Monday 26 April 2010

The Case for DR

To a hammer, every problem looks like a nail, and to the Liberal Democrats, every problem looks like one that can be solved by proportional representation.

The problem that's troubling us all at the moment is the prospect that, based on current opinion polls, Labour could come third in the popular vote but still be the largest party in the Commons and thus could form a government with the support of minor parties, most likely the Lib Dems.

Would proportional representation solve this problem? Well, it would mean that if Labour came third in the popular vote, it would come third in the number of MPs, but that would make little difference as to whether Labour and the Lib Dems could ally and form a government. There would be a different balance within that alliance, but it would still be an alliance of parties neither of which had come first in the popular vote - a result in which none of the electorate gets the result they voted for. PR is not a solution to the current problem.

What we need is to introduce disproportional representation - a system where whichever party gets the highest proportion of the popular vote, regardless of how small the margin, gets given enough seats to form a majority government, thus giving the maximum possible number of voters the result they wanted. The problem then is that the result is the same regardless of the size of the majority. One solution would be that the size of the majority determines how long it is to the next general election - say between two and five years.

Now I admit that this isn't a perfect solution, nor indeed a very thoroughly thought through one (you should hear some of my ideas on organizing Formula One Saturday qualifying), but at least it tries to address the problem we have, rather than being a simple knee-jerk shout of 'PR' regardless of the problem that the electoral system actually faces.

Sunday 25 April 2010

Marathon (and Zeno) Debrief

The news you're all waiting to hear is that I completed the Brighton Marathon in 4 hours 12 minutes 43 seconds.

But I know you're not really interested in how quickly I ran the marathon but, as I asked in an earlier post, what was the distance?

Despite not being able to post diagrams, I suspect Katie had the answer to the question posed (as I guessed she would before I even posted).

The answer is that Mr Tortoise, the slower runner, records the further distance on his GPS.

Suppose the following is the actual route that both runners take on a small fragment of the course. It's a sharp bend, to make things clearer, but the same principals would apply to any bend in the route. (Click the diagrams to enlarge.)
Now Mr Tortoise will complete this section in twice the time of Mr Hare. Given that their GPSs both sample their position once a second, that means Mr Tortoise will have twice as many samples (shown as red crosses) as Mr Hare (green circles).
Given these samples locations, calculation of the actual route taken is done by joining up these locations as shown.
(Note that it's an assumption that the GPS software joins the dot's as straight lines. Conceivably, a much more complex algorithm could be used.)

Clearly the red (Mr Tortoise's) route is going to be longer than Mr Hare's, regardless of the nature of the bend. The only circumstances when this is not the case will be on a straight line, where the intermediate red cross will lie on the green line and the distances will be the same.

So on a perfectly straight route, the total distance will be the same, but if there are any bends at all, the slower runner will always record the further distance.

And of course, both recorded distances will be less than the true route, represented by the black curve.

Which leads me to be puzzled as to why, at the Brighton Marathon, my GPS recorded a distance of 26.41 miles, when an official marathon is only 26.22 miles. Worth further sponsorship, surely?

Wednesday 7 April 2010

Zeno’s Paradox Revisited

Two runners, Mr Hare and Mr Tortoise, run a marathon. Mr Hare is faster and completes the course in, let's say, three hours, whereas Mr Tortoise takes six hours.

Each of them wears an identical GPS running watch, which samples the runner's location once every second and uses the information to calculate speed and distance travelled.

The marathon follows a typical street circuit and each of the two runners follows exactly the same path, without cutting any corners or taking any shortcuts.

The question is, at the end of the race, which of Mr Hare and Mr Tortoise is likely to have travelled the furthest distance according to his GPS watch?

If you have a solution, post it as a comment, showing your working. (Note there are no tricks in this, it's purely an issue of mathematics, or possibly physics.)

I'll be posting my solution after the Brighton Marathon on Sunday April 18th, for which you can still sponsor me at I hope that my time will lie somewhere between those of Messrs Tortoise and Hare.