Not strictly polyominoes but close enough really. I have no idea how I first found out about this set of pieces - I always had a hunch it was the set used in Martin Watson's puzzle '
DemiTri' (which doesn't look like it's on the site any more) but that says 12 pieces. And attempts to create similar sets in Peter Esser's
program by slicing tetrominoes or adding half-squares to triominoes yields 12 and 13 piece sets respectively.
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| Fig. 1: The set, crudely rendered in Microsoft Paint. |
But this looks like a complete set to me, all the ways of putting together three squares and a triangular half-square (if there's a fifteenth one and I've missed it let me know) and it's got a total area of 14 x 3.5 = 49 unit squares, which suggests (among other things) a 7x7 square:
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| Fig. 2: Here's one I made earlier. |
Technique for solving these is a tad unusual. Since they each have one diagonal side, if the outer perimeter of the shape you're filling has no diagonal sides than the pieces are effectively 'paired up' by joining two at the diagonal edge. This results in any shape like this being split into seven heptominoes which can be in turn split in half to give two pieces. So my technique was to first put together a couple of promising looking heptominoes (i.e. ones containing 2x2 or 2x3 rectangles) then trying to fit those together. I used this method to cobble together the shapes below.
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| Fig 3. 10x5 with a bite taken out of it. |
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| Fig. 4: These. Which can be put together to make the shape in Fig. 3. |
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| Fig. 5: More shapes! |
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| ...and this nightmare shape that I found with a solver because there's no way I'd have the patience to do it by hand. |
Sadly, these seem to be more limited with what you can do with them compared to, say, pentominoes or hexiamonds, both of which have a similar number of pieces (that, or I'm just really uncreative. I have a hunch it may be the latter.)
And there's also a scary bonus thought - this set of pieces is just one in a family. There's scope for doing things with the sets of pieces which are four squares and a triangle*, or two squares and two triangles, and so on, and at that point we're approaching just regular sets of polyaboloes or polytans or whatever people generally call them.
But that's going to have to be a post for another time.
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* If I counted correctly there's an odd number of these which might further limit what can be done with them.
A_SUGGESTED_SHAPE:
ReplyDelete14 parts, with one diagonal each, suggests a pair of parallel diagonals, each 7 long. Picture one such line, made of 7 tans, & join them together via 8 squares, so that each tan is between 2 squares. Now, if the long edge of the tans are horizontal, add a square to the SE-edge of each square in the row. Repeat this to add a 3rd, 4th, 5th, & 6th row of 8. Now smooth out the bottom row with the last 7 tans.
Having placed 48 squares & all the tans, you're left with a choice of 6 square holes, to place as you see fit.
I think (if I'm picturing this shape properly) it might not be possible to solve - the piece that's a square tetromino with a pie slice taken out of it, and the three pieces in the middle column of the first image in the post all have the tan placed in a way that would cause a square or squares to reach out over the diagonal edge.
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