TopCoder Statistics - Problem Statement

Problem Statement

Let A be a matrix with N rows and N columns and P be a permutation of integers from 0 to N-1. Consider the following sum: Sum(A, P) = A[0, P[0]] + A[1, P[1]] + ... + A[N-1, P[N-1]], where A[i, j] is the element of A in row i and column j (all indices in this problem are 0-based). The matrix A is called nice if Sum(A, P) is always equal to the same value disregarding of the choice of permutation P.

Fox Ciel wants to draw a nice matrix on the blackboard. She is given a String[] board. It contains N elements and each element contains N characters. If j-th character of board[i] is a digit '0', '1', '2', ..., '9', then A[i, j] must be equal to this digit. If j-th character of board[i] is '-', then A[i, j] can be equal to any non-negative integer (it is allowed to be greater than 9).

Let T be the number of different matrices that satisfy all Ciel's requirements (the constraints will guarantee that the number of such matrices is finite). Compute and return the value of (T modulo 1,000,000,007).

Definition

Class:: EqualSums
Method:: count
Parameters:: String[]
Returns:: int
Method signature:: int count(String[] board)
(be sure your method is public)

Constraints

board will contain between 1 and 50 elements, inclusive.
Each element of board will contain exactly N characters, where N is the number of elements in board.
Each character in board will be one of '-', '0', '1', '2', ..., '9'.
The number of matrices that satisfy all Ciel's requirements will be finite.

Examples

{"1-", "-2"}

Returns: 4

The sum A[0, 1] + A[1, 0] must be equal to 3.
{"123", "4--", "--9"}

Returns: 1
{"9--", "-9-", "--9"}

Returns: 271
{"11", "12"}

Returns: 0

There are no nice matrices that match the given board, so T = 0.
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Returns: 0
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Returns: 0
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Returns: 0
{"855--4---","------7-3","---74-76-","-----4--4","-55---874","63---2-5-","96---5985","---6-265-","4---1---0"}

Returns: 1
{"--77-66-","-7---22-","-6664554","---7----","-5---55-","-766----","-7-3-22-","---8---6"}

Returns: 0
{"--25---2","26-4341-","2-1--41-","-9-76-4-","-57-95-7","2-1--411","5-4767-4","483--6--"}

Returns: 0
{"6---2","96-7-","---6-","-9-75","4-1--"}

Returns: 0
{"----6-","6-----","5--5--","-89-89","--8-7-","5-----"}

Returns: 12
{"---85-874","744743-6-","744--3--3","85585-87-","8--854---","--363-6-2","--696----","633-3---2","-114-043-"}

Returns: 1
{"37--452-","--1----1","-6-43--1","5-4--7-4","8-----7-","-6-43---","-9----4-","---65-33"}

Returns: 1
{"7--85--","-3-----","---9--5","5----4-","4255--1","97---86","--77-5-"}

Returns: 1
{"-7-6--7","9-9-8--","-877-8-","343---4","-9-8-99","89-879-","89---9-"}

Returns: 1
{"382788","7-----","6-----","4-----","3-----","6-----"}

Returns: 1
{"4677585","2------","3------","1------","5------","6------","1------"}

Returns: 1
{"---8--","--4---","-5----","5-----","-----2","----1-"}

Returns: 8120
{"---6---","6------","----6--","-----4-","-0-----","------3","--4----"}

Returns: 34300
{"--7------","-4-------","------2--","-----6---","5--------","-------2-","---6-----","--------8","----8----"}

Returns: 7282800
{"---2-","----3","4----","-3---","--3--"}

Returns: 744
{"----8---","--9-----","4-------","-------1","-----4--","------2-","---6----","-6------"}

Returns: 578556
{"4----","----0","--7--","-7---","---3-"}

Returns: 1280
{"----3-","7-----","--5---","-6----","---4--","-----5"}

Returns: 27720
{"------4","--7----","-7-----","-----6-","---5---","----2--","3------"}

Returns: 126000
{"--2---","-6----","3-----","----6-","-----4","---8--"}

Returns: 19548
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Returns: 433258310
{"---", "111", "112"}

Returns: 0
{"-11", "-11", "-12"}

Returns: 0
{"-4--", "-0-2", "--1-", "4---"}

Returns: 10
{"--2", "02-", "-10"}

Returns: 0
{"4---","-04-","3---","---3"}

Returns: 16
{"71-","-2-","--5"}

Returns: 7
{"0-0","-2-","0-0"}

Returns: 3
{"4---","2---","-2--","--01"}

Returns: 9
{"---1","-4--","2---","--0-"}

Returns: 30
{"-06","2--","8--"}

Returns: 3
{"-4-","--4","04-"}

Returns: 5
{"0-2","-3-","-3-"}

Returns: 4
{"---8","7---","-377","-0--"}

Returns: 8
{"-1-","1-2","-6-"}

Returns: 3
{"--7","-4-","8--"}

Returns: 136
{"-1-","5-4","-3-"}

Returns: 6
{"-5-3-","---6-","2----","---5-","2-3-2"}

Returns: 6
{"--1","9--","-1-"}

Returns: 31
{"--------3","------3--","--0------","-1-------","3---0-01-","-----00--","---2-----","-1----0--","---0-----"}

Returns: 4
{"4-5","-2-","-0-"}

Returns: 5
{"----2","----0","0----","-2---","1-11-"}

Returns: 3
{"--2","--3","34-"}

Returns: 6
{"-0-","1-1","-0-"}

Returns: 2
{"4-5","--1","-1-"}

Returns: 2
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Returns: 1
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Returns: 1
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Returns: 19
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Returns: 1
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Returns: 1
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Returns: 1
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Returns: 190
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Returns: 2
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Returns: 72
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Returns: 72
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Returns: 4
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Returns: 480
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Returns: 16
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Returns: 1920
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Returns: 777600
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Returns: 63642672
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Returns: 10077696
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Returns: 387158345
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Returns: 45094393
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Returns: 523422657
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Returns: 17280
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Returns: 256
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Returns: 70207355
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Returns: 3538944
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Returns: 318399545
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Returns: 10616832
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Returns: 290451936
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Returns: 143327232
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Returns: 440382371
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Returns: 791465068
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Returns: 183999965
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Returns: 406463307
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Returns: 83654870
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Returns: 107376140
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Returns: 631447797
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Returns: 246774373
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Returns: 670275552
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Returns: 813326776
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Returns: 651917829
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Returns: 816252169
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Returns: 377486913
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Returns: 99201653
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Returns: 651614558
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Returns: 832871757
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Returns: 292248064
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Returns: 851893375
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Returns: 960838295
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Returns: 2
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Returns: 9413562
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Returns: 30
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Returns: 1
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Returns: 1
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Returns: 7
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Returns: 84
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Returns: 6
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Returns: 1
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Returns: 1
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Returns: 6
{"-----8----------------------------8---------------","-------7----7---------7-7-----7-7-----7-------7---","---8----------------8---------------------------8-","----8--------------------------------8------8----8","7-7---7--------7--------------------7------7------","7-7---7------------------------------------7------","----------------------------------------8---------","----8--------------------------------8----8-8----8","---------7---7-------------7----------------------","-----------------88-8-----8---------------------8-","---7----------------7-----7---------------------7-","---------8---8------------------------------------","----------8----------8---------8------------------","----------8----------8---------8-------------8----","---8--------------8-8-----8-----------------------","----7-------------------------------------7------7","-------7-----------------7----7-------7-------7---","-------------------------------------7----7-7----7","--------------8--------------8-----------8--------","-----7----------------------------7---------------","-----7----------------------------7-----7---------","-----------------------------7---7-------7--------","777---7-----------------------------7------7---7--","--------------7--------------7---7-------7--------","-------8--------------8-8-----8-8-------------8---","-------7----7---------7--7------7-----7-------7---","-------8----8---------8-88----8-8-------------8---","-----------7----7------7----7----------7----------","-----------------------------8---8-------8--------","----------------------------------7-----7---------","777---7--------7--------------------7------7---7--","----------------------------------7-----7---------","-7----7--------7---------------------------7---7--","--------7------------7---------7-------------7----","---7-------------77-7-----7---------------------7-","-----------------------------7-----------7--------","777---7--------7---------------------------7---7--","---7-------------7--7-----------------------------","-----------8----------------8------8---8----------","----8--------------------------------8----8-8----8","--------7-7----------7---------7-------------7----","-----------8----8--8---8----8------8---8----------","7-7---7--------7--------------------7------7---7--","----7--------------------------------7----7-7-----","--------------7--------------7---7----------------","-------------------------------------7----7------7","-------------------------------------7----7-7----7","----7-------------------------------------7-7-----","-----7----------------------------7-----7---------","-------7----7------------7----7-------------------"}

Returns: 93864121
{"---------1------------2-----------6---------------","--------------------1-----------------------------","---05---1-----3-1------4-------------------4------","------5------------------------------4----6-------","----------8--8-5----------------55----------------","-1---------5---------------7---------------------1","--------------------------0---11-------6----------","------------------6-------3---44------------------","-------3---------------------5-----53---3----7----","---162--2-------2------5-------------------5------","---------4------------5-----------9---------------","-------0-----------3---------------20-------------","---1-2--2-----4-2------5--------------------------","--------------------------1---22-------7----------","-------1-----------4---------3-----31---1----5--4-","------7------------------------------6----8-------","---16---2-----4-2------5-------------------5------","-----------------0------23------------6-----------","---051--1-----3-1------4--------------------------","-2---------6---------------8----------------------","-------------------------3--3---------6-----------","2-1-----------------------------------------5-2---","----------5----------------------2----------------","-------0-----------3---------2-----20---0----4--3-","-3-------------------------9---------------------3","----------7--7------------------------------------","-------0-----------3---------2-----20---0---------","------------------3-------0---11-------6----------","-3---------7-------------------------------------3","------------------4------------2-------7----------","-3---------7---------------9---------------------3","------------8-------3-----------------------------","-------1---------------------3------1---1----5----","------------9-------4-----------------------------","-----------5---------------7---------------------1","-------2-----------5---------4-----42---2----6--5-","------7-----------------------------------8----4--","------7------------------------------6---------4--","------7------------------------------6---------4--","---------3------------4-----------8---------------","-------0---------------------2-----20---0-------3-","--------------------4-----------------------------","------------9-------4-----------------------------","------------------4-------1---22-------7----------","-------------6-3-----------------3----------------","---------2------------------------7---------------","-----------------2------45------------8-----------","2-1------------------3-------------------1--5-----","------------------6-----------44-------9----------","------------------6-------3----4-------9----------"}

Returns: 144

Statistics

Problem Statement for "EqualSums"

Problem Statement

Definition

Constraints

Examples