How many smartphone password combinations can be made using a 3x3 grid?

[Jameson]

Smartphone passwords can be made by submitting a pattern on a 3x3 grid like below.
View attachment 1007

Assuming that the pattern starts in one of the four corners, contains 6 dots and all the dots must be connected, how many different password combinations are there?

Note: You cannot cross the same dot once it is chosen.
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[Jameson]

Congratulation to the following members for their correct solutions:

1) Sudharaka

Solution (from Sudharaka):

Spoiler

Note that each pattern can be considered as a beam with three bending points. Each bending point can take anyone of the four angles, \(45^0,\,90^0\, 135^0\mbox{ and }180^0\). Therefore if we start from one corner we have \(4\times 4\times 4=4^3\) possibilities. However note that we cannot make a pattern with all three angles taking the value \(135^0\). Therefore we have to eliminate that possibility. Hence the number of possibilities becomes \(4^3-1\). Now each pattern has two instances starting from the same corner. For example the L shaped pattern can be made in the following ways, starting from the upper left corner.

* * *
* * *
* * ** * *
* * *
* * *

Hence the number of possibilities becomes \(2(4^3-1)=126\). Since each pattern can be made with each of the four corners we have,

Total number of possibilities \(=4\times 126=504\)