Question

A computer has a keyboard with numbers 0-9. You need to input a 3-digit lock, but the computer follows a special procedure, as follows:

input first digit (any number from 0-9 you want)
add digit 1 to digit 2
input digit 3
add digit 3 to digit 1
input digit 2
add digit 2 to digit 3
loop

The computer calculates in mod 9. Starting from 000, what sequence is needed to make the 3 digits "123"?

Answer 1

Suppose that the lock has three digits, \(a_1\), \(a_2\) and \(a_3\). I assume that the program is initialised with \(a_1=0,\,a_2=0\,a_3=0\). If I execute the first two step in the loop and \(x\) is the input to the first step, is \(a_1=x\) and \(a_2=x\) or is it \(a_1=0\) and \(a_2=x\)?

Answer 2

initially: 0 0 0
1) x 0 0
2) x (x+0) 0
3) x (x+0) y
4) (x+y) x y
5) (x+y) z[reset to another number] y
6) (x+y) z (z+y)

7/1) a[reset to another number] z (z+y)
8/2) a (a+z) (z+y)
9/3) a (a+z) b[reset to another number]
10/4) (a+b) (a+z) b
11/5) (a+b ) c[reset to another number] b
12/6) (a+b) c (c+b)

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