Create a schedule for golf - 12 players, 3 x 4 each day over 5 days. Minimum number of repeat players

[Ottley Kitts]

I'm trying to create a 5 day golfing schedule for 12 players - this means there are 60 cells to fill. I want everyone to play with each other once but then have the minimum of repeats. I've tried the MRAND with INDEX which is great for each day's play but then find there are a lot of repeat groups on other days. Any advice VERY welcome. I'm using A through to L to represent the players names.

Using this formula: =INDEX($A$2:$A$13,MRAND()+1,,TRUE)

To try to explain this graphically I have used the MRAND to randomly generate my players each day so I use all 12 players and then copied it across to the other remaining days. This works to an extent but if you scroll down you will then see, just taking Player A as an example, he plays with the two same players three times (H & I), yet he doesn't player with one player (K) at all.

PLAYERS Wed Thu Fri Sat Sun A L K G H A B E E I E E C J H A L I D K C H A D E F A B C C F G J L G L G D I D F B H I F E I G I H D K J H J C L F B J K A B J K F L B G C D K Player: A A A A A H J I H E C I G E I B F H L D Played with: H 3 times H 3 times I 3 times E 2 times Not Played: K <tbody> </tbody>

<tbody>
</tbody>

Any suggestions would be greatly appreciated. Many thanks Ottley

<tbody>
</tbody>

 

[JackBean]

Both of the previous solutions and the following have STDEV = 0.64775
All 1, 2, 3:

Excel Workbook
	T	U	V
1	ADEJ	BFHL	CGIK
2	ADKL	BCEH	FGIJ
3	ACGL	BHJK	DEFI
4	ABIL	CFJK	DEGH
5	AFHI	BCDG	EJKL
...

Pairing Grid:

Excel Workbook
	G	H	I	J	K	L	M	N	O	P	Q	R	S
1		A	B	C	D	E	F	G	H	I	J	K	L
2	A	X	X	X	X	X	X	X	X	X	X	X	X
3	B	1	X	X	X	X	X	X	X	X	X	X	X
4	C	1	2	X	X	X	X	X	X	X	X	X	X
5	D	2	1	1	X	X	X	X	X	X	X	X	X
6	E	1	1	1	3	X	X	X	X	X	X	X	X
7	F	1	1	1	1	1	X	X	X	X	X	X	X
8	G	1	1	3	2	1	1	X	X	X	X	X	X
9	H	1	3	1	1	2	2	1	X	X	X	X	X
10	I	2	1	1	1	1	3	2	1	X	X	X	X
11	J	1	1	1	1	2	2	1	1	1	X	X	X
12	K	1	1	2	1	1	1	1	1	1	3	X	X
13	L	3	2	1	1	1	1	1	1	1	1	2	X
...

Here are some other possibilities:

Excel Workbook
	A	B	C
1	STDEV	0	2
2	0.64775	1	42
3		2	18
4		3	4
5		4	0
6		5	0
7
8	STDEV	0	2
9	0.64775	1	42
10		2	18
11		3	4
12		4	0
13		5	0
14
15	STDEV	0	0
16	0.815925	1	50
17		2	12
18		3	2
19		4	0
20		5	2
...

It was shown in the previous Post that all 0's can be eliminated from the Pairing Grid with 5 Rows.
The question comes up how many Rows are needed to eliminate all 0's.
The Pairing Matrix uses to Pairings Per Row and Total # Possible Pairings:

Excel Workbook
	A	B	C	D	E	F
1	1	18	0.272727	66	6	18
2	2	36	0.545455
3	3	54	0.818182
4	4	72	1.090909
5	5	90	1.363636
6	6	108	1.636364
7	7	126	1.909091
8	8	144	2.181818
9	9	162	2.454545
10	10	180	2.727273
11	11	198	3
...

This shows that with 11 Rows each can pair 3 times with each.
And there are enough pairs with 4 Rows to cover the squares,
if they can be so distributed.

If the first Row is selected at Random,
how many Pairs must repeat in the next Row?

There are 66 possible pairings.
There are 18 pairings per Row.

The solutions previously posted show that if only 2 Rows of the solution are checked,
3 pairings repeat.
Consider that there are 4 Characters per Group,
and to redistribute 3 of these to other Groups is not possible,
when there are only two other Groups.

So only 15 new pairings can be used per Row.
And 60 pairings can be used in 4 Rows.
So it takes 5 Rows to use more than 60 pairings.

There is also the question with 5 Rows,
whether a solution with all 1's and 2's exists (no 0's).

But these results should be enough to get them out on the course.

 

[Ottley Kitts]

Many thanks. It's great that everyone gets to play each other. Really strange that it should result in 2 players having to play each other on all 5 days but I've put this forward to the group as a solution. I did try to manually tweak it but by reducing the 2 x 5 result I just created some 0's so didn't gain anything. I should have known better than to try. I didn't appreciate it was going to be such a problem when I volunteered to look for a solution. Thanks again.

Ottley

 

[JackBean]

To my way of thinking,
it seems like the all 1, 2, 3 Pairings solution would be more ideal than the one with a 5,
of the ones where each meets each at least once.
But this is a matter of preference.

5 Rows
It is not possible to have all 1's and 2's.
Here is the explanation for that.
In Post 11 the Pairing Matrix shows that there are 6 Pairings per Group.
6*3 Groups per Row = 18 Pairings per Row.
18*5 Rows = 90 Pairings.
There are 66 possible Pairings.
If each pairs once then that is 66.
90-66=24 remaining Pairings.
Since there are 4 per Group,
if the 4 in a Group in the first Row are divided between the 3 Groups of the next Row,
then 1 Pairing must repeat in each of the next 4 Rows, time 3 for 3 Groups.
The number of remaining Rows decreases by one as each is added.
So the number of required extra pairings is:
30=3*(4+3+2+1)
30 is greater than 24.
The number of extra forced pairings is greater than allowed for all 1's and 2's.

4 Rows
If all 1's and 2's is not possible for 5 Rows then it should not be possible for 4 also.

There are 6 Pairings per Group.
6*3 Groups per Row = 18 Pairings per Row.
18*4 Rows = 72 Pairings.
There are 66 possible Pairings.
72-66=6 remaining Pairings.
Since there are 4 per Group,
if the 4 in a Group in the first Row are divided between the 3 Groups of the next Row,
then 1 Pairing must repeat in each of the next 3 Rows, times 3 for 3 Groups.
The number of remaining Rows decreases by one as each is added.
So the number of required extra pairings is:
18=3*(3+2+1)
18 is greater than 6.
The number of extra forced pairings is greater than allowed for all 1's and 2's.

Here is an example of 4 Rows with all 0, 1, 2.

Excel Workbook
	A	B	C
1	ACIK	BDGH	EFJL
2	ABEG	CFHL	DIJK
3	AHKL	BEFI	CDGJ
4	AEHJ	BDIL	CFGK
...

Pairing Grid:

Excel Workbook
	A	B	C	D	E	F	G	H	I	J	K	L	M
1		A	B	C	D	E	F	G	H	I	J	K	L
2	A	X	X	X	X	X	X	X	X	X	X	X	X
3	B	1	X	X	X	X	X	X	X	X	X	X	X
4	C	1	0	X	X	X	X	X	X	X	X	X	X
5	D	0	2	1	X	X	X	X	X	X	X	X	X
6	E	2	2	0	0	X	X	X	X	X	X	X	X
7	F	0	1	2	0	2	X	X	X	X	X	X	X
8	G	1	2	2	2	1	1	X	X	X	X	X	X
9	H	2	1	1	1	1	1	1	X	X	X	X	X
10	I	1	2	1	2	1	1	0	0	X	X	X	X
11	J	1	0	1	2	2	1	1	1	1	X	X	X
12	K	2	0	2	1	0	1	1	1	2	1	X	X
13	L	1	1	1	1	1	2	0	2	1	1	1	X
...

Count of 0, 1, 2:

Excel Workbook
	A	B
1	0	12
2	1	36
3	2	18
...

What about the question (as in Post#11) of whether the squares of the Pairing Grid can be covered with 4 Rows?

There are 66 possible pairings.
54=72-18 Pairings
So the squares cannot be covered with 4 Rows.
Only 54 of the 66 squares can be covered,
if squares have 1's, 2's, but no 3's.
That means 12 blank squares,
and that how many there are in the example above.

But apparently that does not mean that less than 12 blank squares are possible,
if all do not have to be 1's and 2's.
Example:

Excel Workbook
	A	B	C
1	ACDI	BEHK	FGJL
2	AEGJ	BCFK	DHIL
3	ADKL	BEFI	CGHJ
4	ABFH	CDEL	GIJK
...

Pairing Grid:

Excel Workbook
	A	B	C	D	E	F	G	H	I	J	K	L	M
1		A	B	C	D	E	F	G	H	I	J	K	L
2	A	X	X	X	X	X	X	X	X	X	X	X	X
3	B	1	X	X	X	X	X	X	X	X	X	X	X
4	C	1	1	X	X	X	X	X	X	X	X	X	X
5	D	2	0	2	X	X	X	X	X	X	X	X	X
6	E	1	2	1	1	X	X	X	X	X	X	X	X
7	F	1	3	1	0	1	X	X	X	X	X	X	X
8	G	1	0	1	0	1	1	X	X	X	X	X	X
9	H	1	2	1	1	1	1	1	X	X	X	X	X
10	I	1	1	1	2	1	1	1	1	X	X	X	X
11	J	1	0	1	0	1	1	4	1	1	X	X	X
12	K	1	2	1	1	1	1	1	1	1	1	X	X
13	L	1	0	1	3	1	1	1	1	1	1	1	X
...

Count of 0, 1, 2, 3, 4:

Excel Workbook
	A	B
1	0	7
2	1	50
3	2	6
4	3	2
5	4	1
...

This leaves open the question of whether less than 7 blanks are possible,
and if so, how many?</SPAN></SPAN>

 

[cpocpocpo]

I really like the Pairing grid in the discussion that uses 5 rows and 3 foursomes. However, the two tables shown as part of the discussion by JackBean do not match. The Pairing Grid table is what I like. But the smaller 3x5 table of foursome groupings does not match the Pairing grid. Can the correct assignment table be recreated?

This is the one that doesn't match the Pairings table.

ADJL BCGK EFHI
AFGI BEKL CDHJ
AGHK BDFL CEIJ
ACEL BDGI FHJK
ABIJ CDFK EGHL

Thanks in advance for any help.

 

[JackBean]

That is a mistake in Post #8. The groups do not match the pairing grid.

The explanation in Post #13 shows that for all 0, 1, 2's, then the minimum possible number of 0's is 6. I have found up to 13. This raises another question. What is the maximum number of 0's possible with all 0, 1, 2's?

Here is another example of 6 0's.
<b></b><table class="html-maker-worksheet" border="1" cellspacing="0" cellpadding="0"><thead><tr><th></th><th>A</th><th>B</th><th>C</th></tr></thead><tbody><tr ><th>1</th><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">AIKL</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #00FFFF;;">BFGJ</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">CDEH</td></tr><tr ><th>2</th><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #CCFFFF;;">ADEG</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">BHIL</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #CCFFFF;;">CFJK</td></tr><tr ><th>3</th><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">ABFH</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #00FFFF;;">CDJL</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">EGIK</td></tr><tr ><th>4</th><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #CCFFFF;;">AEHJ</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">BDKL</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #CCFFFF;;">CFGI</td></tr><tr ><th>5</th><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">ADFI</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #00FFFF;;">BCEK</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">GHJL</td></tr></tbody></table><b></b><br /><br />
Pairing grid:
<b></b><table class="html-maker-worksheet" border="1" cellspacing="0" cellpadding="0"><thead><tr><th></th><th>A</th><th>B</th><th>C</th><th>D</th><th>E</th><th>F</th><th>G</th><th>H</th><th>I</th><th>J</th><th>K</th><th>L</th><th>M</th></tr></thead><tbody><tr ><th>1</th><td></td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">A</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">B</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">C</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">D</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">E</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">F</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">G</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">H</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">I</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">J</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">K</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">L</td></tr><tr ><th>2</th><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">A</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td></tr><tr ><th>3</th><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">B</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">1</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td></tr><tr ><th>4</th><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">C</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFFF99;;">0</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #00FFFF;;">1</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td></tr><tr ><th>5</th><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">D</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFCC99;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">2</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td></tr><tr ><th>6</th><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">E</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFFF99;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #00FFFF;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFFF99;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #00FFFF;;">2</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td></tr><tr ><th>7</th><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">F</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFCC99;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFCC99;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">0</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td></tr><tr ><th>8</th><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">G</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFFF99;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #00FFFF;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFFF99;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #00FFFF;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFFF99;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #00FFFF;;">2</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td></tr><tr ><th>9</th><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">H</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFCC99;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFCC99;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFCC99;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">1</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td></tr><tr ><th>10</th><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">I</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFFF99;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #00FFFF;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFFF99;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #00FFFF;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFFF99;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #00FFFF;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFFF99;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #00FFFF;;">1</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td></tr><tr ><th>11</th><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">J</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFCC99;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFCC99;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFCC99;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFCC99;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">0</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td></tr><tr ><th>12</th><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">K</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFFF99;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #00FFFF;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFFF99;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #00FFFF;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFFF99;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #00FFFF;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFFF99;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #00FFFF;;">0</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFFF99;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #00FFFF;;">1</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFFF99;;">X</td></tr><tr ><th>13</th><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;;">L</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFCC99;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFCC99;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">0</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFCC99;;">0</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">1</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFCC99;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #FFCC99;;">2</td><td style="font-weight: bold;text-align: right;border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;color: #0000FF;background-color: #CCFFFF;;">2</td><td style="border-top: 1px solid black;border-right: 1px solid black;border-bottom: 1px solid black;border-left: 1px solid black;background-color: #FFCC99;;">X</td></tr></tbody></table><b></b><br /><br />