Jigthoughts

In the other place, at the start of series 1, I gave some thought to classifying the pieces of a jigsaw according to their shape and to the way that they fitted together, but without regard to colour, finish, pattern or material or to their number. I think it is time to give this matter some more thoughts.

We restrict ourselves to two dimensional, one-sided jigsaws which are flat and which are of circular, square or rectangular shape.

We restrict ourselves to jigsaws the pieces of which are essentially quadrilaterals. We specifically exclude the jigsaws which might be made on the basis of a triangle or a hexagon rather than a square. Never come across either, which seems odd, so maybe there is some catch which has not yet reached me.

We consider just the interior pieces. Boundary pieces are subject to boundary conditions, rather different to those of the interior.

Property 1. A jigsaw has property 1 if the pieces may be considered to be making up a rectangular array of so many rows and so many columns, with exactly one piece at each position of the array. Circular puzzles might exhibit a rotary or polar version of this property, although I do not recall one which does; circular puzzles generally being cut more freely than rectangular ones. But nor do I recall a circular puzzle in which the pieces have been cut on a rectangular grid superimposed on the circle; the cut is always at least vaguely polar, with the result that the center of a circle is a singularity which is not shared with rectangles.

Property 2. A jigsaw has property 2 if each piece may be considered to have four sides and for each side to carry either a hole or a prong. With holes being of a shape to hold a corresponding prong, so it should not be possible to slide one piece into position against another; the prong has to be lifted into the hole not pushed. Prongs and holes usually having this not pushing property; the only puzzle which I can remember which systematically failed this test was the foreign puzzle noticed in the other place on 12th April.

Property 3. A jigsaw has property 3 if each piece has a prong-hole-prong-hole configuration. No exotics like prong-prong-hole-hole or hole-hole-hole-hole. The addition of proportion of exotics generally makes a puzzle easier to solve.

Property 4. A jigsaw has property 4 if exactly four pieces meet at every vertex. One makes fewer mistakes with a jigsaw with this property - but on the other hand the interest is fewer too.

Property 5. A jigsaw has property 5 if all the pieces are of a size. The area of the largest is not more than (say) two times the area of the smallest.

Property 6. A jigsaw has property 6 if there is exactly one solution to the puzzle.

Property 7. A jigsaw has property 7 if any one piece fits exactly one position in the jigsaw, without regard to the image. A stronger version of property 6.

Property 8. Property 8 is a combination of property 1 and property 2, with the added qualification that each piece is roughly rectangular and sensibly placed in its position. It's center of gravity is not far from the center of the position. The four prongs and holes are roughly cardinally pointed cruciform with respect to the center of gravity. Variation is permitted, but of a modest and demure variety; no extravagance or whimsy.

Rectangular puzzles from charity shops are generally fairly well behaved with regard to these properties.

The next step is to consider the extent that these properties are independent, one from another. Can a jigsaw have property A while not having property B?