Before I started this blog, I explored polyominoes with cells individually labeled with numbers. I called these sumominoes, as I was looking at sets of all polyominoes with a given sum. Erich Friedman discovered them independently, and called them weightominoes in his July 2009 Math Magic Problem of the Month. I prefer his term for the general concept, as there is no reason they need to be grouped by sum. Both Friedman and I looked at problems where the goal was to overlap these polyominoes in a rectangle so that every cell had the same sum of labels. While I looked at a particular complete set, Friedman looked at pilings of multiple copies of the same cell numbered polyomino. (Aside: “pilings” isn’t a standard term, but it’s a concept we need a term for. We have “packings” for deficient tilings that don’t fill a space, so “pilings” for abundant tilings that fill it with overlap. Then a “tacking” is when there is both empty space and overlap, of course.)
In this kind of problem we looked at sums of cells in the “z” direction. But we could instead look in the x and y directions. There is a common type of figure where we do this already: magic squares!
For this type of problem, excluding 0 as a potential cell label isn’t necessary. Standard numbered dominoes include 0 (blank) as a label, so we might want to do the same for physical puzzles using pips. (Pip patterns are preferable to numerals for physical puzzle pieces since they don’t have a preferred orientation.)
In fact, in the context of standard dominoes, examples have existed for some time. Here is a domino magic rectangle using a full set of double-six dominoes. The row sums are all 24, and the column sums are 21.
Of course, the fact that it can be done doesn’t make it a good puzzle, and working with a full set of dominoes might get tedious. Since I’ve been looking for simple puzzles with small piece sets, I tried to find one in this format. There are two cell numbered dominoes and four L-trominoes with a cell sum of 2. Their total area is 16, good for a 4×4 square,. and their total label sum is 12, giving a row and column sum of 3. One nice thing about a magic square type puzzle is you get an extra challenge for free. Finding a configuration with just row and column magic sums is a fairly light challenge, but getting the main diagonals to also match the magic sum is much harder. I had a small number of these made to give away at the 2022 MOVES conference:
Looking upward in size, there are 12 trominoes with a cell sum of 3, good for a 6×6 square with line sums of 6. I made a prototype:
But this isn’t quite as great as a puzzle, which is why I didn’t bother to conceal the solution as a spoiler. The reason is that it’s easy to make subunits where pip sums are preserved on applying a symmetry action. For example, in the solution above the three I trominoes in the upper left can be permuted in any order without changing row or column sums. Likewise, the two 2×3 rectangles formed from two L trominoes on the bottom can be flipped over one axis. This makes it much easier to turn a semimagic (row and column only) solution into one where the diagonals also work. (Subunits like these can appear in the smaller puzzles here, but there isn’t really enough room for them to dominate a solution.)
What I really want from going one step up from a 4×4 puzzle is a 5×5 one. Well, if we exclude the pieces with 3’s, we have an area of 24, which is almost right. We can make a 5×5, but it would have an unfortunate hole:
Or a fortunate one! Since my pips were lasered out holes to begin with, the big hole should clearly just count as one hole for the purpose of line sums. Now we have 25 holes, and 5 will work as the line sum. (While I normally like rounded corners for pieces since they have a softer tactile feel, if I made more of these, I’d use sharp corners and square pip holes, to visually unify the two different hole sizes.)
Is there anything else we can do with cell numbering? Polyhexes seem promising since there is an extra direction for sums to happen on. And perhaps we can use the numbers for something other than sums. I’m sure there are more creative discoveries waiting to be made!
