INT. JULIAN'S HOUSE – BASEMENT JULIAN, rumpled, tired, and bewildered, sits at a shabby card table poring over BLUEPRINTS of a complicated mechanism. NAOMI hastily enters the room carrying a PAD OF GRAPH PAPER and a BALLPOINT PEN. She's just as sleep-deprived, but she's taking it better. She sits next to Julian. He shoves the blueprints away and leans back in his chair. JULIAN I've got nothing. Even if the duplicator could hit one of these parts, we'd just jam the lock. There's not enough room in here. NAOMI Don't worry about it. We don't have to hit the lock. Julian raises an eyebrow, intrigued… NAOMI (cont'd) The mechanism runs on dot numbers. Like this. On the graph paper she draws
A MULTIPLICATION PROBLEM: ●○●● ●○● ------ ●○●● ○○○○ ●○●● ------ ●○○●●● NAOMI (O.S.) The black circles are dots, and act like ones in binary. The white circles are blanks and act like zeros. Every dot number begins and ends with a dot, and they're multiplied like regular binary numbers—with one exception. See that first blank in the result? That's there 'cause there's no regrouping when the intermediate products are added.
BACK TO THE CONSPIRATORS JULIAN With you so far. NAOMI The lock's cryptosystem needs one dot number as its key, which it reads from a 10-meter punched tape that's fed in… here. The key's supposed to be prime; it should only have lone dots and itself as factors. If it's composite, any combination will get us in. JULIAN So you want to hit some symbols on the tape with the duplicator?
BACK TO THE GRAPH PAPER NAOMI (O.S.) Right. Like, say we hit the blank in this prime. She writes down ●●○● and then the same number with its blank duplicated: ●●○○●. JULIAN (O.S.) Er, that's no good. It's still prime. NAOMI (O.S.) Ah, but the duplicator's power level can be set past 100%. If we fired it at 200% or more, we'd only get that number for the first minute or so. Since there'd still be a full charge's worth of duplication on the original blank, the number'd turn into this— She writes ●●○○○● = ●○●● ✕ ●●●. NAOMI (O.S., cont'd) —after the tape cooled down. There'd be another duplication after another minute if we'd given it 300%, but we'd be in the safe by the time that happened.
BACK TO THE CONSPIRATORS JULIAN Not bad. There's just one problem. NAOMI Yeah? JULIAN The tape is behind the safe's walls, where we can't see it. We won't know what number is punched on it, and, while the material indicator will tell us when we're aiming at the tape, we won't know which symbol the duplicator's pointed at when we fire. Naomi leans back and looks smug. She's already thought of that.
What's Naomi's plan? If at most $n$ symbols can fit on a 10-meter tape, how many shots should the conspirators fire and at what power levels to guarantee that they will eventually get inside?