Elementary S01E03 - S01E08

S01E03 - Child Predator

Nothing special. Just using his iphone lense and a USB stick.

S01E04 - The Rat Race

No computer related stories in this episode.

S01E05 - Lesser Evils

Nothing special either. Holmes uses again the iphone lense, but that's it.

S01E06 - Flight Risk

Finally an opportunity to link my "Enhance" post. Holmes starts with a picture from a security cam from pretty far away (screens taken at ~16:30):
He performs a series of enhancements:
And ends up with the following zoomed in image:

Let's just say that the security camera must be of extraordinary quality. Even more damning would be the cost of storing such a high-quality recording. While storage cost went down significantly in the recent years, it is still unreasonable to store such high quality recordings for a security camera.
Given the fast advances in storage and camera technology I wouldn't be surprised if Holme could do this in real-life in a few years.

S01E07 - One Way to Get Off

At ~7:45 detective Bell explains that they found a bunch of threatening emails (anonymous) on the victims computer. They had traced the IP to a contractor.
The question now: can the police trace emails back to its sender?
All depends on how the emails were sent. In general, sending an email consists of two steps: connecting to the internet, and using some service to send the mail. In order to trace an email back to the source the police needs to undo both steps: the email provider needs to say from which connection (IP) the mail was sent, and the internet provider needs to tell which user was behind that connection at that time. In Europe there are data-retention laws in place that guarantee that the second step ("who is behind the connection?") can be answered. The US doesn't have such a law, but afaik most (all?) providers keep this data, too.
I don't know if email services are required by law to keep their information logged. (If you know, please let us know in the comments). Most definitely do. They might require court-orders (as is the case for GMail and all the other big players), but they do have that data.
That said: it is extremely easy to send real anonymous mails: one can use email services in countries that do not log the incoming IP addresses, and/or one can use internet connections that can not be mapped back to the sender. An easy example is an unprotected wifi-hotspot of a neighbor. If that's not available one can buy prepaid SIM cards anonymously in several countries (for instance Austria, Spain and Croatia). These SIM cards provide data access and is no way to trace the connection back to a specific individual.

Edit: some programs, like Outlook and Kmail skip the email web-service and use a different way to deliver the mail to the recipients inbox (SMTP). In that case the IP address is stored directly in the mail. This just eliminates one step.

At ~12:30 Holmes claims that the "[h]uman eyes are precision instruments. They can detect grooves and lands on a slug more efficiently than any computer."
Computers are extremely good at pattern matching. Unless the slug is severely damaged (in which case some human intuition might help) I don't buy this statement.

At ~30:00 we can see another correct use of lock-picking. The wrench (applying torque) in one hand, and the pick (to push the pins) in the other.
Screenshot taken at ~30:00

S01E08 - The Long Fuse

No computers.

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