One strand, through every helix, once

• 24 helices of 64 bases need 1536 of the 7249-base scaffold — 21% of it ×7
• the shape's 24 helices are one connected piece ×7
• the route visits all 24 helices exactly once, every step between lattice neighbours ×5
• along the top row 3 of the 5 helices have no neighbour off the row, and the two corners have one neighbour each ×4
• so a route must start and end at the two corners and pass along the whole row between them — 5 helices of 25 — and the exhaustive search agrees there is none, although the colours balance 13 to 12 ×4
• the route visits all 30 helices once, and every step is between honeycomb neighbours ×3
• every one of the 12 placements is drawn, not a sample of them ×2
• every shape any cheap test refuses is searched anyway, and none of them has a route — 691,268 shapes on the square lattice ×2
• every shape any cheap test refuses is searched anyway, and none of them has a route — 691,268 shapes on the square lattice, 3,898,510 shapes on the honeycomb lattice ×2
• the exhaustive search refuses all 12 of them, in 748 to 920 partial routes each ×2
• this 11-helix shape passes the colour of every forced end, the exhaustive search finds no route ×2
• 1 shape is balanced and still unroutable — the count is necessary and not sufficient ×1
• 1 shape is refused by one pass over the cells, before any search ×1
• 20 of the 80 honeycomb blocks do not route, against none on the square lattice ×1
• a helix built to the square lattice at 10.5 bases a turn is out by 17.1° over 32 bases and by 137° over 256, and one built to the honeycomb by nothing ×1
• at 10.5 bases a turn every crossover the honeycomb offers in 42 bases lands exactly — 7, 14, 21, 28, 35, 42 ×1
• at one crossover period a design expects 201 of its domains to have somewhere else to go, and at 11 bases it expects fewer than one ×1
• each added test leaves the smallest unrouted shape the same size or larger — 9, 11, 11 helices on the square lattice; 12, 15, 16 helices on the honeycomb lattice — and the honeycomb's is never the smaller ×1
• each added test leaves the smallest unrouted shape the same size or larger — 9, 11, 11, 12 helices on the square lattice; 12, 15, 16, 16 helices on the honeycomb lattice — and the honeycomb's is never the smaller ×1
• each family is grown one step at a time and both ceilings are tested at every step, so which one stops it is found rather than predicted ×1
• every block from one by one to 10 by 8 routes on the square lattice ×1
• every shape any cheap test refuses is searched anyway, and none of them has a route — 3,898,510 shapes on the honeycomb lattice ×1
• every shape is put through the colour count and through the exhaustive search, and the verdict is what the two of them said together ×1
• every test has a smallest survivor on both lattices inside the sizes listed, so every row of the table is a measurement rather than a blank ×1
• every test's survivors are counted at every size from 4 to 16 helices ×1
• no route exists although the colours are balanced 5 to 4 — the parity test is necessary and not sufficient ×1
• no route exists, and the colour counts 1 to 4 said so before the search ran ×1
• on both lattices every set of cheap tests has a smallest shape it passes and no route reaches, within the sizes listed ×1
• on the honeycomb every block odd in both directions, from three by three up, passes the colour count and has no route ×1
• one helix cuts off a piece with no end in it, so the route has to finish there, and the stretch it must cross first has its colours the wrong way round ×1
• over 8 shapes the colour count never refuses one the search can route — the necessary condition holds ×1
• seven bases at ten and a half to a turn is exactly two thirds of a turn, and eight bases at thirty-two thirds is exactly three quarters ×1
• staying bound is the harder of the two demands — 17 bases against 11 for being unique ×1
• the hand-drawn raster is a valid route too, so the abstraction agrees with what a designer draws ×1
• the shortest domain the honeycomb admits that clears the mean leaves 15 domains below the hold on the sequences tried, and the shortest the square lattice admits leaves none ×1
• the shortest lengths really do repeat in one sequence and the longest really do not, so the expectation is checked against a sequence rather than trusted ×1
• the strand's 7249 bases thread 113 helices, and a shape that cannot be routed fails at whatever size it first cannot be ×1
• the tally and the count agree on every survivor ×1
• the two lattices are exact at 10.500 and 10.667 bases a turn, and nowhere else in the range ×1
• the weakest domain of this sequence melts 14.6 degrees below the mean, so the mean is not the number an anneal is held above ×1
• this 11-helix shape passes the steps the ends force, the exhaustive search finds no route, and the colour of every forced end refuses it ×1
• this 12-helix shape passes every test including the corridor count — its junctions need 1 end and it has 1 — and the search finds no route ×1
• this 16-helix shape passes every earlier test and has no route, and the corridor count refuses it: its junctions need 3 ends inside corridors and it has 1 to give ×1
• this 9-helix shape passes the colour count, the ends and the cuts, the exhaustive search finds no route, and the steps the ends force refuses it ×1
• two crossover periods is not enough on either lattice — 37 and 43 degrees against a hold at 45 — and three is enough on both ×1
• while the square lattice's neighbours a quarter turn round are never faced exactly — the nearest bases miss by 4.3° ×1

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