The Double Slit Experiment - delayed measurement I'm not a physicist; I'm just curious about what would happen.
According to this video "Explained ! The Double Slit Experiment" (at 2:50):

"Somebody decided to leave the detectors on but just not take any data. [...] So if it was also going to a magnetic tape - there was no magnetic tape loaded. [...] And what do you think happed? They got this. They got the diffraction pattern."

So the act of measuring (beeing concious of the data) destroys wave-form pattern?
If so lets consider these cases:


*

*What if I'd throw a dice after the experiment to decide if I should look at the data or to destroy it? (e.g. look at the data if the number on dice is even)

*What if the information about measurement would need to travel longer period of time than the particle would need to hit the wall? 


To above cases: Would it mean that information travelled instantly - faster than light or that "the particle can look into the future"?
 A: Both of the experiments that you have proposed have actually been done. The first is called the quantum eraser, and the second is called the delayed choice quantum eraser.
The answer to your question is that if information about which path the particle took is available at the time of measurement then no diffraction pattern will be seen. So, if you take data about the path and then delete it before the experiment is observed (and your apparatus is set up so that the system does not decohere) then you will still see a diffraction pattern.
One way to understand this is to notice that a diffraction pattern is equivalent to measuring the momentum of the photons (since the wavelength is related to the distance between peaks) and a measurement of the path is a position measurement. The uncertainty principle says we can't know both with arbitrary precision at the same time. So, a well constructed quantum eraser experiment guarantees that either you can know the path or you can see the diffraction pattern but never both.
Then, your question becomes "when I observe the experiment am I performing a position measurement or a momentum measurement." The answer to this question, for any quantum mechanical experiment, is that you are performing the microscopic measurement whose different results correspond to distinct macroscopic states (like a dial being in different positions). In the quantum eraser experiment the apparatus is designed so that, depending on exactly how the experiment goes, either the microscopic position or momentum measurement will correspond to distinct macroscopic states. If you can read the data about which path the photon took it is obviously the position measurement that gives you distinct macroscopic states, and if you can see the diffraction pattern it is the momentum measurement. 
A: Well, I also watched the video and was quite amazed by the statement you mentioned at 2:50. Actually, the statement is wrong. Everything he explained in the video is true except that statement. Once the data is recorded, then whatever you do with that data, you will see two dots, not an interference (diffraction) patter. I actually researched about this claim of Tom, below is one of the discussion:
https://www.physicsforums.com/threads/double-slit-experiment-with-detectors-not-recording.414617/
A: There are 2 distinct experiments here..
1) You record all events and measure the distribution on the back plane.
2) You have detectors at each slit and record only those events where one detector fired, and for these events you measure the distribution on the back plane.
Each experiment requests very different information from the system. So it should be no surprise that each experiment gives different results. 
A: does destruction of the "which way" path information change the observation?
Suppose the experiment consists of Bob at hole 2 in the double slit experiment being able to open and close the hole instantly. Let the intensity be so low that on average only one particle at a time is in the apparatus. By closing the hole he ensures the particle must go by route 1 if it is to hit the screen. Now what happens if he manages to re-open hole 2 just before the particle is detected at the screen? By repetition of the experiment the pattern built up at the screen is the interference pattern (case a). If the hole were closed at the time the particle is irreversibly detected at the screen then the pattern built up would not show interference at all (case b). The state of the apparatus at the exact moment of the irreversible detection of the particle at the screen determines if the particle is contributing to case a or case b pattern. Note that in this experiment Bob hasn't detected any particles himself but keeps a record of the time at which he opens or closes the hole which can be correlated with the particle arrival times at the screen. we can thus group the screen observations into two groups- those that occurred with the hole open and those with it closed. The first ones show the interference pattern the second do not. What happens if Bobs records are destroyed before the screen results are analyzed into these two groups? We see a mixed pattern of both a and b so interference fringes on top of high background which tends to wash out the fringes. The point is destroying Bobs information doesn't change the results which were observed at the screen. But those results were determined by the now lost information, they don't suddenly change.
