To find out the permitted routes for Farnborough North/Blackwater to Gerrard's Cross we first look at the routeing validity of the ticket type; in this case you are interested in an Off Peak Day Day Return (
CDR▸ ). There is only one CDR on this flow (Farnborough Stns - Gerrard's Cross), which is routed:
+Any Permitted. The + allows for transfer by Underground between London Termini.
First, the shortest route (or any route that is no more than 3 miles longer than the shortest route) is always valid. I'm having some trouble finding the shortest route, so I'll fall back on what the Routeing Guide allows.
Next we need to find out whether the origin and destination stations are routeing points in their own right or instead associated with one or more routeing points.
Farnborough North and Blackwater are both in the Farnborough Group (along with Farnborough Main) for routeing purposes, so can thus be regarded as a routeing point. Gerrard's Cross is not a routeing point in it's own right nor is it in a station group. Instead it is associated with Banbury Group, Greenford, London Group and Oxford.
Now because Gerrard's Cross is associated with four routeing points we have to use the
fares check rule to find out which of those routeing points is valid. The fares check rule says that an associated routeing point is valid for routeing purposes if the fare for the whole journey (ie Farnborough Stns - Gerrard's Cross) is
not less than the fare to the associated routeing point. Fares from Farnborough Stns - Banbury, Greenford, London or Oxford are all less than the fare from Farnborough Stns - Gerrard's Cross, therefore all four routeing points are valid with an
'Any Permitted' ticket.
Now you need to consult
ATOC» 's (fiendish, esoteric and byzantine) Routeing Guide to find the
map or map combinations that show the permitted route(s) to each of those four routeing points. A route is permitted by tracing the route on the map or maps from origin station/routeing point to destination station/routeing point. Where origin or destination is not a routeing point you must take the shortest route between origin and routeing point or routeing point and destination. You may not double back through
any station, except within a station group or unless an easement permits doubling back. The valid map combinations for Farnborough Stns - Gerrard's Cross are:
For Farnborough Group - Banbury Group: Map BB
For Farnborough Group - Greenford: LONDON*, MW+WR
For Farnborough Group - London Group: CS+WR, WX
For Farnborough Group - Oxford: MW, WW+MW
Then the shortest route between the routeing point and Gerrard's Cross.
(*LONDON means any permitted route to London from the origin station, and any permitted route from London to the destination station.)
From all this I conclude that your chosen route,
Farnborough North/Blackwater --> Reading --> Paddington --> Gerrards Cross --> Marylebone --> Underground --> Waterloo --> Farnborough Main, is permitted with a CDR from Farnborough Stns - Gerrard's Cross. You can use the routeing permission via Greenford on your outward journey to get that bash on the Chiltern 'Ghost Train' and the permission via London Group to Marylebone <Underground> Waterloo to Farnborough Main on your return.
Easy isn't it!!
ATOC's
Routeing Guide in all its complexity can be found here:
http://www.atoc.org/about-atoc/rail-settlement-plan/routeing-guide