Intermittent, car will not start hot or cold?

Hi, the car will start perfectly for about four times then out of the blue it won't start. We have had a new battery installed, a new starter and the mechanic took the "small computer" and had something fixed in it, yet it is still doing the same thing. When it won't start you turn the key fully on and you hear a clicking going around each item that locks in the car so not from the engine side alone. It's as if the car is checking the entire locking system. We tried turning the key to just the first turn of the key and leaving it there for a few moments then tried starting it and it worked for a few times now it won't work. We tried starting it four times in a row and it started perfectly then on the 5th time it would not start. The mechanic has no idea what the problem is and if he carries on replacing more parts I will end up bankrupt but with a new engine :) Please can you suggest what the problem could be. The mechanic said If you bridge the starter out it still does not turn so it cannot be the ignition as we thought. He tried bypassing the ignition and went direct to the starter to no avail. He said the dashboard lights are like disco lights when the car acts up and won't start. Also, it starts when you push it. Thanks