An ant is placed on a large sheet of graph paper on which numerous concentric squares have been marked in ink, indicating each successive odd prime starting with 3. The ant starts moving North, from the centre square of the 3x3 block, and will alternately;
1. Turn right when it reaches an ink line (which seems like a ‘wall’ to the ant); and
2. Resignedly clamber over the ‘wall’ and continue to scurry forwards in its current direction until confronted by another wall.
Hence the first few moves of the ant will be: North 1 step, turn right, East 2 steps, turn right, South 4 steps, turn right, West 7 steps, turn right, and so on.
Question: How many steps will the ant have had to take by the time it climbs over the 10,000th ‘wall’?
(In reply to
computeraided solution by Charlie)
Charlie,
Excellent! but there was no need to trouble with the halfstep (I'd hoped this was clear from the examples I gave).
Perhaps the most interesting thing about Ant’s walk is the area enclosed by his path. If we treat the graph paper as  well, a graph  and ignore the row and column passing through his startingpoint as the ‘axes’, of the graph, then Ant’s walk ‘encloses’ these successive areas, in his current quadrant of the graph.: 1*2, 2*3, 3*5, 5*6, 6*8, 8*9,9*11,11*14, where the multipliers are A130290 in Sloane, and the products are A087427 in Sloane. If, on the other hand, Ant scuttles over the wall, then turns right (i.e. reversing the order) the areas are 2*1,4*2,5*4,7*5,8*7,10*8, where the multipliers are A067076 in Sloane.

Posted by broll
on 20101006 12:31:48 