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 puzzles.

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 5x5 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 6x6 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.