We continue the discussion on the question below:

Let n and k be fixed positive integers of the same parity, k n. We are given 2n lamps
numbered 1 through 2n; each of them can be on or off. At the beginning all lamps are off. We
consider sequences of k steps. At each step one of the lamps is switched (from off to on or from
on to off).
Let N be the number of k-step sequences ending in the state: lamps 1, . . . , n on, lamps
n+1, . . . , 2n off.
Let M be the number of k-step sequences leading to the same state and not touching lamps
n+1, . . . , 2n at all.
Find the ratio N/M.

We showed that every restricted admissible process can be modified in 2 ^ (k-n)  ways which results in as many distinct admissible processes.
What remains to be shown is that every admissible process can be achieved in that way. And surely enough we can replace the switch of  lamp with label l > n that occurs in q by the switch of a lamp with label l - n and consecutively the resulting process p does not have lamps n+1, .. 2n involved.

Moreover the switches for those lamps were even in number and when performing the same on lamps with label < n, the resulting process is admissible and restricted.

Therefore admissible <---> restricted processes are one to one reversible. This completes the proof.

Note that the state of each lamp and it's operations are independent of other lamps, so the combination of ways to operate them is a simple multiplication. And since the state of the lamps are binary, it becomes intuitive that the ratio of the admissible to restricted process has to be a power of two.

Finding Interest in the mundane 

Emails can be boring especially when we are inundated by the tens if not hundreds every day. We are trained to filter what’s relevant and critical to us. We can even recall threads and conversations that may matter to a specific topic of discussion. We spend some time scanning the incoming mails, some more time filtering and most time responding or preparing a response to what matters. Consequently the tools of our trade are the search options on the Email software we use which is Outlook. A cursory look at this search options shows that the query we write are from what we remember such as the subject containing a buzz word or sender or recipient being someone specific. However, there are a few problems with this approach.  

First is we speedboat from the get go. We want to hammer home the wins from each incoming and outgoing responses. There is hardly any technology other than search that we use in our aim to get where we want too and if we have spent a lot of time digging around e-mails, we either request a resend or move on. There may be some tagging and organization involved but these are mostly from what we expect the incoming mails to conform to. Besides with the explosion of containers, the search becomes narrowed leading to higher misses or incorrect assumptions. Secondly, we don’t smell the roses. In our daily routine, there is hardly any learning from the conversations that flow to our inbox unless we have scanned and read each one of them. Many of these conversations may not even be directed at the reader per se and even lesser so at the tangent topics that may boomerang back to us. For example, there is no text analytics or content intelligence that we build from this vast amount of semi-structured data. 

Thirdly, we don’t have appealing visualization of the information that we can have drones retrieve us from the daily grind. Even if we do realize a highly efficient text analytics platform over the emails and manage to build a user interface that can improve our productivity, we may limit ourselves to ultimately showing them as the same search results as we used to see earlier leaving the burden on the user to refine the search results or scan them for items of interest. Yes precision and recall can be improved and even change from being header or metadata based search to full-text but statistical and semantic analysis would not display their immense potential if they presented their results in the same way. What can we then use to render such information ? Take the following visualization as an example: 

 Courtesy: http://vallandingham.me/msd/vis/ 

as an example of what we can organize the results on Outlook. This sort of imagery and interactivity is far more helpful to the user than static endless results. Besides this is complimentary and not a substitute to the traditional search. 

Fourth, even the search tools we use return data not information. Most of the results are the individual emails that meet the query or search criteria whereas the results we are looking for is in terms of concepts or tags that are abstract and may not even be literal in the email. If we look at the following menu from our Outlook software: 

We can see that the query builder window is not the same as the kind of clustering, categorizing and machine learning techniques options we are talking about. There is no way to even build models for search here.  

Finally, we are not talking business intelligence.  There is a lot of overlap between building logic for search and reports to display the results between a business intelligence software add ons and the content intelligence and text analytics platform but the key difference here is that BI is targeted for some kind of specific domain or information whereas we are talking about increasing the panorama of information retrieval tools to glean what passes us by on a day to day basis. Time series data and Big Data techniques have made some of the approaches commercially viable on large scale but to scope it down to personal level and to this repository is altogether different.  

Thus we can explore more from what we get each day.