From: Subject: Regional Problem Set Date: Thu, 26 Jul 2007 03:19:22 +0500 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----=_NextPart_000_0034_01C7CF33.C2E07480" X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 This is a multi-part message in MIME format. ------=_NextPart_000_0034_01C7CF33.C2E07480 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Content-Location: http://www.acm.inf.ethz.ch/ProblemSetArchive/B_EU_SWERC/1995/RegionalProblems.html Regional Problem Set

ACM International Collegiate Programming=20 Contest
Southwestern European Regional Contest
ETH Zurich, = Switzerland,=20 December 9, 1995

Problem Set

Rules=20 and Advice

Problem=20 A: Intersection

Problem=20 B: Synchronous Design

Problem=20 C: Graph Coloring

Problem=20 D: Triangle

Problem=20 E: Anagram

Problem=20 F: Spreadsheet

Problem=20 G: Cube

Problem=20 H: Peter's Calculator

Problem=20 J: Partial Differential Equations

Sponsored by Microsoft
Supported by Union Bank = of=20 Switzerland

Some Rules and Advice

All questions require you to read the test data from a single input = file and=20 to write the results to a single output file. The names of both files = are given=20 in the header of the problem. Your are not allowed to read or write any = other=20 files than the specified ones. Standard input and output are also = considered=20 files.

Output must correspond exactly to the provided sample = output,=20 including spelling and spacing.

All lines (including the last = one)=20 should be terminated with a new-line character, and no whitespace should = appear=20 at the end of a line unless explicitly specified. Tabs should never be = used.=20

All programs will be re-compiled prior to testing with the = judges' data.=20 Non-standard libraries may not be used in your solutions. C programs may = not=20 include any files except: ctype.h, math.h, = stdio.h,=20 stdlib.h, string.h, and strings.h. Pascal = programs=20 may use the extended Reset, Rewrite and the = Close=20 statement, which are not part of the ISO Pascal standard.

After = analyzing=20 a program, the judges will send one of the following messages:

Program accepted. Your program passed all tests and is = accepted as=20 correct.=20
Compile-time error. The judges were not able to = successfully=20 compile your program. The compiler returned an error (not just a = warning).=20
Run-time error. Your program "crashed", i. e. it exited = prematurely=20 due to a run-time error.=20
Wrong answer. The program run through one or more test = cases=20 without a run-time error but the output did not match the expected = output.=20
Presentation error. The output seems to be correct but it = is not=20 presented in the required format. Since it is not always easy to = distinguish=20 this message from the wrong answer message, it is only sent in = obvious=20 cases.=20
Time-limit exceeded. Your program did not finish within the = given=20 amount of time.=20
Contest rule violation. Your program violates a contest = rule like=20 calling non-standard libraries.

Programming style is not considered in this contest. You are free to = code in=20 whatever style you prefer.

The CPU time limit for all problems = is 3=20 minutes except when specified otherwise.

All questions regarding = the=20 contest material should be submitted to the judges by filling in a=20 clarification request form. You can ask the runners in the = room=20 for any non-contest related matters or for getting your printer output. = The=20 helpers will not answer any questions regarding the contest material.=20

Judges' decisions are to be considered final. No cheating will = be=20 tolerated.

Success!

Problem A: Intersection

Source file: intersection.c / intersection.p
Input file:=20 intersection.in
Output file: intersection.out

You are to write = a=20 program that has to decide whether a given line segment intersects a = given=20 rectangle.

An example:

line: start point: (4,9)
end point: (11,2)
rectangle: left-top: = (1,5)
right-bottom: (7,1)

Figure 1: = Line segment=20 does not intersect rectangle

The line is said to intersect = the=20 rectangle if the line and the rectangle have at least one point in = common. The=20 rectangle consists of four straight lines and the area in between. = Although all=20 input values are integer numbers, valid intersection points do not have = to lay=20 on the integer grid.

Input

The input consists of n test cases. The first line of the = input file=20 contains the number n. Each following line contains one test case = of the=20 format: xstart ystart xend yend xleft ytop xright ybottom where=20 (xstart, ystart) is the start and (xend, = yend) the=20 end point of the line and (xleft, ytop) the top left and=20 (xright, ybottom) the bottom right corner of the = rectangle. The=20 eight numbers are separated by a blank. The terms top left and = bottom=20 right do not imply any ordering of coordinates.

Output

For each test case in the input file, the output file should contain = a line=20 consisting either of the letter "T" if the line segment intersects the = rectangle=20 or the letter "F" if the line segment does not intersect the rectangle. =

Example

Input

1
4 9 11 2 1 5 7 1

Output

Problem B: Synchronous Design

Source file: synchronous.c / synchronous.p
Input file:=20 synchronous.in
Output file: synchronous.out

The designers of = digital=20 integrated circuits (IC) are very concerned about the correctness of = their=20 designs because, unlike software, ICs cannot be easily tested. Real = tests are=20 not possible until the design has been finalized and the IC has been=20 produced.

To simulate the behavior of a digital IC and to more or = less=20 guarantee that the final chip will work, all of today's digital ICs are = based on=20 a synchronous design.

Figure 1: = The critical=20 path (dashed line) takes 43ns to settle

In a synchronous = design, an=20 external clock signal triggers the IC to go from a well defined and = stable state=20 to the next one. On the active edge of the clock, all input and output = signals=20 and all internal nodes are stable in either the high or low state. = Between two=20 consecutive edges of the clock, the signals and nodes are allowed to = change and=20 may take any intermediate state. The behavior of a synchronous network = is=20 predictable and will not fail due to hazards or glitches introduced by=20 irregularities of the real circuit.

To analyze whether an IC has = a=20 synchronous design, we distinguish between synchronous and=20 asynchronous nodes. Flip flops are synchronous nodes. On the = active edge=20 of the clock, the output of the flip flop changes to the state of the = input and=20 holds that state throughout the next clock cycle. Synchronous nodes are=20 connected to the clock signal.

Simple gates like ANDs or ORs are=20 asynchronous nodes. Their output changes - with a short delay - whenever = one of=20 their inputs changes. During that transition phase, the output can even = go into=20 some undefined or intermediate state.

For simplicity, we assume = that all=20 inputs of the circuits are directly connected to the output of a = synchronous=20 node outside the circuit and that all outputs of the circuit are = directly=20 connected to the input of a synchronous node outside the circuit. =

For an=20 IC to have a synchronous design, mainly two requirements must be met: =

The signal delay introduced between two synchronous nodes = must be=20 smaller or equal than the clock period so there is enough time for = nodes to=20 become stable. In figure 1, the round-ended boxes are asynchronous = nodes=20 whereas the square boxes are synchronous nodes. The delay introduced = on the=20 dashed path is 43ns and exceeds the given clock period of 30ns.=20
There may be no cycles composed exclusively of asynchronous = nodes.=20 In the real circuit such cycles could oscillate. In figure 2, the = dashed path=20 constitutes a cycle of asynchronous nodes.

Figure 3 shows a circuit with a synchronous design. It does not = contain=20 cycles composed of asynchronous nodes and the longest path between two=20 synchronous nodes is shorter than the clock period of 30ns.

Figure 2: = The design=20 contains a cycle (dashed line)

Figure 3: A = synchronous=20 design

Your are to write a program that decides for a given = IC=20 whether it has a synchronous design or not. You are given a network of=20 synchronous and asynchronous nodes, a delay for each node, some inputs = and=20 outputs and the clock period.

You may safely assume that

the delays introduced between any input and any output of the same = node=20 are equal, i.e. equal to the delay given for that node,=20
synchronous nodes have no delay at all,=20
all connections between two nodes connect an output to an input. =

Input

The input file contains several circuits. The first line gives the = number of=20 circuits in the file.

For each circuit in the file, the first = line=20 contains the clock period for the circuit, given as an integer number in = nanoseconds. The next line gives the number of nodes. The following = lines each=20 contain a node, described by a letter and a integer number. The letter = is 'i'=20 for an input, 'o' for an output, 'a' for an asynchronous node and 's' = for a=20 synchronous node. The number gives the delay introduced by the node as = an=20 integer number in nanoseconds (only meaningful for an asynchronous = node). Nodes=20 are implicitly numbered, starting at zero.

After the nodes, the = number of=20 connections for the circuit follows. Each following line contains a pair = of=20 integer numbers denoting a connection between the output and the input = of two=20 nodes. The connection links an output of the node given by the first = number and=20 an input of the node given by the second number.

The clock signal = is not=20 given in the input file. We assume that all synchronous nodes are = properly=20 connected to the clock signal.

Output

For each circuit in the input file, your output file should contain a = line=20 with one of the following messages:

"Synchronous design. Maximum delay: <ss>." = if the=20 circuit has a synchronous design. <ss> should be replaced = by the=20 longest delay found on any path between two synchronous nodes.=20
"Circuit contains cycle." if the circuit contains a cycle = composed exclusively of asynchronous nodes.=20
"Clock period exceeded." if there is a path between two=20 synchronous nodes that is longer than the given clock period and there = are no=20 cycles composed of asynchronous nodes.

Example

Input

Output

Synchronous design. Maximum delay: 28

Problem C: Graph Coloring

Source file: coloring.c / coloring.p
Input file: = coloring.in
Output=20 file: coloring.out

You are to write a program that tries to find = an=20 optimal coloring for a given graph. Colors are applied to the nodes of = the graph=20 and the only available colors are black and white. The coloring of the = graph is=20 called optimal if a maximum of nodes is black. The coloring is = restricted by the=20 rule that no two connected nodes may be black.

Figure 1: An = optimal graph=20 with three black nodes

Input

The graph is given as a set of nodes denoted by numbers 1...n, = n <=3D 100, and a set of undirected edges denoted by = pairs of=20 node numbers (n1, n2), n1 !=3D n2. The = input file=20 contains m graphs. The number m is given on the first = line. The=20 first line of each graph contains n and k, the number of = nodes and=20 the number of edges, respectively. The following k lines contain = the=20 edges given by a pair of node numbers, which are separated by a space. =

Output

The output should consists of 2m lines, two lines for each = graph found=20 in the input file. The first line of should contain the maximum number = of nodes=20 that can be colored black in the graph. The second line should contain = one=20 possible optimal coloring. It is given by the list of black nodes, = separated by=20 a blank.

Example

Input

Output

3
1 4 5

Problem D: Triangle

Source file: triangle.c / triangle.p
Input file: = triangle.in
Output=20 file: triangle.out

A triangle is a basic shape of planar = geometry. It=20 consists of three straight lines and three angles in between. Figure 1 = shows how=20 the sides and angles are usually labeled.

Figure 1:=20 Triangle

A look into a book about geometry shows that many = formulas=20 for triangles exist:

The values of a, b, c, alpha, = beta, and=20 gamma form a set of six parameters that fully define a triangle. = If a=20 large enough subset of these parameters is given, the missing ones can = be=20 calculated by using the formulas above.

You are to write a = program that=20 calculates the missing parameters for a given subset of the six = parameters of a=20 triangle. For some sets of parameters, it is not possible to calculate = the=20 triangle because either too few is known about the triangle or the = parameters=20 would lead to an invalid triangle. The sides of a valid triangle are = greater=20 than 0 and the angles are greater than 0 and less than pi. Your program = should=20 detect this case and output: "Invalid input." The same phrase = should be=20 output if more than the minimal set needed to compute the triangle is = given but=20 the parameters conflict with each other, e.g. all three angles are given = but=20 their sum is greater than pi.

Other sets of parameters can lead = to more=20 than one but still a finite number of valid solutions for the triangle. = In such=20 a case, your program should output: "More than one = solution."

In=20 all other cases, your program should compute the missing parameters and = output=20 all six parameters.

Input

The first line of the input file contains a number indicating the = number of=20 parameter sets to follow. Each following line consists of six numbers, = separated=20 by a single blank character. The numbers are the values for the = parameters=20 a, alpha, b, beta, c, and = gamma=20 respectively. The parameters are labeled as shown in figure 1. A value = of -1=20 indicates that the corresponding parameter is undefined and has to be=20 calculated. All floating-point numbers include at least eight = significant=20 digits.

Output

Your program should output a line for each set of parameters found in = the=20 input file. If a unique solution for a valid triangle can be found for = the given=20 parameters, your program should output the six parameters a,=20 alpha, b, beta, c, gamma, separated = by a=20 blank character. Otherwise the line should contain the phrase "More = than one=20 solution." or "Invalid input." as explained = above.

The=20 numbers in the output files should include at least eight significant = digits.=20 Your calculations should be precise enough to get the six most = significant=20 digits correct.

Example

Input

3
62.72048064 2.26853639 -1.00000000 0.56794657 -1.00000000 -1.00000000
15.69326944 0.24714213 -1.00000000 1.80433105 66.04067877 -1.00000000
72.83685175 1.04409241 -1.00000000 -1.00000000 -1.00000000 =
-1.00000000

Output

62.72048064 2.26853639 44.02668698 0.56794657 =
24.58722491 0.30510970
Invalid input.
Invalid input.

Problem E: Anagram

Source file: anagram.c / anagram.p
Input file: = anagram.in
Output file:=20 anagram.out

You are to write a program that has to generate all = possible=20 words from a given set of letters.

Example

Given the word "abc", your program should - by exploring all = different=20 combination of the three letters - output the words "abc", "acb", "bac", = "bca",=20 "cab" and "cba".

In the word taken from the input file, some = letters may=20 appear more than once. For a given word, your program should not produce = the=20 same word more than once, and the words should be output in = alphabetically=20 ascending order.

Input

The input file consists of several words. The first line contains a = number=20 giving the number of words to follow. Each following line contains one = word. A=20 word consists of uppercase or lowercase letters from A to Z. Uppercase = and=20 lowercase letters are to be considered different.

Output

For each word in the input file, the output file should contain all = different=20 words that can be generated with the letters of the given word. The = words=20 generated from the same input word should be output in alphabetically = ascending=20 order.

Example

Input

2
abc
acba

Output

abc
acb
bac
bca
cab
cba
aabc
aacb
abac
abca
acab
acba
baac
baca
bcaa
caab
caba
cbaa

Hint

The number of possible combinations raises very quickly with the = number of=20 given letters. Make sure you do not use long words for testing your = program=20 because the output file could become very big, waste a lot of space on = the disk=20 and degrade the performance of the network.

Problem F: Spreadsheet

Source file: spreadsheet.c / spreadsheet.p
Input file:=20 spreadsheet.in
Output file: spreadsheet.out

In 1979, Dan = Bricklin and=20 Bob Frankston wrote VisiCalc, the first spreadsheet application. It = became a=20 huge success and, at that time, was the killer application for the Apple = II=20 computers. Today, spreadsheets are found on most desktop = computers.

The=20 idea behind spreadsheets is very simple, though powerful. A spreadsheet = consists=20 of a table where each cell contains either a number or a formula. A = formula can=20 compute an expression that depends on the values of other cells. Text = and=20 graphics can be added for presentation purposes.

You are to write = a very=20 simple spreadsheet application. Your program should accept several = spreadsheets.=20 Each cell of the spreadsheet contains either a numeric value (integers = only) or=20 a formula, which only support sums. After having computed the values of = all=20 formulas, your program should output the resulting spreadsheet where all = formulas have been replaced by their value.

 A1    B1     C1    =
 D1     E1     F1     ...   =20
 A2    B2     C2     D2     E2     F2     ...   =20
 A3    B3     C3     D3     E3     F3     ...   =20
 A4    B4     C4     D4     E4     F4     ...   =20
 A5    B5     C5     D5     E5     F5     ...   =20
 A6    B6     C6     D6     E6     F6     ...   =20
 ...   ...    ...    ...    ...    ...    ...

Figure 1: Naming of the top left cells

Input

The first line of the input file contains the number of spreadsheets = to=20 follow. A spreadsheet starts with a line consisting of two integer = numbers,=20 separated by a space, giving the number of columns and rows. The = following lines=20 of the spreadsheet each contain a row. A row consists of the cells of = that row,=20 separated by a single space.

A cell consists either of a numeric = integer=20 value or of a formula. A formula starts with an equal sign (=3D). After = that, one=20 or more cell names follow, separated by plus signs (+). The value of = such a=20 formula is the sum of all values found in the referenced cells. These = cells may=20 again contain a formula. There are no spaces within a = formula.

You may=20 safely assume that there are no cyclic dependencies between cells. So = each=20 spreadsheet can be fully computed.

The name of a cell consists of = one to=20 three letters for the column followed by a number between 1 and 999 = (including)=20 for the row. The letters for the column form the following series: A, B, = C, ...,=20 Z, AA, AB, AC, ..., AZ, BA, ..., BZ, CA, ... ZZ, AAA, AAB, AAC, ... AAZ, = ABA,=20 ..., ABZ, ACA, ..., ZZZ. These letters correspond to the number from 1 = to 18278.=20 The top left cell has the name A1. See figure 1.

Output

The output of your program should have the same format as the input, = except=20 that the number of spreadsheets and the number of columns and rows are = not=20 repeated. Furthermore, all formulas should be replaced by their value. =

Example

Input

1
4 3
10 34 37 =3DA1+B1+C1
40 17 34 =3DA2+B2+C2
=3DA1+A2 =3DB1+B2 =3DC1+C2 =3DD1+D2

Output

10 34 37 81
40 17 34 91
50 51 71 172

Problem G: Cube

Source file: cube.c / cube.p
Input file: None
Output = file:=20 cube.out

There was once a 3 by 3 by 3 cube built of 27 smaller = cubes. It=20 has fallen apart into seven pieces:

Figure 1: = The seven pieces=20 that once formed a cube

The seven pieces can be assembled in = many=20 ways to again form the cube. Figure 2 shows one of these possibilities. = The=20 first square stands for the front plane, the next one for the middle = plane and=20 the last one for the back plane of the cube. The letters in the cells = stand for=20 the name of piece filling out the corresponding space in the cube. The = name of=20 the seven pieces can be found in figure 1.

a   d   c      d   d =
  g      d   g   g  =20
a   c   c      b   f   g      b   f   e  =20
a   a   c      f   f   e      b   e   e  =20


a   a   b      f   f   e      f   e   e  =20
a   b   b      g   f   c      g   g   e  =20
a   d   c      d   d   c      d   g   c

Figure 2: Two possibilities of assembling the cube

You = are to=20 write a program that outputs all possibilities of assembling the cube = but=20 suppress solutions that are mere rotations of another solution. The time = limit=20 for this problem is 15 minutes!

Input

No input is needed.

Output

For each solution found, your program should output a line containing = the=20 solution as a string. The string is a linearized form of the cube. Each = letter=20 stands for the piece filling out the corresponding space in the cube. It = is=20 linearized as follows:

The string consists of substrings representing the front, middle = and back=20 plane.=20
Each substring consists of substrings representing the top, middle = and=20 bottom row.=20
Each row substring consists of letters representing the left, = middle and=20 right cell.

The solutions in figure 2 would be represented like this: =

adcaccaacddgbfgffedggbfebee
aababbadcffegfcddcfeeggedgc

It is very important that your program uses the naming convention = given in=20 figure 1 and linearizes the cube as explained above.

Figure 3: = Positions of the=20 cells in the string

Figure 3 again shows how the cells of = the cube=20 are linearized.

Example

The output of your program could start like this: =

aababbadcggeffcddcgeegfedfc
aababbadceffgdcgdceefedfggc
aababbadcffegfcddcfeeggedgc
...

Hint

Piece a is the only part that, by rotation and translation, = cannot be=20 transformed into itself. In order to avoid solutions that are mere = rotations of=20 an already found solution, you may restrict transformations of piece = a to=20 translations.

Problem H: Peter's Calculator

Source file: calculator.c / calculator.p
Input file:=20 calculator.in
Output file: calculator.out

Unfortunately, = Peter's=20 Calculator broke down last week. Now Peter is left with his computer, = which has=20 no calculator application, and paper and pencil, which is too tiresome = for an=20 engineer. As one of Peter's friends, you are asked to write him a = calculator=20 application. After talking to him, you figure out the following:

Peter does only integer arithmetic. The operations he needs are = addition,=20 subtraction and multiplication.=20
He would like to use an arbitrary number of variables whose names = are not=20 longer than 50 characters.=20
His main way of doing calculations are to type in a few formulas = and to=20 assign them to variables. Some formulas are complicated expressions, = which can=20 refer to yet undefined variables, while other formulas consist of a = single=20 number. Then Peter asks for the value of some variables, i.e. he = evaluates the=20 formulas.=20
Peters wants to redefine some variables and then to reevaluate = formulas=20 that depend on these variables.

The input strictly adheres to the following syntax (given in EBNF): =

file =3D line { line } <EOF>.
line =3D [ assignment | print | reset ] <CR>.
assignment =3D var ":=3D" expression.
print =3D "PRINT" var.
reset =3D "RESET".
expression =3D term { addop term }.
term =3D factor { mulop factor }.
factor =3D "(" expression ")" | var | number.
addop =3D "+" | "-".
mulop =3D "*".

In the Extended Backus-Naur Formalism (EBNF), A =3D B C = declares that=20 the grammatical construct A consists of a B followed = by a=20 C. A =3D B | C means that A consists of a = B=20 or, alternatively, of a C. A =3D [ B ] defines = construct=20 A to be either a B or nothing and A =3D { B } = tells you=20 that A consists of the concatenation of any number of = Bs=20 (including none).

The production var stands for the name = of a=20 variable, which starts with a letter followed by up to 49 letters or = digits.=20 Letters may be uppercase or lowercase. The production number = stands for=20 a integer number. The precise syntax for these productions are given = below. The=20 case of letters is important for both variables and statements. =

var =3D letter { letter | digit }.
number =3D [ "-" ] digit { digit }.
letter =3D "A" | "B" | ... | "Z" | "a" | "b" | ... | "z".
digit =3D "0" | "1" | ... | "8" | "9".

Between the parts of a grammatical construct but not within the names = of=20 variables or integer numbers, any number of spaces may appear. = <EOF>=20 stands for the end of the input file and <CR> stands for the = new-line=20 character. All lines in the input file are shorter than 200=20 characters.

The value of a variable is said to be undefined:

if it has not yet been defined or it refers to a variable, which = has not=20 yet been defined;=20
if the definition of the variable contains a cycle.

Your are to write a program that implements Peter's calculator. It = should=20 store all variable definitions and for each "PRINT" statement evaluate = the=20 specified variable based on the latest variable definitions. If your = program=20 encounters a "RESET" statement, it should delete all stored variables so = that=20 all variables become undefined.

Input

The input file contains calculations adhering to the syntax given = above. Each=20 line contains either an assignment to a variable, a "PRINT" statement, a = "RESET"=20 statement or nothing.

Output

For each "PRINT" statement found in the input file, your program = should=20 output a line containing the numerical value of the specified variable = or the=20 word "UNDEF" if the variable is undefined.

Example

Input

a :=3D b + c
b :=3D 3
c :=3D 5
PRINT d
PRINT a
b :=3D 8
PRINT a
RESET
PRINT a

Output

UNDEF
8
13
UNDEF

Problem J: Partial differential equations =

Source file: equation.c / equation.p
Input file: = equation.in
Output=20 file: equation.out

In engineering sciences, partial differential=20 equations play an important and central role. For example, the = temperature of a=20 metal plate can be expressed as a partial differential equation if the=20 temperature on the boundaries is known. This is called a boundary value=20 problem.

Usually, it is not easy to solve these problems. = Analytical=20 solutions exist only in very special cases. But there are some more or = less=20 "good" numerical ways to solve boundary value problems.

We now = will look=20 at one method which works with finite difference approximations for the=20 derivatives of a function. For this approach, we do not look at an = analytical=20 function u(x) but we are only interested in the values of = u=20 at a finite set of discrete points x_i: = u_i =3D=20 u(xi). The distance between two adjacent points,=20 x_i and x_i+1, is constant: h = =3D=20 x_i+1 - x_i (cf. figure 1). =

Figure 1: = u(x) at some=20 discrete points x_i

The finite difference = approximation of=20 a first derivative of the function u(x) is

(1)

The second derivative is approximated by

(2)

This approximation works with 2-dimensional functions = u(x,=20 y) as well. For simplicity we only work on square problems, i.e.=20 (x, y) is element of [0,1] x [0,1]. Again, the area of the = function is discretized in a similar way: x_i+1 -=20 x_i =3D y_i+1 - y_i = =3D h=20 =3D 1 / n, for some integer n >=3D 2. We only look at = the values of=20 u(x, y) at the discrete points P_k = =3D=20 (x_i, y_j): = u_i,j =3D=20 u(P_k). With this discretization, we have a = function=20 u_i,j as shown in figure 2:

Figure 2: = Function=20 u_i,j in the discretization area

On the boundary,=20 u(x_i, y_j) is given by 4 known = functions:

(3)

The points P_k cover the inner points of the = discretization=20 area, i.e. the area without the boundary. They are numbered from left to = right=20 and from top to bottom like English text.

What we now want to do = is to=20 solve the poisson-equation in the area [0,1] x [0,1]:

(4)

with the above boundary conditions. f(x, y) is a = given=20 2-dimensional function. With equation (2) and the above discretization, = the=20 poisson-equation can be approximated at

, (5)

where f_i,j is the function = f(x,=20 y), evaluated at the discrete points (x_i,=20 y_j).

Formula (5) can be written in a more = readable=20 form, depending on the position of the discrete points:

(6a)

A similar equation, which we will use as an example below, is: =

(6b)

We call the 3x3 matrix on the left hand side v and the 3x3 = matrix on=20 the right hand side g. Now, equation (6b) can be formulated in = every=20 point of the discrete area of figure 2:

(7)

(7) is a linear equation system for the values of u(x,=20 y) at the points P₁, P₂,=20 P₃ and P₄.

By rearranging and = adding=20 the terms on each line, the linear equation system can be formulated as: =

az =3D b (8)

where a is a 4x4 matrix and b is a vector with 4 elements. = Vector=20 z represents the unknown values of u(x, y) = at the=20 points P₁, P₂, P₃ = and=20 P₄.

You are to write a program that creates = the linear=20 equation system (7) in the form (8) for any two matrices v and = g=20 (6). As input, the two matrices v and g and the functions=20 b₁, b₂, b₃,=20 b₄, and f are given. Also, a parameter n = is=20 given as the number of discretization intervals. Thus, h =3D 1/n. As the = result,=20 your program should calculate the matrix a and the vector = b. For=20 this more general case, there are (n-1)² inner points = and=20 a and b must be sized accordingly.

Input

The input file consists of m tests. The number m is = given in=20 the first line of the file. The first line of each test contains the = number=20 n which gives the number of discretizations intervals as defined = above.=20 You may assume that 2 <=3D n <=3D 30. Then the 3x3 matrices = v and=20 g follow. The following four lines contain the functions=20 b₁, b₂, b₃ and=20 b₄, each given as a vector of order n+1, = containing the=20 values for 0, h, 2h, ..., 1. Finally, the function = f is=20 given as a n+1 by n+1 matrix. Like the vectors before, it contains the = values=20 for x, y =3D 0, h, 2h, ..., 1. Each row = contains from=20 left to right the function values for increasing x values while = each=20 column contains from top to bottom the function values for increasing = y=20 values.

A vector occupies one line. Its values are given in = ascending=20 order, separated by a space. A n by n matrix occupies = n=20 lines. Its rows are given in ascending order as vectors, which occupy = one line=20 each. All values found in the input file are integer values.

Output

For each test found in the input file, your program should output the = matrices a and b. Matrix a is a = (n-1)² x=20 (n-1)²matrix (the discretization area (cf. figure 2) = contains=20 (n-1)²inner points, which are unknown). The vector = b=20 is of order (n-1)². They should be output in the same = format=20 as the vectors and matrices in the input file. Your output should only = contain=20 integer values. Note that the expression 1 / h² yields = an=20 integer number and that all other calculations can also be done using = integer=20 numbers.

Example

Input

Output

-36 0 0 36
0 -36 27 0
0 18 -36 0
9 0 0 -36
-8 -152 -198 -333

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ACM International Collegiate Programming=20 ContestSouthwestern European Regional ContestETH Zurich, = Switzerland,=20 December 9, 1995

Problem Set

Contents

Some Rules and Advice

Problem A: Intersection

An example:

Input

Output

Example

Input

Output

Problem B: Synchronous Design

Input

Output

Example

Input

Output

Problem C: Graph Coloring

Input

Output

Example

Input

Output

Problem D: Triangle

Input

Output

Example

Input

Output

Problem E: Anagram

Example

Input

Output

Example

Input

Output

Hint

Problem F: Spreadsheet

Input

Output

Example

Input

Output

Problem G: Cube

Input

Output

Example

Hint

Problem H: Peter's Calculator

Input

Output

Example

Input

Output

Problem J: Partial differential equations =

Input

Output

Example

Input

Output

ACM International Collegiate Programming=20 Contest
Southwestern European Regional Contest
ETH Zurich, = Switzerland,=20 December 9, 1995