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   <TITLE>1991 ACM Finals, Problem D - The Domino Effect</TITLE>
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<H2>
<I>1991 ACM Scholastic Programming Contest Finals</I></H2></CENTER>

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<H4>
<I>sponsored by AT&amp;T Computer Systems</I></H4></CENTER>

<CENTER>
<H2>
Problem D<BR>
The Domino Effect</H2></CENTER>
A standard set of Double Six dominoes contains 28 pieces (called bones)
each displaying two numbers from 0 (blank) to 6 using dice-like pips. The
28 bones, which are unique, consist of the following combinations of pips:
<PRE>Bone #&nbsp; Pips&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Bone #&nbsp; Pips&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Bone #&nbsp; Pips&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Bone #&nbsp; Pips

1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0 | 0&nbsp;&nbsp;&nbsp;&nbsp; 8&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1 | 1&nbsp;&nbsp;&nbsp;&nbsp; 15&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2 | 3&nbsp;&nbsp;&nbsp;&nbsp; 22&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3 | 6
2&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0 | 1&nbsp;&nbsp;&nbsp;&nbsp; 9&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1 | 2&nbsp;&nbsp;&nbsp;&nbsp; 16&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2 | 4&nbsp;&nbsp;&nbsp;&nbsp; 23&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 4 | 4
3&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0 | 2&nbsp;&nbsp;&nbsp;&nbsp; 10&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1 | 3&nbsp;&nbsp;&nbsp;&nbsp; 17&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2 | 5&nbsp;&nbsp;&nbsp;&nbsp; 24&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 4 | 5
4&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0 | 3&nbsp;&nbsp;&nbsp;&nbsp; 11&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1 | 4&nbsp;&nbsp;&nbsp;&nbsp; 18&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2 | 6&nbsp;&nbsp;&nbsp;&nbsp; 25&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 4 | 6
5&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0 | 4&nbsp;&nbsp;&nbsp;&nbsp; 12&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1 | 5&nbsp;&nbsp;&nbsp;&nbsp; 19&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3 | 3&nbsp;&nbsp;&nbsp;&nbsp; 26&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 5 | 5
6&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0 | 5&nbsp;&nbsp;&nbsp;&nbsp; 13&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1 | 6&nbsp;&nbsp;&nbsp;&nbsp; 20&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3 | 4&nbsp;&nbsp;&nbsp;&nbsp; 27&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 5 | 6
7&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0 | 6&nbsp;&nbsp;&nbsp;&nbsp; 14&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2 | 2&nbsp;&nbsp;&nbsp;&nbsp; 21&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3 | 5&nbsp;&nbsp;&nbsp;&nbsp; 28&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 6 | 6</PRE>
All the Double Six dominoes in a set can he laid out to display a 7 x 8
grid of pips. Each layout corresponds at least one "map" of the dominoes.
A map consists of an identical 7 x 8 grid with the appropriate bone numbers
substituted for the pip numbers appearing on that bone. An example of a
7 x 8 grid display of pips and a corresponding map of bone numbers is shown
below.
<PRE>&nbsp;&nbsp;&nbsp; 7 x 8 grid of pips&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; map of bone numbers

&nbsp; 6&nbsp; 6&nbsp; 2&nbsp; 6&nbsp; 5&nbsp; 2&nbsp; 4&nbsp; 1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 28 28 14&nbsp; 7 17 17 11 11
&nbsp; 1&nbsp; 3&nbsp; 2&nbsp; 0&nbsp; 1&nbsp; 0&nbsp; 3&nbsp; 4&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 10 10 14&nbsp; 7&nbsp; 2&nbsp; 2 21 23
&nbsp; 1&nbsp; 3&nbsp; 2&nbsp; 4&nbsp; 6&nbsp; 6&nbsp; 5&nbsp; 4&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 8&nbsp;&nbsp; 4 16 25 25 13 21 23&nbsp;
&nbsp; 1&nbsp; 0&nbsp; 4&nbsp; 3&nbsp; 2&nbsp; 1&nbsp; 1&nbsp; 2&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 8&nbsp;&nbsp; 4 16 15 15 13&nbsp; 9&nbsp; 9
&nbsp; 5&nbsp; 1&nbsp; 3&nbsp; 6&nbsp; 0&nbsp; 4&nbsp; 5&nbsp; 5&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 12 12 22 22&nbsp; 5&nbsp; 5 26 26
&nbsp; 5&nbsp; 5&nbsp; 4&nbsp; 0&nbsp; 2&nbsp; 6&nbsp; 0&nbsp; 3&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 27 24 24&nbsp; 3&nbsp; 3 18&nbsp; 1 19
&nbsp; 6&nbsp; 0&nbsp; 5&nbsp; 3&nbsp; 4&nbsp; 2&nbsp; 0&nbsp; 3&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 27&nbsp; 6&nbsp; 6 20 20 18&nbsp; 1 19</PRE>

<H3>
Input and Output</H3>
The input file will contain several of problem sets. Each set consists
of seven lines of eight integers from 0 through 6, representing an observed
pattern of pips. Each set is corresponds to a legitimate configuration
of bones (there will be at least one map possible for each problem set).
There is no intervening data separating the problem sets.

<P>Correct output consists of a problem set label (beginning with Set #1)
followed by an echo printing of the problem set itself. This is followed
by a map label for the set and the map(s) which correspond to the problem
set. (Multiple maps can be output in any order.) After all maps for a problem
set have been printed, a summary line stating the number of possible maps
appears. At least three lines are skipped between the output from different
problem sets while at least one line separates the labels, echo printing,
and maps within the same problem set. A sample input file of two problem
sets along with the correct output are shown on the reverse of this page.
<H3>
Sample Input</H3>

<PRE>5 4 3 6 5 3 4 6
0 6 0 1 2 3 1 1
3 2 6 5 0 4 2 0
5 3 6 2 3 2 0 6
4 0 4 1 0 0 4 1
5 2 2 4 4 1 6 5
5 5 3 6 1 2 3 1
4 2 5 2 6 3 5 4
5 0 4 3 1 4 1 1
1 2 3 0 2 2 2 2
1 4 0 1 3 5 6 5
4 0 6 0 3 6 6 5
4 0 1 6 4 0 3 0
6 5 3 6 2 1 5 3</PRE>

<H3>
Sample Output</H3>

<PRE>Layout #1:

&nbsp;&nbsp; 5&nbsp;&nbsp; 4&nbsp;&nbsp; 3&nbsp;&nbsp; 6&nbsp;&nbsp; 5&nbsp;&nbsp; 3&nbsp;&nbsp; 4&nbsp;&nbsp; 6
&nbsp;&nbsp; 0&nbsp;&nbsp; 6&nbsp;&nbsp; 0&nbsp;&nbsp; 1&nbsp;&nbsp; 2&nbsp;&nbsp; 3&nbsp;&nbsp; 1&nbsp;&nbsp; 1
&nbsp;&nbsp; 3&nbsp;&nbsp; 2&nbsp;&nbsp; 6&nbsp;&nbsp; 5&nbsp;&nbsp; 0&nbsp;&nbsp; 4&nbsp;&nbsp; 2&nbsp;&nbsp; 0
&nbsp;&nbsp; 5&nbsp;&nbsp; 3&nbsp;&nbsp; 6&nbsp;&nbsp; 2&nbsp;&nbsp; 3&nbsp;&nbsp; 2&nbsp;&nbsp; 0&nbsp;&nbsp; 6
&nbsp;&nbsp; 4&nbsp;&nbsp; 0&nbsp;&nbsp; 4&nbsp;&nbsp; 1&nbsp;&nbsp; 0&nbsp;&nbsp; 0&nbsp;&nbsp; 4&nbsp;&nbsp; 1
&nbsp;&nbsp; 5&nbsp;&nbsp; 2&nbsp;&nbsp; 2&nbsp;&nbsp; 4&nbsp;&nbsp; 4&nbsp;&nbsp; 1&nbsp;&nbsp; 6&nbsp;&nbsp; 5
&nbsp;&nbsp; 5&nbsp;&nbsp; 5&nbsp;&nbsp; 3&nbsp;&nbsp; 6&nbsp;&nbsp; 1&nbsp;&nbsp; 2&nbsp;&nbsp; 3&nbsp;&nbsp; 1

Maps resulting from layout #1 are:

&nbsp;&nbsp; 6&nbsp; 20&nbsp; 20&nbsp; 27&nbsp; 27&nbsp; 19&nbsp; 25&nbsp; 25
&nbsp;&nbsp; 6&nbsp; 18&nbsp;&nbsp; 2&nbsp;&nbsp; 2&nbsp;&nbsp; 3&nbsp; 19&nbsp;&nbsp; 8&nbsp;&nbsp; 8
&nbsp; 21&nbsp; 18&nbsp; 28&nbsp; 17&nbsp;&nbsp; 3&nbsp; 16&nbsp; 16&nbsp;&nbsp; 7
&nbsp; 21&nbsp;&nbsp; 4&nbsp; 28&nbsp; 17&nbsp; 15&nbsp; 15&nbsp;&nbsp; 5&nbsp;&nbsp; 7
&nbsp; 24&nbsp;&nbsp; 4&nbsp; 11&nbsp; 11&nbsp;&nbsp; 1&nbsp;&nbsp; 1&nbsp;&nbsp; 5&nbsp; 12
&nbsp; 24&nbsp; 14&nbsp; 14&nbsp; 23&nbsp; 23&nbsp; 13&nbsp; 13&nbsp; 12
&nbsp; 26&nbsp; 26&nbsp; 22&nbsp; 22&nbsp;&nbsp; 9&nbsp;&nbsp; 9&nbsp; 10&nbsp; 10

There are 1 solution(s) for layout #1.


Layout #2:

&nbsp;&nbsp; 4&nbsp;&nbsp; 2&nbsp;&nbsp; 5&nbsp;&nbsp; 2&nbsp;&nbsp; 6&nbsp;&nbsp; 3&nbsp;&nbsp; 5&nbsp;&nbsp; 4
&nbsp;&nbsp; 5&nbsp;&nbsp; 0&nbsp;&nbsp; 4&nbsp;&nbsp; 3&nbsp;&nbsp; 1&nbsp;&nbsp; 4&nbsp;&nbsp; 1&nbsp;&nbsp; 1
&nbsp;&nbsp; 1&nbsp;&nbsp; 2&nbsp;&nbsp; 3&nbsp;&nbsp; 0&nbsp;&nbsp; 2&nbsp;&nbsp; 2&nbsp;&nbsp; 2&nbsp;&nbsp; 2
&nbsp;&nbsp; 1&nbsp;&nbsp; 4&nbsp;&nbsp; 0&nbsp;&nbsp; 1&nbsp;&nbsp; 3&nbsp;&nbsp; 5&nbsp;&nbsp; 6&nbsp;&nbsp; 5
&nbsp;&nbsp; 4&nbsp;&nbsp; 0&nbsp;&nbsp; 6&nbsp;&nbsp; 0&nbsp;&nbsp; 3&nbsp;&nbsp; 6&nbsp;&nbsp; 6&nbsp;&nbsp; 5
&nbsp;&nbsp; 4&nbsp;&nbsp; 0&nbsp;&nbsp; 1&nbsp;&nbsp; 6&nbsp;&nbsp; 4&nbsp;&nbsp; 0&nbsp;&nbsp; 3&nbsp;&nbsp; 0
&nbsp;&nbsp; 6&nbsp;&nbsp; 5&nbsp;&nbsp; 3&nbsp;&nbsp; 6&nbsp;&nbsp; 2&nbsp;&nbsp; 1&nbsp;&nbsp; 5&nbsp;&nbsp; 3

Maps resulting from layout #2 are:

&nbsp; 16&nbsp; 16&nbsp; 24&nbsp; 18&nbsp; 18&nbsp; 20&nbsp; 12&nbsp; 11
&nbsp;&nbsp; 6&nbsp;&nbsp; 6&nbsp; 24&nbsp; 10&nbsp; 10&nbsp; 20&nbsp; 12&nbsp; 11
&nbsp;&nbsp; 8&nbsp; 15&nbsp; 15&nbsp;&nbsp; 3&nbsp;&nbsp; 3&nbsp; 17&nbsp; 14&nbsp; 14
&nbsp;&nbsp; 8&nbsp;&nbsp; 5&nbsp;&nbsp; 5&nbsp;&nbsp; 2&nbsp; 19&nbsp; 17&nbsp; 28&nbsp; 26
&nbsp; 23&nbsp;&nbsp; 1&nbsp; 13&nbsp;&nbsp; 2&nbsp; 19&nbsp;&nbsp; 7&nbsp; 28&nbsp; 26
&nbsp; 23&nbsp;&nbsp; 1&nbsp; 13&nbsp; 25&nbsp; 25&nbsp;&nbsp; 7&nbsp; 21&nbsp;&nbsp; 4
&nbsp; 27&nbsp; 27&nbsp; 22&nbsp; 22&nbsp;&nbsp; 9&nbsp;&nbsp; 9&nbsp; 21&nbsp;&nbsp; 4

&nbsp; 16&nbsp; 16&nbsp; 24&nbsp; 18&nbsp; 18&nbsp; 20&nbsp; 12&nbsp; 11
&nbsp;&nbsp; 6&nbsp;&nbsp; 6&nbsp; 24&nbsp; 10&nbsp; 10&nbsp; 20&nbsp; 12&nbsp; 11
&nbsp;&nbsp; 8&nbsp; 15&nbsp; 15&nbsp;&nbsp; 3&nbsp;&nbsp; 3&nbsp; 17&nbsp; 14&nbsp; 14
&nbsp;&nbsp; 8&nbsp;&nbsp; 5&nbsp;&nbsp; 5&nbsp;&nbsp; 2&nbsp; 19&nbsp; 17&nbsp; 28&nbsp; 26
&nbsp; 23&nbsp;&nbsp; 1&nbsp; 13&nbsp;&nbsp; 2&nbsp; 19&nbsp;&nbsp; 7&nbsp; 28&nbsp; 26
&nbsp; 23&nbsp;&nbsp; 1&nbsp; 13&nbsp; 25&nbsp; 25&nbsp;&nbsp; 7&nbsp;&nbsp; 4&nbsp;&nbsp; 4
&nbsp; 27&nbsp; 27&nbsp; 22&nbsp; 22&nbsp;&nbsp; 9&nbsp;&nbsp; 9&nbsp; 21&nbsp; 21

There are 2 solution(s) for layout #2.</PRE>

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