# Problem of the week #68 - July 15th, 2013

Status
Not open for further replies.

#### Jameson

Staff member
Smartphone passwords can be made by submitting a pattern on a 3x3 grid like below.

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.
--------------------

#### Jameson

Staff member
Congratulation to the following members for their correct solutions:

1) Sudharaka

Solution (from Sudharaka):
Note that each pattern can be considered as a beam with three bending points. Each bending point can take any one 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 $$=4\times 126=504$$

Status
Not open for further replies.