. Sudoku) is a two-time World Sudoku Champion and five-time US Puzzle Champion, as well as the author of several books of puzzles. His puzzles are hand-crafted, with artistic themes, serving as a kind of “cure for the common sudoku.” Each week he posts a new puzzle on his blog, The Art of Puzzles. This week’s prescription deals with Chaotic Calcu-doku, a twisted variation of the familiar number puzzle that breaks the normal rules.

Of the many complaints I hear about other sources of Calcu-doku puzzles (too easy/repetitive, too many clues/too many small cages, poor use of subtraction/division, …), one frequent comment is that the default 1-N number sets used in the puzzle eventually get boring. Once you’ve learned all the basics for the most common sets like 1-5 and 1-6, there isn’t much room for the puzzle to grow in difficulty without apparently needing to grow in size.

There are some simple fixes to this problem. First, there is no reason to use 1-N all the time. From introducing 0 as the first number, having different number sets, or even having completely unknown number sets, there is plenty of room to change the flavor of a puzzle with a unique set of numbers that isn’t 1-N. One of my favorite special number sets, used in my book TomTom Puzzles, was the first 6 Fibonacci numbers where having two 1’s in the set of possible numbers led to a lot of unusual possibilities compared to the standard puzzle.

But even then, the fact that a certain set of numbers must appear once in every row and column still constrains the puzzle a lot so that after getting a few numbers in, the values of the remaining cages are no longer as crucial compared to doing “sudoku-like” elimination steps. I’ve often wondered if using more open number sets would contain an interesting puzzle space, and some experimentation in this direction is the subject of this week’s prescription.

In Chaotic Calcu-doku, a range of numbers is defined with more possible members than cells in each row or column. While the normal rule that “no number repeats in a row/column” is maintained, there is not the same certainly that the last number must be X, because there will be more options for that last number. In the first 5×5 puzzle below, any number from 1-6 can be put into a cell, with an unknown quantity of each number used (there could be zero 6’s, resulting in a standard calcu-doku, or there could be one, two, three, four, or even five 6’s – you don’t know). In the second 6×6 puzzle, exactly four instances of each number from 1-9 must appear, obeying all other rules. Both puzzles should offer quite different challenges than standard calcu-doku puzzles. Enjoy!

Rules: Enter (the indicated quantity of) numbers from the given range into the grid so that each cell contains a number and no number repeats in any row or column. The sum or product of the numbers in each cage must match the indicated value given in the upper-left corner of the cage.

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