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Example

Rules- 1) At-least 4 and at-max 9 dots must be connected. 2) There can be no jumpsenter image description here

3) Once a dot is crossed, you can jump over it.

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    $\begingroup$ See OEIS: A163889. The total number of unlock patterns is A163889(4) + ... + A163889(9) = 389130. [Edit: the OEIS might have A163889(4) wrong -- can anyone verify if that term is 1624 or 1642?] $\endgroup$
    – Nathaniel Johnston
    Commented Jan 10, 2014 at 17:36
  • $\begingroup$ Start by enumerating legal first arrows. If I understand the rules, there are at most 40 such. Then do legal second arrows (If you use symmetry, there are 7 branches to explore). You now have a handle on a good upper bound, as there will be a little more than 6! ways to finish each branch. This question is better asked at a different forum. $\endgroup$
    – The Masked Avenger
    Commented Jan 10, 2014 at 17:43
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    $\begingroup$ As a follow-up to my previous comment (which I can't edit anymore) -- the 4th term of the OEIS entry was indeed incorrect (it was 1642 rather than the correct 1624). I have since corrected it. The final answer to this problem is 389112 (not 389130). $\endgroup$
    – Nathaniel Johnston
    Commented Jan 10, 2014 at 17:52
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    $\begingroup$ Regardless of the number of combinations, this isn't a very secure way to lock a phone unless you thoroughly clean the screen after touching it. $\endgroup$
    – Ryan Budney
    Commented Jan 10, 2014 at 18:07

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