This website details how Marc Gawley broke a Guinness World Record^{TM}  you can watch a video of the day here
It chronicles my attempt at the Tube Challenge  visiting every station on the London Underground network (270) in as short a time as possible. The challenge was first recognised in 1960, and according to Guinness the record had remained unbroken since 2009 at a time of 16hrs 44mins 16secs. This is how I went about planning my assault on the record, a mixture of web research, on the ground reconnaissance, and some hefty analytics, in collaboration with Martin East.  I had an unexpected 15 minutes of fame, with radio, tv, newspaper stories, and an article in Mensa magazine. Click a logo to see more: More recently the BBC made some fun of me here. 
Curious to know more? Planning your own attempt? You can get in touch with me here.
I liked this recent paper showing that 5% of London's commuters do not travel by the optimal route.
The Fastest Tube Challenge possible:
What is the minimum time to travel to all 270 London Underground stations?
Being sure to have found the real shortest time to visit all 270 tube stations is pretty difficult without calculating the time for every conceivable route (this is a travelling salesman type problem). However, we can calculate a time which we are confident the shortest time must be greater than  a minimum minimum [sic] time.
Travelling along all lines from end to end would certainly meet the requirements of the record, but takes too long to get a record (currently 16h 20m) especially when you add on minor details such as incorporating connection times between lines.
A big issue is that multiple stations are visited multiple times, which is not efficient.
Let's reduce the map to the minimum (ish) amount of track that one would have to travel by tube in order to visit each station once, according to the Guinness rules:
Obviously there are numerous (though finite) ways to go about this, especially for the zone 1 and 2 stations, but this is a fair attempt. This version has 44 track segments.
The important feature is that both ends of every segment (unless the end of a line) can be connected to at least one other segment by travelling along an erased piece of track that would in real life connect two adjacent stations. So the segment that ends at Waterloo and the one that begins at Lambeth North are in real life connected by one stop on the Bakerloo line. Similarly, between Waterloo and Southwark by one stop on the Jubilee line.
Ignoring these erased sections of track, TFL journey planner gives the total time to travel along these 44 segments of track as 9h 24m:
Without actually trying to do it, let’s assume that we can optimally connect all the 44 segments, i.e., 43 simple connections, with no backtracking or duplication whatsoever:
 Backtrack means go from one end of a segment to the other and then back again
 Duplication means later on travelling along a segment (in either direction) that has previously been done
In this optimal world we enter one end of a segment and leave at the other end to connect onto a new segment that we’ve not been on before, 43 times.
Remember, each of these segments is just a single stop from connecting to another one. In central London two adjacent stations might be 2 mins apart on a train, in outer London as much as 5 mins. We’ll take 3 mins. (We could run between stations, but it’s rare that this will be less than 3 mins, so I’m comfortable with this estimate as a conservative average for 43 connections).
So that’s another 43 x 3 = 129 mins of connections, taking us up to 11h 33m.
Now there must be some backtracking/exiting the network at the outer segments. A Tube Challenger needs to do something much more time consuming than the assumed 3minute simple connection at these stations at least:
 Heathrow area
 Amersham/Chesham area
 Watford
 Stanmore
 Edgware
 Mill Hill East
 High Barnet
 Cockfosters
 Epping
 Walthamstow Central
 Upminster
 Morden
 Richmond
 Wimbledon
This is not exhaustive: What does she do at Uxbridge, West Ruislip, Brixton, etc.? Hence the list  and in turn the cumulative duration in the next section  is conservative.
For each of these difficult connections she has to either get back to the start of the segment, or get to a station on another segment/line.
We’ll also allow running outside the network at a speed of 3.5 minutes/km, which would be faster than the majority of the population.
As we're trying to get a minimum theoretical time I’m not considering if any of the moves in the table is actually sensible, or if it will join neatly to the end of another segment that hasn't yet been done.
Station 
Shortest possible move 
Time taken (minutes) 
Heathrow 
HEX to Paddington 
15 
Amersham/Chesham 
Chiltern to Marylebone 
39 
Watford 
To Rickmansworth (w/ north curve) 
8 
Stanmore 
To Preston Road 
15 
Edgware 
To Camden and change branch 
22 
High Barnet 
To Camden and change branch 
23 
Cockfosters 
Finsbury Park and use Victoria 
21 
Epping 
Stratford and use Jubilee 
28 
Upminster 
West Ham on a train 
15 
Walthamstow Central 
Finsbury Park 
14 
Morden 
To Wimbledon running 
10 
Wimbledon 
To Morden running 
10 
Richmond 
To Turnham Green 
9 


229 
That’s a total of 3h 49m.
We could make the worst two of these the first and last two segments of the day, which removes two back track connection times. Subtracting 39m and 28m, that comes down to 2h 42m additional. (I believe Amersham/Chesham start and Epping finish is tried by some people).
Some of these connection solutions clash: You can’t do the change to the other line at Finsbury Park for both the Walthamstow Central and Cockfosters backtracks. So one of these (at least) must have a worse solution. But I'll keep it as it is, to be conservative for the minimum time.
Alternatively some of these solutions combine usefully: For Morden and Wimbledon, if you run to the other, you only need to do it once. This is actually very efficient, and removes one of the backtrack problems (indeed this is the kind of insight she needs to have a go at breaking the world record). Not all of them will combine like this, but let's say a couple can.
Also, these difficult connections do mean we avoid an equivalent number of 3minute simple connections that we already added on.
Net I'll say "only" 2h to be added on for difficult connections (I think this is the crucial variable, and the extent to which this can be optimised makes or breaks a route).
We’re at 13h 33m.
There’ll be some waiting time during the day: we won’t arrive at each new platform with 0 seconds to spare. Anecdotally it seems that the typical number of change of trains during a day is 40 to 60 in a full attempt. These were the stats when I did mine in 2011:
The mode, weighted average, and weighted average excluding outliers all come out to be 4 minutes.
These were the TFL timetabled changes and would have varied on the day according to the actual service. (The negative number comes from one of the runs I did: At ‘normalfast’ speed I’d be one minute late to make the departing train, so had to be killmyselffast on that one. The gamble paid off).
Conservatively, let’s say 35 changes at 3 minutes each, that’s another 1h 45m to add on.
In total we’re up to 15h18m.
This is my suggestion for the theoretical fastest time to visit all 270 London Underground stations in one day.
Given that the assumptions have generally been on the low side, the actual possible minimum time is probably higher than this.