How the cousins are made

Cut a circle from the square that circumscribes it and four offcuts are left, one at each corner. Every cousin is that circle with some of its own offcuts put back — one, two, three or four of them, in a new position. Nothing is added and nothing is thrown away, so a cousin with all four pieces returned has exactly the area of the square it came from: 4r².

Where a piece may go. An offcut has one concave arc and two straight legs. Its arc can lie back along the circle — which forces its arc-corner onto the circle’s centre, so there are exactly four such positions — or a leg can be fixed flat against the leg of a piece already placed. Those two moves generate all 73 cousins. The offcut is its own mirror image across its diagonal, so it is only ever turned, never flipped: four turns, no reflections.

Why some can never tile. A cousin with k pieces, h of them on the circle, has 4−h quarter discs and k−h offcuts. The difference is 4−k, so anything with fewer than four pieces carries spare quarter discs that no other cousin can absorb. Those shapes leave one offcut-shaped hole per missing piece, per tile — and the hole is a corner piece, the same thing that was cut out to make it.