Rack Leave in Scrabble

The computer program Quackle is one of the strongest scrabble players in the world. It uses the following 'Superleave' valuation:

http://web.mit.edu/~jasonkb/Public/scrabble/superleaves
(warning: huge-assed file)
http://vizsage.com/other/data/superleaves.xls
(The above, sorted by value, only up-to-4 leaves)

To find "synergies" and "anti-synergies" (dysphoria?), I calculated the marginal valuation for each combo. Basically, how much of the value for each two-leave is explained by the consituent one-leaves, etc? For example:

From S (7.35) and M (0.08), the joint valuation of MS is 7.44, a marginal gain of 0.1: the joint valuation is almost entirely from the individual letters S&M. (<-- will lead to interesting google hits). This combination has no synergy.
From Q (-9.0) and U (-5.1), the joint valuation of QU is 0.2, a marginal gain of 14.3. This is by far the largest synergy; next is ZO at +3.2.

I also played with three-letter synergies, defined as "3-leave valuation marginally different from the most explanatory 2-leave+1leave value."

General Lessons:

Get a feel for the 1-leave list, and the learn these:
- Synergy:
  QU OZ JU CH GN WY IN DE JK ER EV GIN JKY JKU ERS KWY HWY ?IN EST JOW ?AL ?EL ?IL IST
- Anti-Synergy:
  BP CG FP MV PV CW CQ QS SX LQ BV SZ QR BC CZ VZ MQ RX GQ
  [most things with blank]
  BPV CGQ BCG LQR FPV LNQ SVZ CMQ CLQ BCV BNV KTV LMQ GKT CFV GMQ FSV LNR DGT
Worth keeping with a blank: The letters in "Lei an orc DT mug" + the following digrams
ES RS ST HS AS CS NS LS IS ER MS OS PS DS DE EL CH EN SS IN CE AR AL AC ET AN CI CR AM SZ BS KS AT IL CN RT EM GN EZ IR CO SU GS SY OR CL EH IT AD DR NR NT NO DI EP CT DN IM IZ LO AZ HR AE MR LY PR HT CK MO AH LR DL LT LN DO FS OT MN OZ CD EI BE AP AB HN BL LP BR LM AI RY HO HI SX SW EG IP HL AG CM EK EO GI OP DH DT EX CP RZ MT HP DM GR NZ HM NY GL FR NP PT FL
Double letters are bad (duh), except FF, which is good.

See the spreadsheet at http://vizsage.com/other/data/superleaves.xls). Don't go betting the house on these results....

One-tile valuation
?	25.6
S	 8.0
Z	 5.1
X	 3.3
R	 1.1
H	 1.1
C	 0.9
M	 0.6
D	 0.5
E	 0.4
N	 0.2
T	-0.1
L	-0.2
P	-0.5
K	-0.5
Y	-0.6
A	-0.6
J	-1.5
B	-2.0
I	-2.1
F	-2.2
O	-2.5
G	-2.9
W	-3.8
U	-5.1
V	-5.6
Q	-6.8


Two leaves with 2 or more points of synergy (marginal gain in valuation):
		1+1 vs 2
Leave	Val	marg.val
QU	 0.20	14.34
OZ	 4.06	 3.24
JU	-5.49	 3.22
CH	 4.53	 3.18
GN	 1.16	 3.18
WY	-3.22	 2.95
IN	 1.93	 2.75
DE	 3.73	 2.64
JK	-2.51	 2.62
ER	 4.79	 2.59
EV	-2.80	 2.54
JY	-2.68	 2.39
HW	-2.07	 2.37
KY	-0.72	 2.34
IQ	-8.20	 2.29
AM	 1.92	 2.24
IV	-5.26	 2.23
IZ	 3.64	 2.21
KW	-4.05	 2.18
JW	-6.09	 2.15
JO	-3.55	 2.12
AQ	-7.32	 2.08
CK	 1.63	 2.07
GI	-2.17	 2.01


Non-double letter 2-leaves with more than two points of anti-synergy:
		 1+1 vs 2
Leave	Val	marg.val
?J	 17.30	-5.74
?Q	 12.74	-4.87
?X	 21.88	-4.59
BP	- 7.75	-4.42
CG	- 5.79	-4.22
?Z	 25.85	-3.68
FP	- 7.59	-3.47
MV	- 9.34	-3.42
PV	-10.26	-3.27
CW	- 6.80	-3.25
CQ	-10.97	-3.09
QS	-4.68	-3.03
SX	  4.23	-2.98
LQ	-11.54	-2.96
BV	-11.18	-2.84
SZ	  7.47	-2.80
QR	-10.13	-2.67
BC	- 3.86	-2.64
CZ	  1.41	-2.63
VZ	- 5.62	-2.54
MQ	-11.44	-2.52
RX	-1.12	-2.52
GQ	-14.20	-2.51
FV	-11.61	-2.48
NQ	-10.75	-2.42
CJ	- 4.86	-2.41
GK	- 6.64	-2.39
KV	- 9.90	-2.34
HV	- 8.07	-2.30
?W	 19.66	-2.28
JR	- 4.31	-2.28
GP	- 5.95	-2.27
GX	- 5.06	-2.23
KT	- 3.57	-2.20
?V	 18.45	-2.16
CF	- 4.15	-2.14
IU	- 8.77	-2.14
JS	  1.65	-2.13
SY	  3.78	-2.07


All blank-tile 2leaves:
		 1+1 vs 2
Leave	Val	marg.val
?L	 28.53	 1.50
?E	 28.73	 1.46
?I	 26.49	 1.37
?A	 27.56	 1.35
?N	 28.61	 1.33
?O	 25.59	 1.08
?R	 29.14	 0.99
?C	 28.67	 0.94
?T	 27.65	 0.85
?D	 27.47	 0.43
?M	 26.84	 0.15
?G	 24.04	 0.12
?U	 21.54	 0.07
?B	 24.07	-0.20
?H	 26.56	-0.28
?P	 25.30	-0.32
?S	 33.60	-0.36
?F	 22.03	-1.45
?Y	 23.65	-1.46
?K	 23.06	-1.99
?V	 18.45	-2.16
?W	 19.66	-2.28
?Z	 25.85	-3.68
?X	 21.88	-4.59
?Q	 12.74	-4.87
?J	 17.30	-5.74
??	 38.38	-14.84


All double-letter 2leaves:
		 1+1 vs 2
Leave	Val	marg.val
??	 38.38	-14.84
SS	  6.63	-8.07
RR	- 4.78	-7.86
NN	- 5.85	-7.19
CC	- 4.50	-6.74
LL	- 5.66	-6.50
TT	- 5.67	-6.05
II	- 8.84	-5.86
DD	- 4.97	-5.83
HH	- 5.16	-5.62
YY	- 8.15	-5.15
AA	- 5.90	-5.10
MM	- 4.89	-5.05
UU	-15.17	-4.89
EE	- 3.56	-4.88
GG	- 9.31	-3.93
OO	- 7.83	-3.63
PP	- 5.33	-3.35
VV	-14.96	-2.96
WW	-12.23	-2.89
BB	- 7.45	-2.77
FF	- 4.73	 1.53    !!!!!  FF is synergistic. 

Notable 3-Leave Synergies			
		 1+1+1	 2+1 vs 3 
Leave	Val	m.val	marg.val
GIN	  7.1	 10.65	 7.5
JKY	  1.3	 7.88	 5.3
JKU	-2.8	 7.43	 4.2
ERS	 16.8	 7.25	 4.7
KWY	- 0.8	 6.90	 4.0
HWY	  0.8	 6.77	 3.8
?IN	 32.3	 6.48	 3.7
EST	 14.4	 6.19	 4.2
JOW	- 4.4	 5.91	 3.8
?AL	 32.0	 5.35	 3.5
?EL	 32.7	 5.03	 3.5
?IL	 30.6	 5.01	 3.5
IST	 11.1	 5.00	 3.5


Notable 3-Leave Anti-Synergies, no double letter or blank			
		 1+1+1	 2+1 vs 3 
Leave	Val	m.val	marg.val
BPV	-17.9	-8.57	-4.2
CGQ	-18.8	-8.24	-4.0
BCG	-11.6	-7.70	-3.5
LQR	-14.5	-7.46	-4.5
FPV	-17.5	-7.37	-3.9
LNQ	-15.0	-7.05	-4.1
SVZ	- 2.6	-6.86	-4.1
CMQ	-14.5	-6.70	-3.6
CLQ	-14.1	-6.63	-3.5
BCV	-13.8	-6.57	-3.7
BNV	-14.2	-6.52	-3.7
KTV	-13.8	-6.43	-4.1
LMQ	-14.9	-6.42	-3.5
GKT	-10.4	-6.34	-4.0
CFV	-14.1	-6.04	-3.6
GMQ	-17.6	-5.98	-3.5
FSV	- 7.7	-5.96	-3.5
LNR	- 2.9	-5.48	-3.7
DGT	- 7.5	-5.43	-3.6


Notable 3-Leave Anti-Synergies, blank			
		1+1+1	2+1 vs 3 
Leave	Val	m.val	marg.val
?JZ	 16.2	-9.74	-4.0
?VZ	 13.8	-9.73	-6.1
?KX	 16.0	-8.94	-4.4
?KV	 10.6	-8.46	-6.1
?XZ	 21.0	-8.36	-3.8
?GX	 15.5	-8.24	-3.7
?FV	  9.7	-7.76	-5.3
?WZ	 17.5	-7.32	-3.6
?KZ	 20.7	-7.30	-3.6
?FP	 15.4	-7.13	-3.7
?PV	 12.5	-7.08	-3.8
?MV	 13.8	-6.93	-3.5
?BV	 11.5	-6.77	-3.9
?HV	 14.1	-6.72	-4.4



Notable 3-Leave Anti-Synergies, double letter			
		1+1+1	2+1 vs 3 
Leave	Val	m.val	marg.val
SSS	  2.7	-19.39	-11.3
RRR	-14.0	-18.57	-10.7
NNN	-14.9	-16.91	-9.7
LLL	-14.0	-15.24	-8.7
TTT	-13.9	-14.46	-8.4
III	-17.7	-13.27	-7.4
DDD	-11.7	-12.95	-7.1
CCG	-13.0	-12.53	-5.8
LRR	- 8.7	-12.23	-4.4
SSZ	  5.4	-12.23	-4.2
EEE	-10.1	-12.04	-7.2
SSX	  2.7	-11.85	-3.8
AAA	-13.0	-11.83	-6.7
QSS	- 5.9	-11.63	-3.6
?YY	 12.4	-11.19	-6.0
LNN	- 9.1	-10.90	-3.7
CCQ	-17.2	-10.39	-3.7
UUU	-25.5	-10.06	-5.2
LLN	- 8.5	-10.03	-3.5
CGG	-14.0	-9.69	-5.5
?HH	 17.9	-9.22	-3.6
GGG	-16.8	-8.75	-4.8
MMV	-14.6	-8.72	-3.7
OOO	-15.0	-8.66	-5.0
BPP	-12.7	-8.40	-4.0
IUU	-20.2	-8.39	-3.5
?WW	  9.2	-8.06	-5.2
PPV	-15.8	-7.79	-4.4
GGT	-12.7	-7.53	-3.6
FPP	-12.3	-7.20	-3.7
?VV	  7.5	-7.11	-4.2
PVV	-19.8	-6.83	-3.6
BBV	-17.3	-6.59	-3.8
ESS	 11.0	-4.36	-3.7
ERR	  0.4	-3.35	-4.5
CCO	- 3.1	-3.23	-3.5
INN	- 3.1	-2.92	-4.3
CCH	- 0.2	-2.70	-4.0
IIX	- 5.4	-2.24	-3.5
IIN	- 4.5	-2.22	-3.6
IIZ	- 1.6	-1.57	-3.8
IIV	-10.5	-1.48	-3.7
DDE	  0.1	-1.40	-4.0
AAM	- 2.0	-1.32	-3.6
WYY	- 8.8	-1.08	-4.0
EER	  1.8	-1.03	-3.6


Notable 3-Leave Synergies with blank			
		1+1+1	2+1 vs 3 
Leave	Val	m.val	marg.val
?ES	37.4	2.76	0.8
?RS	36.0	0.52	0.5
?ST	35.4	1.22	0.3
?HS	35.2	1.04	0.6
?AS	35.2	1.67	0.3
?CS	35.1	0.00	0.4
?NS	35.0	0.39	0.0
?LS	35.0	0.59	0.4
?IS	34.9	2.47	1.1
?ER	34.5	5.65	3.1
?MS	34.0	-0.02	0.0
?OS	33.6	1.70	0.6
?PS	33.3	0.30	0.4
?DS	33.2	-1.23	0.3
?DE	33.1	5.35	2.7
?EL	32.7	5.03	3.5
?CH	32.7	4.72	1.5
?EN	32.4	4.50	3.0
?SS	32.4	-8.91	0.8
?IN	32.3	6.48	3.7
?CE	32.1	3.68	2.2
?AR	32.0	4.29	2.9
?AL	32.0	5.35	3.5
?AC	32.0	4.63	3.0
?ET	31.9	4.42	2.8
?AN	31.3	4.45	3.1
?CI	31.0	4.78	3.1
?CR	31.0	1.68	0.7
?AM	30.9	4.56	2.3
?SZ	30.8	-6.11	2.4
?BS	30.7	-0.89	0.0
?KS	30.6	-1.76	0.2
?AT	30.6	4.17	2.7
?IL	30.6	5.01	3.5
?CN	30.4	1.99	0.7
?RT	30.3	1.92	0.9
?EM	30.3	2.91	1.5
?GN	30.2	5.58	2.4
?EZ	30.2	-0.03	1.4
?IR	30.1	3.46	2.1
?CO	30.1	4.42	2.7
?SU	30.0	1.13	0.8
?GS	29.9	-1.38	0.2
?SY	29.8	-2.65	0.6
?OR	29.7	3.66	2.6
?CL	29.7	1.55	0.0
?EH	29.6	2.13	0.7
?IT	29.6	4.27	2.9
?AD	29.5	2.88	1.5
?DR	29.5	0.89	0.1
?NR	29.4	0.55	0.4
?NT	29.3	1.86	0.5
?NO	29.3	4.11	2.8
?DI	29.2	3.66	2.3
?EP	29.1	2.85	1.4
?CT	29.1	1.18	0.2
?DN	29.0	1.33	0.0
?IM	29.0	3.78	2.1
?IZ	29.0	0.93	0.4
?LO	29.0	4.03	2.5
?AZ	28.9	-0.24	1.6
?HR	28.9	0.49	0.5
?AE	28.9	2.00	0.5
?MR	28.7	0.49	0.3
?LY	28.7	3.15	1.5
?PR	28.6	1.46	0.5
?HT	28.5	1.50	0.7
?CK	28.4	2.26	0.2
?MO	28.4	3.82	2.1
?AH	28.4	1.93	0.6
?LR	28.3	-0.23	1.2
?DL	28.3	0.80	0.4
?LT	28.2	1.00	0.1
?LN	28.2	0.49	0.8
?DO	28.2	3.21	2.1
?FS	28.1	-2.69	1.0
?OT	28.1	3.43	2.4
?MN	28.1	0.76	0.6
?OZ	28.0	0.58	0.5
?CD	28.0	-0.19	0.6
?EI	27.9	2.11	0.6
?BE	27.7	2.72	1.3
?AP	27.6	2.42	1.1
?AB	27.3	3.40	1.8
?HN	27.2	-0.28	0.0
?BL	27.1	2.42	0.9
?LP	27.0	1.00	0.5
?BR	27.0	1.19	0.2
?LM	27.0	-0.16	0.3
?AI	26.8	2.08	0.7
?RY	26.8	0.14	0.5
?HO	26.8	2.05	1.0
?HI	26.7	1.36	0.0
?SX	26.7	-7.16	2.6
?SW	26.7	-2.63	0.3
?EG	26.7	2.08	0.6
?IP	26.6	2.49	1.1
?HL	26.6	-0.65	0.4
?AG	26.5	3.02	1.7
?CM	26.5	-1.32	0.5
?EK	26.5	0.77	0.1
?EO	26.5	1.29	0.2
?GI	26.5	4.02	2.0
?OP	26.4	2.88	1.4
?DH	26.4	-0.92	0.1
?DT	26.3	-0.96	0.6
?EX	26.3	-0.88	2.3
?CP	26.2	-0.54	0.2
?RZ	26.1	-4.94	1.3
?MT	26.1	-0.75	0.6
?HP	26.1	0.25	0.4
?DM	26.1	-1.02	0.2
?GR	26.0	0.58	0.4
?NZ	26.0	-4.20	0.5
?HM	25.9	-1.06	0.4
?NY	25.8	0.00	0.2
?GL	25.8	1.43	0.1
?FR	25.8	0.73	0.3
?NP	25.7	-0.55	0.2
?PT	25.7	-0.11	0.2
?FL	25.7	1.78	0.3
?LZ	25.5	-4.42	0.7
?CY	25.5	-0.72	0.4
?BO	25.5	3.29	1.6

Rules of Thumb for Rack Leave in Scrabble

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