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Domino arrangements (Posted on 2006-05-27) Difficulty: 5 of 5
Take the 15 smallest dominoes in a set (double blank through double four.)

In how many ways can they be arranged in a row such that the numbers on consecutive pieces match.

Count the two directions separately.

See The Solution Submitted by Jer    
Rating: 3.6667 (3 votes)

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Solution Fully computerized solution | Comment 6 of 7 |

In the set up to double fours, there are five possibilities for the number of pips at either end: 0 - 4.  Each is paired with each of the other four possibilities and with itself, and therefore appears 6 times in all.  As this is an even number, and internally within the row of dominoes they are paired, the two ends of the row must have a matching pair of pip numbers.  In this way, it is as if it were a circular arrangement; the difference is that a starting point is marked.

For a computer solution, we could rearrange all 15 pairs of numbers, but this would take a considerable amount of computing time. So rather than do that, we can rearrange the known end-to-end conditions, rather than the dominoes themselves. Letting a vertical bar (|) represent the boundary between one domino and another, we know that, to take 2 pips as an example, 2|22|2 must occur somewhere (considering the arrangement as circular; otherwise the exception is if a 2 piece is at an end); also, a simple 2|2 must occur, but not adjacent to the 2|22|2.  Since each has two adjacencies to non-2's, all the dominoes containing a 2-pip are accounted for, and similarly for the other numbers of pips.

Let a represent 0|0 and A represent 0|00|0.
Let b represent 1|1 and B represent 1|11|1.
Let c represent 2|2 and C represent 2|22|2.
Let d represent 3|3 and D represent 3|33|3.
Let e represent 4|4 and E represent 4|44|4.

Then one particular arrangement can be represented as:

abcdeACEBD

translated literally that is
0|01|12|23|34|40|00|02|22|24|44|41|11|13|33|3

but since a domino is not to be broken, that last 3 is brought to the front (as a convention; we could have brought the initial zero to the back if we had chosen the alternative convention):
30|01|12|23|34|40|00|02|22|24|44|41|11|13|33

It can be seen, however, that with this scheme, we will miss all those rows that begin with a double domino. We will have to adjust the numbers, to count those that end with a double (the last character in the representing string is capital) twice.

So the brute force method is to go through all permutations of abcdeABCDE and verify that no two cases (upper and lower) of the same letter, such as aA, occur together, and no two pairs of the same two letters are together. For example, abcdeABCDE itself cannot be used as bc and BC are both together, as are the other pairs, including eA and Ea, even though the latter is adjacent end-to-beginning.  Be assuring, for example that de doesn't occur in the same string as ED (the reversal), we also assure that every possible pair does occur.

We take care of the wrap-around nature of this by appending a copy of the first character at the end, for comparison purposes.  This isn't to be confused with our conceptual taking of the last pip number from the back to the front.


DECLARE FUNCTION tr$ (x$)
DECLARE SUB permute (a$)
CLS
DEFLNG A-Z

a$ = "abcdeABCDE": h$ = a$

DO
 a1$ = LCASE$(a$ + LEFT$(a$, 1))
 good = 1
 FOR i = 1 TO 10
   IF MID$(a1$, i, 1) = MID$(a1$, i + 1, 1) THEN good = 0: EXIT FOR
 NEXT
 IF good THEN
   FOR i = 1 TO 10
    ad$ = MID$(a1$, i, 2)
    ad2$ = RIGHT$(ad$, 1) + LEFT$(ad$, 1)
    IF INSTR(i + 1, a1$, ad$) > 0 OR INSTR(a1$, ad2$) > 0 THEN good = 0: EXIT FOR
   NEXT
   IF good THEN
     PRINT a$
     PRINT tr$(RIGHT$(a$, 1));
     FOR i = 1 TO 9
      PRINT tr$(MID$(a$, i, 1)); "|"; tr$(MID$(a$, i, 1));
      IF UCASE$(MID$(a$, i, 1)) = MID$(a$, i, 1) THEN
        PRINT tr$(MID$(a$, i, 1)); "|"; tr$(MID$(a$, i, 1));
      END IF
     NEXT
     PRINT tr$(RIGHT$(a$, 1));
     IF UCASE$(RIGHT$(a$, 1)) = RIGHT$(a$, 1) THEN
       PRINT "|"; tr$(RIGHT$(a$, 1)); tr$(RIGHT$(a$, 1))
     ELSE
       PRINT
     END IF

     ct = ct + 1
   END IF
 END IF
 permute a$
LOOP UNTIL a$ = h$

PRINT ct


DEFSNG A-Z
FUNCTION tr$ (x$)
   tr$ = LTRIM$(STR$(INSTR("abcde", LCASE$(x$)) - 1))
END FUNCTION

(The permute subroutine is found elsewhere on this site.)


The count arrived at is 84,480. But remember that this excludes rows that begin with a double piece. We can simply count those that end in a double piece twice, and thereby consider that last piece to be in the front.  By our scheme, half the rows end in a double (the last character in the representing string is a capital letter). So we multiply 84,480 by 3/2 giving a total of 126,720 possible rows.  And, rightly, now only 1/3 of them end in a double piece, and 1/3 begin with a double piece. None can do both, as double pieces can't be adjacent (to be split by cutting the circular arrangement) as they have different numbers on them. This jibes with the fact that 1/3 of the pieces are double.

In the end, the final count is 126,720 possible rows.

A random sampling of 291 of the 16,896 (i.e., 84,480/5) possible rows with blanks on the ends (but excluding double at the first position) follows. It was made by checking, for each string, the last character was a or A, and a pseudorandom number between 0 and 16895 was less than 300. This serves as a check that what we're allowing through is correct. Approximately half (133) do end in a double domino. The total number (291) is compatible with the sought 300. Seeing these results showed me I was missing leading doubles:

bCADcEBdea
01|12|22|20|00|03|33|32|24|44|41|11|13|34|40
bCAecDEBda
01|12|22|20|00|04|42|23|33|34|44|41|11|13|30
bCEBDcAdea
01|12|22|24|44|41|11|13|33|32|20|00|03|34|40
bCdEADBeca
01|12|22|23|34|44|40|00|03|33|31|11|14|42|20
bCeBdcAEDa
01|12|22|24|41|11|13|32|20|00|04|44|43|33|30
bCedcAEBDa
01|12|22|24|43|32|20|00|04|44|41|11|13|33|30
bDCBEcAdea
01|13|33|32|22|21|11|14|44|42|20|00|03|34|40
bDCBeAcEda
01|13|33|32|22|21|11|14|40|00|02|24|44|43|30
bDCedAEBca
01|13|33|32|22|24|43|30|00|04|44|41|11|12|20
bDEBceACda
01|13|33|34|44|41|11|12|24|40|00|02|22|23|30
bDacBEdCeA
01|13|33|30|02|21|11|14|44|43|32|22|24|40|00
bDacEBCdeA
01|13|33|30|02|24|44|41|11|12|22|23|34|40|00
bEAdceDBCa
01|14|44|40|00|03|32|24|43|33|31|11|12|22|20
bECDeAcBda
01|14|44|42|22|23|33|34|40|00|02|21|11|13|30
bEaCedcBDA
01|14|44|40|02|22|24|43|32|21|11|13|33|30|00
bcDBEaCedA
01|12|23|33|31|11|14|44|40|02|22|24|43|30|00
bcDeBdACEa
01|12|23|33|34|41|11|13|30|00|02|22|24|44|40
bcEdBeaDCA
01|12|24|44|43|31|11|14|40|03|33|32|22|20|00
bcaDeCdBEA
01|12|20|03|33|34|42|22|23|31|11|14|44|40|00
bcaeCDBEdA
01|12|20|04|42|22|23|33|31|11|14|44|43|30|00
bceBdAEDCa
01|12|24|41|11|13|30|00|04|44|43|33|32|22|20
bdAEBCDeca
01|13|30|00|04|44|41|11|12|22|23|33|34|42|20
bdEAcBeCDa
01|13|34|44|40|00|02|21|11|14|42|22|23|33|30
bdaeBcEDCA
01|13|30|04|41|11|12|24|44|43|33|32|22|20|00
bdeADcEBCa
01|13|34|40|00|03|33|32|24|44|41|11|12|22|20
beAcEDCBda
01|14|40|00|02|24|44|43|33|32|22|21|11|13|30
beaCdBcEDA
01|14|40|02|22|23|31|11|12|24|44|43|33|30|00
beadCEDBcA
01|14|40|03|32|22|24|44|43|33|31|11|12|20|00
cBADECdbea
02|21|11|10|00|03|33|34|44|42|22|23|31|14|40
cBADbedCEa
02|21|11|10|00|03|33|31|14|43|32|22|24|44|40
cBEAdCeDba
02|21|11|14|44|40|00|03|32|22|24|43|33|31|10
cBaDbedCEA
02|21|11|10|03|33|31|14|43|32|22|24|44|40|00
cBaEbdCeDA
02|21|11|10|04|44|41|13|32|22|24|43|33|30|00
cBedbaECDA
02|21|11|14|43|31|10|04|44|42|22|23|33|30|00
cDaBCEbdeA
02|23|33|30|01|11|12|22|24|44|41|13|34|40|00
cDbAeBCEda
02|23|33|31|10|00|04|41|11|12|22|24|44|43|30
cDbCEBadeA
02|23|33|31|12|22|24|44|41|11|10|03|34|40|00
cDbECBadeA
02|23|33|31|14|44|42|22|21|11|10|03|34|40|00
cDeCBEadbA
02|23|33|34|42|22|21|11|14|44|40|03|31|10|00
cDebaECBdA
02|23|33|34|41|10|04|44|42|22|21|11|13|30|00
cEABDebCda
02|24|44|40|00|01|11|13|33|34|41|12|22|23|30
cEADCBedba
02|24|44|40|00|03|33|32|22|21|11|14|43|31|10
cEbCdeADBa
02|24|44|41|12|22|23|34|40|00|03|33|31|11|10
cEbDCBaedA
02|24|44|41|13|33|32|22|21|11|10|04|43|30|00
cEbaeDCBdA
02|24|44|41|10|04|43|33|32|22|21|11|13|30|00
cEdBeADCba
02|24|44|43|31|11|14|40|00|03|33|32|22|21|10
cEdCBDAbea
02|24|44|43|32|22|21|11|13|33|30|00|01|14|40
cEdCbDABea
02|24|44|43|32|22|21|13|33|30|00|01|11|14|40
cbADECdBea
02|21|10|00|03|33|34|44|42|22|23|31|11|14|40
cbDaeCdEBA
02|21|13|33|30|04|42|22|23|34|44|41|11|10|00
cbDeCdaEBA
02|21|13|33|34|42|22|23|30|04|44|41|11|10|00
cbEDCeABda
02|21|14|44|43|33|32|22|24|40|00|01|11|13|30
cbEDCeadBA
02|21|14|44|43|33|32|22|24|40|03|31|11|10|00
cbdEBADCea
02|21|13|34|44|41|11|10|00|03|33|32|22|24|40
cbdeBaDCEA
02|21|13|34|41|11|10|03|33|32|22|24|44|40|00
cbeCdaEDBA
02|21|14|42|22|23|30|04|44|43|33|31|11|10|00
cbeDaECdBA
02|21|14|43|33|30|04|44|42|22|23|31|11|10|00
cdAECbDeBa
02|23|30|00|04|44|42|22|21|13|33|34|41|11|10
cdAEDbeCBa
02|23|30|00|04|44|43|33|31|14|42|22|21|11|10
cdECBDabeA
02|23|34|44|42|22|21|11|13|33|30|01|14|40|00
cdECbeABDa
02|23|34|44|42|22|21|14|40|00|01|11|13|33|30
cdECbeaBDA
02|23|34|44|42|22|21|14|40|01|11|13|33|30|00
cdEbDAeCBa
02|23|34|44|41|13|33|30|00|04|42|22|21|11|10
cdbECBADea
02|23|31|14|44|42|22|21|11|10|00|03|33|34|40
cdbeCBaDEA
02|23|31|14|42|22|21|11|10|03|33|34|44|40|00
cdeBAECbDa
02|23|34|41|11|10|00|04|44|42|22|21|13|33|30
cdeCbEaBDA
02|23|34|42|22|21|14|44|40|01|11|13|33|30|00
ceABCDEbda
02|24|40|00|01|11|12|22|23|33|34|44|41|13|30
ceAbdCBEDa
02|24|40|00|01|13|32|22|21|11|14|44|43|33|30
cebadBCDEA
02|24|41|10|03|31|11|12|22|23|33|34|44|40|00
dBCDEbAeca
03|31|11|12|22|23|33|34|44|41|10|00|04|42|20
dBECDeAcba
03|31|11|14|44|42|22|23|33|34|40|00|02|21|10
dBcAeDCEba
03|31|11|12|20|00|04|43|33|32|22|24|44|41|10
dBcEbaeDCA
03|31|11|12|24|44|41|10|04|43|33|32|22|20|00
dCBAeDbEca
03|32|22|21|11|10|00|04|43|33|31|14|44|42|20
dCEABeDbca
03|32|22|24|44|40|00|01|11|14|43|33|31|12|20
dCEAbeDBca
03|32|22|24|44|40|00|01|14|43|33|31|11|12|20
dCEBcabDeA
03|32|22|24|44|41|11|12|20|01|13|33|34|40|00
dCbeDBAEca
03|32|22|21|14|43|33|31|11|10|00|04|44|42|20
dCeAbDEBca
03|32|22|24|40|00|01|13|33|34|44|41|11|12|20
dCeacbEDBA
03|32|22|24|40|02|21|14|44|43|33|31|11|10|00
dEAbecDBCa
03|34|44|40|00|01|14|42|23|33|31|11|12|22|20
dEBDCAbcea
03|34|44|41|11|13|33|32|22|20|00|01|12|24|40
dEBaeCbDcA
03|34|44|41|11|10|04|42|22|21|13|33|32|20|00
dECAebDcBa
03|34|44|42|22|20|00|04|41|13|33|32|21|11|10
dEbaeCBDcA
03|34|44|41|10|04|42|22|21|11|13|33|32|20|00
dbAcEDCBea
03|31|10|00|02|24|44|43|33|32|22|21|11|14|40
dbacBEDCeA
03|31|10|02|21|11|14|44|43|33|32|22|24|40|00
dbcAECDeBa
03|31|12|20|00|04|44|42|22|23|33|34|41|11|10
dbcDEACeBa
03|31|12|23|33|34|44|40|00|02|22|24|41|11|10
dbcDEBACea
03|31|12|23|33|34|44|41|11|10|00|02|22|24|40
dbcDEBAeCa
03|31|12|23|33|34|44|41|11|10|00|04|42|22|20
dbeDCABcEa
03|31|14|43|33|32|22|20|00|01|11|12|24|44|40
dbeaBcEDCA
03|31|14|40|01|11|12|24|44|43|33|32|22|20|00
dcABDeCbEa
03|32|20|00|01|11|13|33|34|42|22|21|14|44|40
dcAEDBCeba
03|32|20|00|04|44|43|33|31|11|12|22|24|41|10
dcAEbDeCBa
03|32|20|00|04|44|41|13|33|34|42|22|21|11|10
dcBACEDbea
03|32|21|11|10|00|02|22|24|44|43|33|31|14|40
dcBEACeDba
03|32|21|11|14|44|40|00|02|22|24|43|33|31|10
dcBEabDeCA
03|32|21|11|14|44|40|01|13|33|34|42|22|20|00
dcBeAbDECa
03|32|21|11|14|40|00|01|13|33|34|44|42|22|20
dcEBCabDeA
03|32|24|44|41|11|12|22|20|01|13|33|34|40|00
deCDBcaEbA
03|34|42|22|23|33|31|11|12|20|04|44|41|10|00
deCbAEBDca
03|34|42|22|21|10|00|04|44|41|11|13|33|32|20
deacbECDBA
03|34|40|02|21|14|44|42|22|23|33|31|11|10|00
debCDBaEcA
03|34|41|12|22|23|33|31|11|10|04|44|42|20|00
debCaEcDBA
03|34|41|12|22|20|04|44|42|23|33|31|11|10|00
decaEBCDbA
03|34|42|20|04|44|41|11|12|22|23|33|31|10|00
decbDCAEBa
03|34|42|21|13|33|32|22|20|00|04|44|41|11|10
eBACdEcbDa
04|41|11|10|00|02|22|23|34|44|42|21|13|33|30
eBCEDbacdA
04|41|11|12|22|24|44|43|33|31|10|02|23|30|00
eBCEDcAdba
04|41|11|12|22|24|44|43|33|32|20|00|03|31|10
eBCabdcEDA
04|41|11|12|22|20|01|13|32|24|44|43|33|30|00
eBDabcdECA
04|41|11|13|33|30|01|12|23|34|44|42|22|20|00
eBcADCEdba
04|41|11|12|20|00|03|33|32|22|24|44|43|31|10
eBdCEDabcA
04|41|11|13|32|22|24|44|43|33|30|01|12|20|00
eCABDcbEda
04|42|22|20|00|01|11|13|33|32|21|14|44|43|30
eCAbcDEBda
04|42|22|20|00|01|12|23|33|34|44|41|11|13|30
eCDbcAdEBa
04|42|22|23|33|31|12|20|00|03|34|44|41|11|10
eCbEdcADBa
04|42|22|21|14|44|43|32|20|00|03|33|31|11|10
eCdBAcbEDa
04|42|22|23|31|11|10|00|02|21|14|44|43|33|30
eCdEBDAbca
04|42|22|23|34|44|41|11|13|33|30|00|01|12|20
eCdbcaBEDA
04|42|22|23|31|12|20|01|11|14|44|43|33|30|00
eDAcbECdBa
04|43|33|30|00|02|21|14|44|42|22|23|31|11|10
eDAcdbCEBa
04|43|33|30|00|02|23|31|12|22|24|44|41|11|10
eDBEcdACba
04|43|33|31|11|14|44|42|23|30|00|02|22|21|10
eDBEcdAbCa
04|43|33|31|11|14|44|42|23|30|00|01|12|22|20
eDaBEcdbCA
04|43|33|30|01|11|14|44|42|23|31|12|22|20|00
eDaCbdcEBA
04|43|33|30|02|22|21|13|32|24|44|41|11|10|00
eDcEBdabCA
04|43|33|32|24|44|41|11|13|30|01|12|22|20|00
eDcabECBdA
04|43|33|32|20|01|14|44|42|22|21|11|13|30|00
ebCaBdEcDA
04|41|12|22|20|01|11|13|34|44|42|23|33|30|00
ebDECadcBA
04|41|13|33|34|44|42|22|20|03|32|21|11|10|00
ecADBCdEba
04|42|20|00|03|33|31|11|12|22|23|34|44|41|10
ecadbEDCBA
04|42|20|03|31|14|44|43|33|32|22|21|11|10|00
ecbAdBEDCa
04|42|21|10|00|03|31|11|14|44|43|33|32|22|20
ecdEbADBCa
04|42|23|34|44|41|10|00|03|33|31|11|12|22|20
edABCEbDca
04|43|30|00|01|11|12|22|24|44|41|13|33|32|20
edACEbcDBa
04|43|30|00|02|22|24|44|41|12|23|33|31|11|10
edBCEbAcDa
04|43|31|11|12|22|24|44|41|10|00|02|23|33|30
edBEcAbCDa
04|43|31|11|14|44|42|20|00|01|12|22|23|33|30
edBaDcbECA
04|43|31|11|10|03|33|32|21|14|44|42|22|20|00
edBcDaCEbA
04|43|31|11|12|23|33|30|02|22|24|44|41|10|00
edCEbDaBcA
04|43|32|22|24|44|41|13|33|30|01|11|12|20|00
edbECDABca
04|43|31|14|44|42|22|23|33|30|00|01|11|12|20
edcBECaDbA
04|43|32|21|11|14|44|42|22|20|03|33|31|10|00
edcEBDACba
04|43|32|24|44|41|11|13|33|30|00|02|22|21|10
edcEBDaCbA
04|43|32|24|44|41|11|13|33|30|02|22|21|10|00
edcEBaDbCA
04|43|32|24|44|41|11|10|03|33|31|12|22|20|00
edcbDABECa
04|43|32|21|13|33|30|00|01|11|14|44|42|22|20
BCDEcAebda
01|11|12|22|23|33|34|44|42|20|00|04|41|13|30
BCDbEdaceA
01|11|12|22|23|33|31|14|44|43|30|02|24|40|00
BCDeAdbEca
01|11|12|22|23|33|34|40|00|03|31|14|44|42|20
BCEbDcadeA
01|11|12|22|24|44|41|13|33|32|20|03|34|40|00
BCaecdEbDA
01|11|12|22|20|04|42|23|34|44|41|13|33|30|00
BCdEAcebDa
01|11|12|22|23|34|44|40|00|02|24|41|13|33|30
BCdbEDaecA
01|11|12|22|23|31|14|44|43|33|30|04|42|20|00
BDAcdebCEa
01|11|13|33|30|00|02|23|34|41|12|22|24|44|40
BDEbcAdCea
01|11|13|33|34|44|41|12|20|00|03|32|22|24|40
BDcAEbCeda
01|11|13|33|32|20|00|04|44|41|12|22|24|43|30
BDcebCaEdA
01|11|13|33|32|24|41|12|22|20|04|44|43|30|00
BECbDeAcda
01|11|14|44|42|22|21|13|33|34|40|00|02|23|30
BEDaCdbceA
01|11|14|44|43|33|30|02|22|23|31|12|24|40|00
BEDbceAdCa
01|11|14|44|43|33|31|12|24|40|00|03|32|22|20
BEDcbdaeCA
01|11|14|44|43|33|32|21|13|30|04|42|22|20|00
BEacbdeCDA
01|11|14|44|40|02|21|13|34|42|22|23|33|30|00
BEadeCDbcA
01|11|14|44|40|03|34|42|22|23|33|31|12|20|00
BEadecbDCA
01|11|14|44|40|03|34|42|21|13|33|32|22|20|00
BEcAeDbCda
01|11|14|44|42|20|00|04|43|33|31|12|22|23|30
BEcbdaCDeA
01|11|14|44|42|21|13|30|02|22|23|33|34|40|00
BEdbcAeCDa
01|11|14|44|43|31|12|20|00|04|42|22|23|33|30
BEdcADbCea
01|11|14|44|43|32|20|00|03|33|31|12|22|24|40
BcAdCebDEa
01|11|12|20|00|03|32|22|24|41|13|33|34|44|40
BcAedbECDa
01|11|12|20|00|04|43|31|14|44|42|22|23|33|30
BcDbedaCEA
01|11|12|23|33|31|14|43|30|02|22|24|44|40|00
BcDbedaECA
01|11|12|23|33|31|14|43|30|04|44|42|22|20|00
BcEDCAdbea
01|11|12|24|44|43|33|32|22|20|00|03|31|14|40
BcEaCdbeDA
01|11|12|24|44|40|02|22|23|31|14|43|33|30|00
BcEbDeaCdA
01|11|12|24|44|41|13|33|34|40|02|22|23|30|00
BcaEbdCeDA
01|11|12|20|04|44|41|13|32|22|24|43|33|30|00
BcdEACebDa
01|11|12|23|34|44|40|00|02|22|24|41|13|33|30
BcebdaCDEA
01|11|12|24|41|13|30|02|22|23|33|34|44|40|00
BdCaebcEDA
01|11|13|32|22|20|04|41|12|24|44|43|33|30|00
BdCbEDaecA
01|11|13|32|22|21|14|44|43|33|30|04|42|20|00
BdEADCbeca
01|11|13|34|44|40|00|03|33|32|22|21|14|42|20
BdEbcADCea
01|11|13|34|44|41|12|20|00|03|33|32|22|24|40
BdceACbEDa
01|11|13|32|24|40|00|02|22|21|14|44|43|33|30
BeCDaEdbcA
01|11|14|42|22|23|33|30|04|44|43|31|12|20|00
BeDbcEaCdA
01|11|14|43|33|31|12|24|44|40|02|22|23|30|00
BedbCEacDA
01|11|14|43|31|12|22|24|44|40|02|23|33|30|00
CBDcebadEA
02|22|21|11|13|33|32|24|41|10|03|34|44|40|00
CBaDEbdceA
02|22|21|11|10|03|33|34|44|41|13|32|24|40|00
CBaecdbEDA
02|22|21|11|10|04|42|23|31|14|44|43|33|30|00
CDAEbcedBa
02|22|23|33|30|00|04|44|41|12|24|43|31|11|10
CDAEdbceBa
02|22|23|33|30|00|04|44|43|31|12|24|41|11|10
CDBEabcedA
02|22|23|33|31|11|14|44|40|01|12|24|43|30|00
CDaeBdEcbA
02|22|23|33|30|04|41|11|13|34|44|42|21|10|00
CDebdABcEa
02|22|23|33|34|41|13|30|00|01|11|12|24|44|40
CDecBAdbEa
02|22|23|33|34|42|21|11|10|00|03|31|14|44|40
CEADBedcba
02|22|24|44|40|00|03|33|31|11|14|43|32|21|10
CEAbDeBcda
02|22|24|44|40|00|01|13|33|34|41|11|12|23|30
CEBcDbAdea
02|22|24|44|41|11|12|23|33|31|10|00|03|34|40
CEbdcBaeDA
02|22|24|44|41|13|32|21|11|10|04|43|33|30|00
CbAEBDceda
02|22|21|10|00|04|44|41|11|13|33|32|24|43|30
CbAdEcDBea
02|22|21|10|00|03|34|44|42|23|33|31|11|14|40
CbDEadceBA
02|22|21|13|33|34|44|40|03|32|24|41|11|10|00
CbDEcdAeBa
02|22|21|13|33|34|44|42|23|30|00|04|41|11|10
CbEcdBaeDA
02|22|21|14|44|42|23|31|11|10|04|43|33|30|00
CdAbDEcBea
02|22|23|30|00|01|13|33|34|44|42|21|11|14|40
CdAeDbEcBa
02|22|23|30|00|04|43|33|31|14|44|42|21|11|10
CdBcEbAeDa
02|22|23|31|11|12|24|44|41|10|00|04|43|33|30
CdEcBeADba
02|22|23|34|44|42|21|11|14|40|00|03|33|31|10
CdbaDEBceA
02|22|23|31|10|03|33|34|44|41|11|12|24|40|00
CdbaeBcEDA
02|22|23|31|10|04|41|11|12|24|44|43|33|30|00
CdeabcEBDA
02|22|23|34|40|01|12|24|44|41|11|13|33|30|00
CdecbADBEa
02|22|23|34|42|21|10|00|03|33|31|11|14|44|40
CeBcDbadEA
02|22|24|41|11|12|23|33|31|10|03|34|44|40|00
CeDcBdaEbA
02|22|24|43|33|32|21|11|13|30|04|44|41|10|00
CeDcbdaEBA
02|22|24|43|33|32|21|13|30|04|44|41|11|10|00
CebAEdcBDa
02|22|24|41|10|00|04|44|43|32|21|11|13|33|30
CedAEbDcBa
02|22|24|43|30|00|04|44|41|13|33|32|21|11|10
DBCdecAbEa
03|33|31|11|12|22|23|34|42|20|00|01|14|44|40
DBCebaEdcA
03|33|31|11|12|22|24|41|10|04|44|43|32|20|00
DBcAEdCeba
03|33|31|11|12|20|00|04|44|43|32|22|24|41|10
DBcdebACEa
03|33|31|11|12|23|34|41|10|00|02|22|24|44|40
DBcedCabEA
03|33|31|11|12|24|43|32|22|20|01|14|44|40|00
DBedCAbcEa
03|33|31|11|14|43|32|22|20|00|01|12|24|44|40
DCBEdbaecA
03|33|32|22|21|11|14|44|43|31|10|04|42|20|00
DCBaEbdecA
03|33|32|22|21|11|10|04|44|41|13|34|42|20|00
DCBdecAEba
03|33|32|22|21|11|13|34|42|20|00|04|44|41|10
DCBedbacEA
03|33|32|22|21|11|14|43|31|10|02|24|44|40|00
DCEAbedBca
03|33|32|22|24|44|40|00|01|14|43|31|11|12|20
DCEaBdebcA
03|33|32|22|24|44|40|01|11|13|34|41|12|20|00
DCbEAcedBa
03|33|32|22|21|14|44|40|00|02|24|43|31|11|10
DCbEdBaceA
03|33|32|22|21|14|44|43|31|11|10|02|24|40|00
DCeBcaEdbA
03|33|32|22|24|41|11|12|20|04|44|43|31|10|00
DEABcdbeCa
03|33|34|44|40|00|01|11|12|23|31|14|42|22|20
DECdBcAbea
03|33|34|44|42|22|23|31|11|12|20|00|01|14|40
DECdbeABca
03|33|34|44|42|22|23|31|14|40|00|01|11|12|20
DEbceaCdBA
03|33|34|44|41|12|24|40|02|22|23|31|11|10|00
DEbdCaBceA
03|33|34|44|41|13|32|22|20|01|11|12|24|40|00
DEcdBACbea
03|33|34|44|42|23|31|11|10|00|02|22|21|14|40
DbACBEdcea
03|33|31|10|00|02|22|21|11|14|44|43|32|24|40
DbAcBeCdEa
03|33|31|10|00|02|21|11|14|42|22|23|34|44|40
DbCeacdEBA
03|33|31|12|22|24|40|02|23|34|44|41|11|10|00
DbEdcAeCBa
03|33|31|14|44|43|32|20|00|04|42|22|21|11|10
DbcAeCdEBa
03|33|31|12|20|00|04|42|22|23|34|44|41|11|10
DbcdeBaCEA
03|33|31|12|23|34|41|11|10|02|22|24|44|40|00
DbeAcEdCBa
03|33|31|14|40|00|02|24|44|43|32|22|21|11|10
DcAedbECBa
03|33|32|20|00|04|43|31|14|44|42|22|21|11|10
DcBaedbECA
03|33|32|21|11|10|04|43|31|14|44|42|22|20|00
DcaBCEdbeA
03|33|32|20|01|11|12|22|24|44|43|31|14|40|00
DcaEdbCeBA
03|33|32|20|04|44|43|31|12|22|24|41|11|10|00
DeacdBECbA
03|33|34|40|02|23|31|11|14|44|42|22|21|10|00
DebdcAECBa
03|33|34|41|13|32|20|00|04|44|42|22|21|11|10
DecABCdbEa
03|33|34|42|20|00|01|11|12|22|23|31|14|44|40
DecdbEACBa
03|33|34|42|23|31|14|44|40|00|02|22|21|11|10
EBACDbceda
04|44|41|11|10|00|02|22|23|33|31|12|24|43|30
EBAcbdeCDa
04|44|41|11|10|00|02|21|13|34|42|22|23|33|30
EBCabDecdA
04|44|41|11|12|22|20|01|13|33|34|42|23|30|00
EBCedcabDA
04|44|41|11|12|22|24|43|32|20|01|13|33|30|00
EBDcadeCbA
04|44|41|11|13|33|32|20|03|34|42|22|21|10|00
EBdabCDecA
04|44|41|11|13|30|01|12|22|23|33|34|42|20|00
ECAbcDeBda
04|44|42|22|20|00|01|12|23|33|34|41|11|13|30
ECBeDbAdca
04|44|42|22|21|11|14|43|33|31|10|00|03|32|20
ECDAcBdeba
04|44|42|22|23|33|30|00|02|21|11|13|34|41|10
ECDeBdabcA
04|44|42|22|23|33|34|41|11|13|30|01|12|20|00
ECDebcadBA
04|44|42|22|23|33|34|41|12|20|03|31|11|10|00
ECbDeBAcda
04|44|42|22|21|13|33|34|41|11|10|00|02|23|30
EDAbeCBdca
04|44|43|33|30|00|01|14|42|22|21|11|13|32|20
EDAcdbeCBa
04|44|43|33|30|00|02|23|31|14|42|22|21|11|10
EDBCebadcA
04|44|43|33|31|11|12|22|24|41|10|03|32|20|00
EDCBecAdba
04|44|43|33|32|22|21|11|14|42|20|00|03|31|10
EDCeBdAcba
04|44|43|33|32|22|24|41|11|13|30|00|02|21|10
EDcaBeCbdA
04|44|43|33|32|20|01|11|14|42|22|21|13|30|00
EbcdeCABDa
04|44|41|12|23|34|42|22|20|00|01|11|13|33|30
EbcdeCaDBA
04|44|41|12|23|34|42|22|20|03|33|31|11|10|00
EbdABCDeca
04|44|41|13|30|00|01|11|12|22|23|33|34|42|20
EcDBCabedA
04|44|42|23|33|31|11|12|22|20|01|14|43|30|00
EcDBadebCA
04|44|42|23|33|31|11|10|03|34|41|12|22|20|00
EcDBedaCbA
04|44|42|23|33|31|11|14|43|30|02|22|21|10|00
EcbedaBDCA
04|44|42|21|14|43|30|01|11|13|33|32|22|20|00
EcdBACbeDa
04|44|42|23|31|11|10|00|02|22|21|14|43|33|30
EcdeBCADba
04|44|42|23|34|41|11|12|22|20|00|03|33|31|10
EcdeBCAbDa
04|44|42|23|34|41|11|12|22|20|00|01|13|33|30
EdAbCeBDca
04|44|43|30|00|01|12|22|24|41|11|13|33|32|20
EdBeCaDcbA
04|44|43|31|11|14|42|22|20|03|33|32|21|10|00
EdCBaDbecA
04|44|43|32|22|21|11|10|03|33|31|14|42|20|00
EdCaDbceBA
04|44|43|32|22|20|03|33|31|12|24|41|11|10|00
EdaBDCebcA
04|44|43|30|01|11|13|33|32|22|24|41|12|20|00
EdaCDbceBA
04|44|43|30|02|22|23|33|31|12|24|41|11|10|00


  Posted by Charlie on 2006-05-28 14:19:59
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