Cutting capers – a setter’s tale by Ploy
The idea of a Henry Ernest Dudeney crossword had been in the back of my mind for some time, and his minimal triangle-square dissection seemed a possible candidate. Dudeney (1857-1930) was a prolific English inventor of puzzles, many of which were published in magazines and newspapers of the time, and in his own puzzle collections. He was particularly good at finding solutions to plane dissection problems, where one shape was to be cut into pieces and reassembled into another.
As published in his “Canterbury Puzzles”, The Haberdasher’s Problem is to find a minimal dissection of an equilateral triangle into a square. For the purposes of a crossword, I decided that starting with a square, rather than a triangular, grid would be more tractable. Exploring the geometry of the dissection soon revealed that many of the dimensions would be irrational numbers, especially as the fourth root of 3 was involved! The quest was then to see if this could be brought into the restricted world of the crossword grid with reasonable accuracy. Testing square grids with sizes 10×10 to 16×16, I was delighted to find that a 14×14 grid could be cut from external cell junctions to a point roughly half way along an internal cell edge with an overall dimensional accuracy of 0.7% or better for the dissection. (One solver calculated the angles of the resulting “equilateral” triangle and found them to be within 0.8% of 60°.)
It was then a matter of turning this into a hopefully entertaining puzzle which would celebrate Dudeney’s achievement. I wanted the word HABERDASHER to appear in the completed grid as a hint, and was very pleased to note that the word BERDASH would be very helpful in this respect. The draft puzzle was then sent to my trusty test-solver. This brought to light a feature which was entirely serendipitous – there was a second way of reassembling the four pieces which also gave Dudeney’s full name in the grid, but which used a different “N”. However, the resulting shape looked a little like a boat, so did not correspond to Dudeney’s dissection. The Magpie editors also discovered this, which led to the strengthening of the preamble wording to clearly guide solvers to the correct solution.
Another aspect was that the final grid to be submitted was triangular, whereas the solution grid in the Magpie entry form needed to be the original, square grid. The warning notice on these entry forms points to such possibilities.
Overall I was very pleased with comments made by solvers, and am grateful to anyone who expressed their views. That it’s impossible to please everyone was neatly summed up by these two comments: “I really did groan when ‘cut’ appeared in the instructions – I hate these ones” and “A welcome opportunity to get the scissors out”!
Returning to Dudeney and this particular dissection, I wondered how he might have arrived at it. A well-known approach to some of these dissections is to superimpose suitable, equal area, tessellations of the two shapes in question and look for an alignment that minimises the number of cut pieces. After a little experimentation, I found the following:
I feel it must be unique, but have no way of proving it, and would like to think that Dudeney was aware of it, but will probably never know!
As for puzzle’s title, “Turn, Turn, Turn”, this reflected the fact that any three of the four pieces can be turned, as if on hinges, to effect the transformation. The animation at http://en.wikipedia.org/wiki/Dudeney is particularly striking.