## May 11th, 2005

### Puzzle 8: Magnetic Field

I just discovered this puzzle design today - sadly, I do not know its origin. The instances I found all had even-sided grid lengths, gave all numbers for all rows and columns, and always gave at least one domino filled in to start. I promptly did away with all that silliness in trying to compose my own, and I rather like the result. This is a cunning puzzle design, seemingly simple but possessing room for cleverness and a peculiar aesthetic. It is primarily a puzzle of parity.

Left side puzzle, right side unique solution - and it's not just the forest green "ink" that makes it unique.

The object is to determine the polarization - or lack thereof - of each domino in the grid: either positive in one half and negative in the other, or neutral (non-polar) in both. Cells with matching polarity cannot be orthogonally adjacent. Numbers to the left of each row and above each column must match the number of positive cells in that row or column, respectively; likewise, numbers to the right of each row and below each column must match the number of negative cells in that row or column, respectively.

And, for those ever-so-curious list lovers:

1) The puzzle grid is divided into dominoes - that is, pairs of cells. Some of these dominoes can be "polarized" - that is, just like a magnet, they have a positive end and a negative end: one of their two cells is "positive" and is to be marked with a plus sign, and the other is "negative" and to be marked with a minus sign. Any domino that isn't polarized is neutral: neither end is positive or negative, and the whole domino is to be shaded. If all dominoes are so marked without violating any of the following rules, the puzzle is solved. [Any "black cells" in the starting grid are actually just holes in the pattern - you can think of them as neutral cells without hurting anything.]
2) In much the same way two magnets will repel each other when ends of matching polarity are close together, the grid wouldn't be stable if polarized dominoes pushed each other. For this reason, two cells with the same polarity can't share a side. (So you can't have two minuses next to each other, although they can touch at corners. Same goes for plusses.)
3) See the plus sign outside the upper-left corner? Each of the numbers to its right - each above a column of the grid - is the number of plus signs that must be placed in its column, no more, no less. The numbers below that plus sign do the same thing for the rows they're in front of.
4) And yes, that's a minus sign outside the bottom-right corner, and the numbers above it and to its left must match the number of minus signs in their respective rows and columns.

)

I hope you find this entertaining. It was a fascinating exercise trying to take this discovery and make my own working prototype of it in such a short period, having to learn the rules and techniques myself - and quickly - and then apply them to build a non-trivial example worthy of sharing with others. Let me know what you think of it here, and of course, email me your solution if you solve it. - ZM