Square Grid Puzzles

by Alexandre Owen Muñiz

I don't tend to spend much time on puzzles like Sudoku or KenKen. There's a universe of new mathematical problems waiting to be discovered, so why should I waste time trying to solve yet another Sudoku? Occasionally, however, I get an inspiration to design my own puzzle variant. The ones I've come up with so far are below. It's probably to be expected that they all involve poyominoes in some way.

Magic Square Sudoku

When I learned about Sudoku, it seemed like it would make an obvious match with magic 45-ominos, since both use a 3×3 grid of 3×3 blocks. I wanted to create a Sudoku puzzle on a grid with a magic 45-omino which would be relevant to the puzzle. Aaron Humphries came up with the crucial idea that numbers inside the polyomino should be required to be no larger than the number of polyomino squares inside that 3×3 block. After that, the magic square of givens seemed like the most natural way to give enough information to make working puzzles. (It wasn't quite, so I added the constraint that the diagonals must also contain all digits.) David Eppstein wrote a program to find arrangements of givens; the following made the easiest puzzle, but I still found it to be quite a serious challenge.

In addition to the normal Sudoku rules, the following rules apply:

• Each of the main diagonals must include each digit between 1 and 9.
• The digits in the shaded cells must be no larger than the number of shaded cells in the 3×3 blocks they are in. (This number is the given in each 3×3 block.)

Uwe Wiedemann has come up with his own variation on my Magic Square Sudoku, which uses different constraints on the placement of the blue squares. A site called printsudoku.com also has so-called Magic Sudokus, but they don't use any restriction at all on the placement of the blue squares, and moreover, they have extra givens, and no magic square of givens. That hardly seems all that magic to me! (There are also a number of puzzles out there calling themselves Magic Square Sudoku that utilize magic squares in different ways than I have, and these have just as much right to the name as mine does.)

Pentominous

The Rules

Divide the squares of the puzzle (white squares only) into a number of pieces. Each piece must contain at least one letter. It must be possible to place each pentomino in exactly one position such that:

• it is entirely within one of the pieces, and
• it contains the letter corresponding to itself.

(And if you don't remember off the top of your head which pentomino corresponds to which letter, Wikipedia's pentomino page may help.)

This was inspired by Zotmeister's polyominous puzzles.

Ice Farm

This one comes with a story.

Sator Arepo is a Martian ice-pyramid farmer. Martian ice-pyramids naturally align themselves either in one of the four cardinal directions, or pointing straight up. A wondrous property of ice-pyramids is that when they grow in a five by five grid, each row and column will contain exactly one pyramid of each orientation.

Arepo has divided his field into five plots, each with a separate pyramid colony. When a pyramid is pointing toward a neighboring colony, it radiates mystical ice-energy onto that colony. (The mystical ice-energy does not continue on to other colonies beyond the nearest neighbor, but it will blow past other pyramids belonging to the same colony on its way.) A standing pyramid warms its colony, removing a unit of mystical ice-energy from it. Arepo has carefully made sure that each colony is netting one or two units of ice-energy, as is preferred for optimal ice-pyramid growth, and has kept a map showing how much mystical ice-energy each colony receives.

Now a storm has blown dust over Arepo's field, completely covering all but three of the ice-pyramids. Help Sator Arepo clear the dust off of his field, and find out how his pyramids are aligned!

If you have questions, comments, solutions to unsolved problems, or ideas for new, related problems, please email me.

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