Googon's Garage Gadget

[googon97]

If Googon has a normal garage opener with buttons 0-9 and enter and a code of 3 digits, it takes 3,000 button pushes to ensure the garage to open. Googon's Garage Gadget Guys builds a garage opener with 10 buttons. Googon realizes that there is no room for an enter button (he can't just take out a number 0-9). His garage opener opens if the code is pushed at any time. For example pushing 0123 could open the garage if the code is 012 or 123. Googon is worried that the garage opener is not safe. He needs your help on the issue (seriously he doesn't know the answer). How many times does someone need to push a button to ensure the garage will open? Can you prove that? Is their a pattern to the button pushing?

Edited February 26, 2013 by googon97

[vistaptb]

1,002 pushes. There are 1,000 possible combinations, and there are 1,000 strings of 3 numbers in 1,002 button pushes. The real question is: can one achieve this "optimal" solution with no repetition of a combination?

[augustinus]

the answer 1002 might actually be performed.
i only just tested myself with simple 1 and 2 and 3 combination, which starts at 111, and find the largest possible number which starts from 11, which is 113, and so on and so on, with exception of getting other duplicate starting number (starts with 22, or 33); before using the largest possible number, fill it first with the same numbers, then after 222 or 333 removed, continued as before

the result of combination using 123 should be like this:

11133323313222321312311221211

therefore, with some computing algorithm and decent database to store and validate inputted combination to avoid duplicates, it might works. i havent tested with 10 numbers combinations though, i will try to write the algorithm and test with 10 numbers later

Edited February 26, 2013 by augustinus