PPATH - Prime Path

The ministers of the cabinet were quite upset by the message from the Chief of Security stating that they would all have to change the four-digit room numbers on their offices.
— It is a matter of security to change such things every now and then, to keep the enemy in the dark.
— But look, I have chosen my number 1033 for good reasons. I am the Prime minister, you know!
— I know, so therefore your new number 8179 is also a prime. You will just have to paste four new digits over the four old ones on your office door.
— No, it's not that simple. Suppose that I change the first digit to an 8, then the number will read 8033 which is not a prime!
— I see, being the prime minister you cannot stand having a non-prime number on your door even for a few seconds.
— Correct! So I must invent a scheme for going from 1033 to 8179 by a path of prime numbers where only one digit is changed from one prime to the next prime.

Now, the minister of finance, who had been eavesdropping, intervened.
— No unnecessary expenditure, please! I happen to know that the price of a digit is one pound.
— Hmm, in that case I need a computer program to minimize the cost. You don't know some very cheap software gurus, do you?
— In fact, I do. You see, there is this programming contest going on...

Help the prime minister to find the cheapest prime path between any two given four-digit primes! The first digit must be nonzero, of course. Here is a solution in the case above.

The cost of this solution is 6 pounds. Note that the digit 1 which got pasted over in step 2 can not be reused in the last step – a new 1 must be purchased.


One line with a positive number: the number of test cases (at most 100). Then for each test case, one line with two numbers separated by a blank. Both numbers are four-digit primes (without leading zeros).


One line for each case, either with a number stating the minimal cost or containing the word Impossible.


1033 8179
1373 8017
1033 1033


hide comments
abhinav_rai: 2015-12-17 23:00:29

Cool Problem. Made me remember bfs and heights along with sieve and how to check for single digit change. Indeed great problem!!

KAUSHAL AGRAWAL: 2015-12-08 23:23:15

First problem of Graph Theory, solved in one go

Last edit: 2016-02-09 20:02:56
rocode0001: 2015-12-05 13:51:33

first problem of bfs.....14hours....right in two go!!!!!!

kejriwal: 2015-11-03 18:47:00

Amazing problem :D !!

Ankush Sharma: 2015-10-15 20:38:17

Nothing is impossible :) !!

:.Mohib.:: 2015-09-17 21:49:19

Awsm que..!!

thirupathireddy: 2015-09-17 07:39:01

nice one on bfs.............

ALi Ibrahim: 2015-08-01 22:22:17

Nice problem, first Accepted :)

sHaShAnK sHeKhAr: 2015-05-21 11:21:23

Nice question, can be solved from both Dijkstra and simple BFS :) since cost of each edge is 1($).

i_am_looser: 2015-05-08 06:55:03

Literally very nice problem : )
Hint - Dijkstra + Implementation.

Added by:overwise
Time limit:2s
Source limit:50000B
Memory limit:1536MB
Cluster: Cube (Intel G860)
Languages:All except: ERL JS-RHINO NODEJS PERL6 VB.NET
Resource:ACM ICPC NWERC 2006