London Transport Interlinking (Sieve Method)

Here's a question:

How many bus routes serve all major transport modes in London?

Those transport modes being the Underground, Overground, DLR, Trams, Crossrail, and National Rail. You could hypothetically break the Underground and National Rail into separate branded lines, but that would be a bit confusing - so let's stick with the original premise. 

You could go through each bus route one by one, endlessly searching for a route to meet all six, but you wouldn't find any - the answer is zero. Technically. 

I did this a while back because my life isn't all too exciting, before stopping whilst writing up the prefixed routes because even this wasn't that exciting. However, if you want a solution, you could bypass all that and simply come up with a few rules, from which you can eliminate a substantial number of routes. That is:

• the route stops by the trams, and;
• the route stops by the DLR.
  

This is as these two transport networks are concentrated in different areas of London - the trams to the south, the DLR to the east. So any possible routes should cover both these areas roughly. There's thus no point in considering the H98 because it goes nowhere near the DLR. Obviously this would only work if you knew exactly where each route went, which technically goes against my point of not having to write out each possibility, but either way. 

Or in other words, the route needs to go from Beckenham/Croydon/Mitcham/Wimbledon to Lewisham/Central London/the Docklands. There are only a few of these, such as the 54, 75 and the N133*. From here, the 54 doesn't serve an Overground station, whilst the 75 doesn't serve a Crossrail station. So the N133 wins...on a technicality. 

It turns out it doesn't serve Bank station like the 133 route does, instead going direct to Liverpool Street. However, it does serve Monument station, which is an interchange with Bank, so I suppose you could go via station entrances into Bank. The links, if you're wondering, are:

Tram - Mitcham; National Rail - Streatham Common; Underground - Brixton; DLR* - Bank/Monument; Overground/Crossrail - Liverpool Street.

A late finish for sure, with three interchanges coming in the last mile of the route. You'll also notice that it doesn't serve separate stations for the Overground and Crossrail, so it now feels even more like a technicality than anything else. Also, the N133 only runs at night, when the trams and DLR don't operate, and the answer is more of a cheat than it should have been.

Time for an easier question, on the same principle of reducing possibilities:

Which Underground line serves all major London transport modes?

It must serve the trams and DLR, and only one line does so - the District line, so it's the only choice. And it works, with its interchanges at:

Trams/National Rail - Wimbledon; Overground - Richmond; Underground - Ealing Broadway; DLR* - Bank/Monument; Crossrail - Whitechapel

If you included out of station interchanges, the Northern line would also work, but that's even more of a technicality and I don't want to delve into that.

Here's one final question:

Are there any transport services which meet every National Rail line in London?

Since there are many London railway termini, all mostly centred in Central London, you might expect the answer to be yes. First, let's establish some rules as to what a National Rail line is. I'm going to define the various branding schema as separate National Rail lines regardless of who operates the lines, of which there are a few, from which the rules can be further refined. 

The Gatwick Express, for example, only serves Victoria in London, so this mode of transport must go there. Thus, the Trams, DLR, Overground, Crossrail, and the majority of tube lines are out already. Then you've got the Stansted Express, which operates services from Liverpool Street. You're now down to the Circle line and the 26/N26 bus routes. Alas, neither serve a station operated by Chiltern Railways, and so we can answer the question by saying no transport service meets every National Rail line.

I suppose I answered these questions by virtue of a sieve - remove as many options as possible to focus on a limited number of possibilities, which you can also use to find the total number of prime numbers and other listed concepts. Obviously it's not a certain method - sieve too broadly and you've removed perhaps too many options, sieve too narrowly and you've removed too few options. But why let an imperfect method get in the way of writing a blogpost?

If my answers were wrong, feel free to correct me.