NY10B - Nim-B Sum

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Note: This problem has nothing to do with siting sewage plants, power lines or wind farms. NIM is an ambigram.

The game of NIM is played with any number of piles of objects with any number of objects in each pile. At each turn, a player takes one or more (up to all) objects from one pile. In the normal form of the game, the player who takes the last object is the winner. There is a well-known strategy for this game based on the nim-2 sum.

The Nim-B sum (nim sum base B) of two non-negative integers X and Y (written NimSum(BXY)) is computed as follows:

  1. Write each of X and Y in base B.
  2. Each digit in base B of the Nim-B sum is the sum modulo B of the corresponding digits in the base B representation of X and Y.

For example:

 NimSum(2, 123, 456) = 1111011 ¤ 111001000 = 110110011 = 435

NimSum(3, 123, 456) = 11120 ¤ 121220 = 102010 = 300

NimSum(4, 123, 456) = 1323 ¤ 13020 = 10303 = 307

The strategy for normal form Nim is to compute the Nim-2 sum T of the sizes of all piles. If at any time, you end your turn with T = 0, you are guaranteed a WIN. Any opponent move must leave T not 0 and there is always a move to get T back to 0. This is done by computing NimSum(2, TPS) for each pile; if this is less than the pile size (PS), compute the difference between the PS and the Nim-2 sum and remove it from that pile as your next move.

Write a program to compute NimSum(BXY).


The first line of input contains a single integer P, (1$ \le$P$ \le$1000), which is the number of data sets that follow. Each data set is a single line that contains the data set number, followed by a space, followed by three space separated decimal integers, BX and Y. 2$ \le$B$ \le$2000000, 0$ \le$X$ \le$2000000, 0$ \le$Y$ \le$2000000.


For each data set there is one line of output. It contains the data set number followed by a single space, followed by N, the decimal representation of the Nim sum in base B of X and Y.

Sample Input

1 2 123 456
2 3 123 456
3 4 123 456
4 5 123 456

Sample Output

1 435
2 300
3 307
4 429

hide comments
DHEERAJ KUMAR: 2016-03-21 04:11:54

What's the point of taking input of first number?? Should be moved to tutorial :P

Ankit!!: 2013-06-21 06:11:48

more test cases plzzzz..

$!:D: 2013-06-04 16:00:28

Last edit: 2013-06-04 17:49:01
Naman: 2012-01-19 17:04:02

keep it simple.. Ignore unnecessary details...

Santiago Palacio: 2011-06-17 06:54:33

Weird, VERY weird, as every other testcase is correct, why would that be wrong...?

Suprabh Shukla: 2011-06-16 12:03:21

@Santiago: The answer for 4 is 429 just as the author has written. Maybe that case wasn't there in the test data if you got AC with answer as 427.

Santiago Palacio: 2011-06-13 05:52:13

int (32 bit) is enough?
Yes, it is, got AC, and there's a problem with the test cases, answer for 4 is 427, NOT 429.

Last edit: 2011-06-13 06:08:45
jobless: 2011-05-10 16:55:34

why am i getting wa??
my submission id is 5069253,plzz help

Gurpreet Singh: 2011-05-08 19:06:06

why there is so much unnecessary details in this question??

cegprakash: 2011-04-09 11:47:27

@Rach: i too had that doubt first

Added by:egaeus
Time limit:1s
Source limit:50000B
Memory limit:1536MB
Cluster: Cube (Intel G860)
Languages:All except: ASM64
Resource:Greater New York Region 2010