I find this happens a lot with "thought" problems: I can't solve it (and can often prove that) because the rules of the problem are inadequately explained.
The problem specifically says they are communicating using the internet, so why not?
"Hey, I sent you a box. It's locked with a combo lock. Call me when you get it: I want to be talking to you when you see the surprise! I'll tell you the combo on the phone!"
The only way to avoid this possibility (of a man in the middle attack) is to be "introduced" by a third party you already trust. See e.g. https://en.wikipedia.org/wiki/Needham–Schroeder_protocol
In the absence of a trusted third party, you could be talking to anybody, through any number of lurking intermediaries. If you've got a trusted third party then the problem, as posed, is moot.
10 if it's the sequence of natural numbers
13 if it's the sequence of N such that N=2^n for natural numbers n, where N does not contain a 0
153 if it's sequence of N such that the the sum of the digits of N each raised to the power of the number of digits in N equals N.
However, i'm not sure that I understand your second example.
If N=2^n (assuming 2 to the power of n, not 2 xor n) then one would expect the sequence to be 1, 2, 4, 8, 16, etc.
If N=2^n using the C xor notation, then we would expect a sequence 3, 0, 1, 6, 7, 4, etc
Your third example is pure evil, and I'm glad that I never had you as a maths teacher ;-)
"Identify as many sequences as you can which fit this set of numbers, and tell me the next number in each sequence" is the non-trick actually being asked.
Someone recently pointed out to me that if you were designing hardware to (e.g.) refresh all memory locations in a system, or otherwise visit a set of 2^n sequential locations in no particular order, then using a maximal-length shift register of the appropriate width is actually simpler than a 'simple' adder or subtractor. That could be seen as a literal interpretation of Kolmogorov complexity.
11 if it's the sequence of N such that N=2^n for natural numbers n, where N does not contain a 24
https://www.media-partners.com/upload/i20121017160441/img1.j...
1. Bolt Cutters 2. Break the box 3. Send the key design via the internet and a link to https://www.youtube.com/watch?v=8M-HYptOMJI
1. these are regular padlocks;
2. Alice and Bob already have a secure communications channel along which information, but not objects, can flow (the Internet);
...then the puzzle has a "cheese" solution: just communicate the metrics of the key along your secure channel, and have your counterparty reproduce the key using them! Padlock keys are essentially number sequences :)
I don't waste much time with them to be honest.
A sends their open padlock and a box to B. B puts their open padlock in the box, and locks it with A's padlock and sends it back. A opens his padlock, puts the ring inside, locks it with B's padlock and sends it back.
Maybe it was just too hard for you? :P
And they are both promptly stolen, because neither is inside a padlocked box. "anything sent through the mail will be stolen unless it is enclosed in a padlocked box". Anything includes padlocks and boxes. In fact, even the padlocked box would be stolen because it's not inside a padlocked box. And if it were, the outer box would just be stolen. The puzzle as written is just turtles all the way down.
The proper form of the riddle would be that anything except for a padlocked box will be stolen from the mail.
That’s why — A’s open padlock will be stolen.