# Help a man to recover his PIN

A man forgot his 6 digits PIN, but fortunately he remembered the clue for his PIN.

• The n-th digit is the first digit of (product of other digits multiplied by n)
• The number is not 000000

If the PIN is abcdef, then :

The 1st digit = The first digit of $1 × b × c × d × e × f$
The 2nd digit = the first digit of $2 × a × c × d × e × f$
...
The 6th digit = The first digit of $6 × a × b × c × d × e$

What is his PIN?

• PIN number, ATM machine, NIC card ... uggh! It's just a PIN. It stands for Personal Identification Number. :) – Rubio Dec 17 '16 at 8:24
• I've often thought it would be amusing to write a story in which all those phrases appear but they are actually used correctly. A gang of criminals have an ATM machine (a machine that makes ATMs) to make special fake ATMs that record PINs, so then if they have a PIN number they can look up that position on the list to find the PIN. Etc. But it was never amusing enough for me actually to sit down and write it. – Gareth McCaughan Dec 17 '16 at 11:03

His PIN is

529311

a=5; 1×b×c×d×e×f = 1x2x9x3x1×1 = 54
b=2; 2×a×c×d×e×f = 2x5x9x3x1×1 = 270
c=9; 3×a×b×d×e×f = 3x5x2x3x1×1 = 90
d=3; 4×a×b×c×e×f = 4x5x2x9x1×1 = 360
e=1; 5×a×b×c×d×f = 5x5x2x9x3×1 = 1350
f=1; 6×a×b×c×d×e = 6x5x2x9x3×1 = 1620

• +1. Brute force alone, or was there an analytic step? – IanF1 Dec 17 '16 at 16:58
• I believe this is solvable, but it looked much easier to brute-force so I did. – Rubio Dec 17 '16 at 19:12