The Geograph Project

Six years ago, I signed up to take part in the Geograph project in the U.K. http://www.geograph.org.uk/

This is a mixture of eduaction and fun, and a challenge and a game. Photographs of places (not people) are linked to the Ordnance Survey (O.S.) grid-square in which the picture was taken. The O.S. maps are divided into squares with side 1km (i.e.metric) and any place in the U.K. can be located by a grid reference. I am sitting at SX 9309 9189 which locates me to within 10 metres. Some webpages will accept that and locate my home on a map or satellite.

The aim of those who signed up at first was to be the first to obtain a photograph for a grid square, and I recorded first "Hits" for about 100 of them. Most of mainland Britain has now been "Geographed" so participation for many people means extending the range of pictures. Within many grid squares there are numerous sites and sights to record, and those who are part of the project try to extend the range in various ways, and to add more squares to their personal tally. At the time of writing, I have photographs recorded in 1118 squares with a further 36 where my picture is only of a close-up detail.

So where does the link with O.R. apply? It first comes with the problem of planning an expedition to add further squares to that total. This could be seen as a variant of the orienteering problem, of finding ones way around check-points in the shortest time. Except that there are no check-points, all one wants are pictures from a square. So in an ideal world, one could stand at the corner of four grid-squares, and turn north-east, south-east, south-west and north-west and take pictures in each direction with negligible distance covered. To get a further two squares, you would have to walk one kilometre to the next intersection and take two more squares. Or you could walk 1.414.km diagonally and photograph three more squares. (I say walk, but obviously, you can travel in any way that you like.) So, travelling horizontally or vertically means that you obtain 4+2N pictures with a distance of N kilometres, i.e. an average of 2+4/N pictures per km. travelling diagonally gives 4+3N pictures for a walk of 1.414N kilometres, an average of
2.121 +2.828/N pictures per km. If you want 6 pictures, walk along a grid line.
If you want 7, go diagonally, If you want 8, go along a grid line, If you want 9 or 10, go diagonally. My reckoning is that for 21 pictures or more, the diagonal is best, but below that, you need to compare the strategies.

Letting H be along a grid line, D be diagonal
5H, 6H, 7D, 8H, 9D, 10D, 11H, 12H, 13D, 14H, 15D, 16D, 17H, 18H, 19D, 20H,

The problem is more serious than this because roads and paths do not allow one to wander at will. So the problem becomes more realistic when you start to impose such constraints, and to impose the obvious condition of returning to the start point. That is left for a future occasion.