Susan couldn't quite remember her bank PIN code.

She knew:

• It had 4 or 5 digits.
• It used 4 or 5 different digits.
• The first digit is 4 or 5.
• The second digit is 4 or 5 and none of the other digits is smaller.
• The number is 4 or 5 times a prime number.
• If you reverse the order of the digits, the resulting number has 4 or 5 prime factors, all different.
• In 4 or 5 of these facts, the "4" is the correct number.

What is Susan's PIN code?

There are seven bullet points, each of which gives a value 4 or 5.  I Made a tree which potentially would have 2^7 branches but many of them get culled as impossible.  I proceeded in the order they are given sometimes pruning back.

I cant accomplish a tree here so I will just list the full branches.

Bullet 1 (4 or 5 digits)
{4, 5}

Bullet 2 (4 or 5 different digits)
{44, 54, 55} [note 45 is impossible]

Bullet 3 (1st digit 4 or 5)
{444, 445, 544, 545, 554, 555}

Bullet 4 (2d digit 4 or 5, none smaller)  [this will cause back-pruning]
{4454, 5444, 5454, 5455, 5554}

We now know the number looks like one of the following:
54 _ _, 44 _ _ _, 54 _ _ _, 55 _ _ _

Bullet 5 (4 or 5 times a prime)  note: if 5, the number ends in 5 which very limiting
{44544, 54444, 54445, 54544, 54545, 54554}

Bullet 6 (5 or 4 prime factors of reversal)  note: doesnt limit things until we start looking at candidate numbers
{445444, 544444, 544454, 545444, 545454, 545544,
445445, 544445, 544455, 545445, 545455, 545545}

Bullet 7 (4 or 5 fours in these branches)  very limiting.  Only if there are 3 fours add a 4th, only if there are 5 fours add a 5.
{4454445, 5444445, 5444554, 5454454, 5454544, 5455444}

We are down to 6 cases.

Lets look at the only 4 digit case: 54 _ _ and 4 times a prime and 4 factors of reversal.  The prime must be from 1350 to 1375 and end in 7 or 9.  1367 fits the bill.  1367*4 = 5468.  Its reversal is 8645 = 5*7*13*19

Got it!  The PIN is 5468
(4 digits, 4 different, 1st is 5, 2nd is 4 none smaller, 4 times the prime 1367, reversal has 4 factors, "4" is the correct number 5 times)

I dont have time to check the 5 digit cases right now.  Maybe Ill come back to this.

Posted by Jer on 2010-11-18 15:55:00