Bowling versus the strong counties in 1920

Over my long obsession with old county cricket, I have gradually become aware that the old English spin bowlers who dominated the bowling averages and top wicket-takers before the “Revolution of 1959” — not to mention allowed county cricket to pay its way when fast bowling was weak enough —were not, objectively, nearly so good as I presumed before reading ‘Woodcock’s Hundred’ in the 1998 Wisden. The most striking thing to me upon reading that list was that only three of the thirteen bowlers who had taken over 2,500 first-class wickets were included in Woodcock‘s list. Moreover all those three — Wilfrid Rhodes, George Hirst, and W.G. Grace — were largely included for their batting, not their bowling!

As I have noted previously, since the middle 1890s there has not been a single English spin bowler who was a matchwinner at Test level in Australia. No doubt, this fact tied the hands of Wisden writers in my youth because it produced an irreconcilable conflict between their two desires for more spin bowling and for England winning more. Wisden in the Preston, Woodcock and Wright eras was thus unclear about what it really wanted and unwilling to look at the issue of how good old English spin bowlers really were altogether. Between 1910 and 1950, it was taken for granted that nature dictated English spin could never succeed in Australia, although why this was so was never discussed. Even after the 1951/1952 tour of India, which demonstrated English spinners of the era were not as good as Wisden wished to believe:

“...even second-class Indian slow bowlers often looked more dangerous than the Englishmen [English spinners bowling in India]” 

The same year, Harold Dale in his Cricket Crusaders said that:

“English spin bowlers have failed in Australia by reason of their upbringing. On the average English wicket during an average English summer, the merest finger-action imparts sufficient spin to the ball to give positive results at the other end. Thus Englishmen come to Australia unversed and unpractised in the very emphatic effort required to turn a ball on wickets where climate [as much or more, actually, soils] and groundsmen combine more to protect the batsmen”

Studying county cricket of the 1920s reveals immediately that the disparities between the counties were in batting rather than bowling: the 1922 Wisden noted how bowlers constantly gained false reputations from cheaply dismissing weak counties — Northamptonshire, Derbyshire, Worcestershire, Glamorgan, Leicestershire, Gloucestershire, Warwickshire, and to a smaller extent Sussex and Somerset.

For this reason I have compiled county bowling averages against only the “strong” counties — the “Big Six” of Yorkshire, Surrey, Lancashire, Nottinghamshire, Middlesex and Kent, plus the southern counties of Hampshire and Essex who were very rich in amateur batting when at full strength.

In the table below spin bowlers are shaded in gold, and bowlers who went on the (disastrous) 1920/1921 Ashes tour are in bold. Only bowlers who bowled minimally 1,000 balls (166.4 overs) against the “strong” counties have been included.

Bowling in 1920 Against Strong Counties (Qualification 1,000 Balls):

		Overs	Maidens	Runs	Wickets	Average	5 w/i	10 w/m
Mr. E.R. Wilson	Yorkshire	291.4	139	431	31	13.90	3	0
W. Rhodes	Yorkshire	577.3	146	1,243	75	16.57	7	3
J.W. Hearne	Middlesex	594.2	114	1,547	93	16.63	8	2
H. Dean	Lancashire	484.1	126	1,166	70	16.66	6	2
Mr. V.W.C. Jupp	Sussex	360	78	1,000	59	16.95	5	1
Mr. J.J. Bridges	Somerset	243.5	77	628	37	16.97	1	0
L. Cook	Lancashire	550.2	132	1,277	75	17.03	4	3
C.W.L. Parker	Gloucestershire	322.1	103	773	44	17.57	3	0
A.E. Relf	Sussex	261	116	389	22	17.68	1	0
F.J. Durston	Middlesex	592.4	160	1,526	83	18.39	6	0
F.E. Woolley	Kent	591.5	157	1,418	76	18.66	5	1
T. Rushby	Surrey	392.1	113	879	47	18.70	1	0
Mr. P.G.H. Fender	Surrey	416.4	72	1,281	68	18.84	3	1
W.E. Astill	Leicestershire	404.3	94	921	47	19.60	4	0
Mr. J.C. White	Somerset	347.5	96	697	35	19.91	1	0
H.E. Roberts	Sussex	189.4	25	623	30	20.77	2	1
A.S. Kennedy	Hampshire	663.3	158	1,724	82	21.02	7	3
J.D. Tyldesley	Lancashire	172.3	31	486	23	21.13	2	0
J.H. King	Leicestershire	319.5	72	870	41	21.22	2	1
C.H. Parkin	Lancashire	228.2	55	581	27	21.52	3	0
A.P. Freeman	Kent	407.1	103	1,126	52	21.65	2	0
R.K. Tyldesley	Lancashire	269.5	51	744	34	21.88	2	0
Mr. J.W.H.T. Douglas	Essex	349.3	59	1,265	56	22.59	4	0
T.L. Richmond	Nottinghamshire	573.1	89	1,917	84	22.82	8	2
G.R. Cox	Sussex	410.1	117	964	42	22.95	2	0
W.E. Benskin	Leicestershire	257.1	56	735	32	22.97	1	1
Mr. G.M. Reay	Surrey	230.3	57	558	24	23.25	0	0
J.W. Hitch	Surrey	417.4	82	1,190	50	23.80	2	1
F. Barratt	Nottinghamshire	368	92	977	41	23.83	3	0
W.J. Fairservice	Kent	437.4	104	1,097	46	23.85	1	1
H. Howell	Warwickshire	406.4	71	1,156	48	24.08	4	2
A. Morton	Derbyshire	363.1	89	883	36	24.53	3	0
Mr. A.E.R. Gilligan	Sussex	247.5	49	739	29	25.48	1	0
W. Wells	Northamptonshire	219	38	736	28	26.29	2	0
A. Waddington	Yorkshire	518.3	129	1,322	50	26.44	1	0
Mr. R.C. Robertson-Glasgow	Somerset	170	33	488	18	27.11	1	0
E. Robinson	Yorkshire	323.5	88	788	29	27.17	1	0
S.J. Staples	Nottinghamshire	202.3	39	547	20	27.35	1	0
A.E. Thomas	Northamptonshire	239.5	55	638	23	27.74	1	1
M.W. Tate	Sussex	312.1	89	762	27	28.22	0	0
Mr. G.T.S. Stevens	Middlesex	210.3	20	785	27	29.07	1	0
Mr. N.E. Haig	Middlesex	212.1	62	504	17	29.65	1	0
S.W.A. Cadman	Derbyshire	249.4	78	564	19	29.68	0	0
Hon. F.S.G. Calthorpe	Warwickshire	395.1	73	1,219	40	30.48	2	0
H.W. Lee	Middlesex	260.2	55	665	21	31.67	0	0
Mr. G.M. Louden	Essex	329.4	42	1,131	34	33.26	3	0
Dr. C.H. Gunasekara	Middlesex	244.2	68	543	16	33.94	0	0
J.A. Newman	Hampshire	548.1	89	1,783	49	36.39	2	0
V. Murdin	Northamptonshire	317.1	64	1,131	31	36.48	2	0
C.R. Preece	Worcestershire	247	58	764	20	38.20	0	0
F.P. Ryan	Hampshire	166.5	20	612	16	38.25	1	0
R. Kilner	Yorkshire	206	64	424	11	38.55	0	0
F. Pearson	Worcestershire	185.3	52	544	14	38.86	1	0
C.N. Woolley	Northamptonshire	383.2	74	1,017	25	40.68	2	0
Mr. G.G.F. Greig	Worcestershire	191.5	35	653	16	40.81	1	0
Mr. J.G. Dixon	Essex	167	15	669	14	47.79	1	0

Glancing the table I can tentatively make these conclusions regarding the 1920/1921 tour:

1. the Australian pitches rather than batting account more than I thought for the failures of English spinners
1. Rhodes and J.W. Hearne, two of the top four bowlers in this table, went to Australia and failed completely as bowlers
2. apart from Rhodes in the abnormal 1903/1904 summer, every bowler averaging below 18.75 against “strong” counties was absolutely hopeless in Australia
• Jupp, Dean, Durston, Cook and Bridges never went to Australia, and only Jupp was ever asked (and likely for his batting)
2. England was even more severely affected than I assumed by weakness in pace bowling (which of course accounts for the profitability of county cricket after the two Wars)
1. Harry Howell and Abe Waddington, thought likely to do well in Australia based upon their overall county records and styles, actually owed more to bad batting than the leading spinners
3. The amateur bowlers who would be suggested as the best available during the 1921 English summer — with the exception of George Louden — did not bowl enough against the strong counties to be on the list
1. Of those who defeated the Australians in 1921 for A.C. MacLaren’s England XI, Michael Falcon did not bowl at all against “strong” counties, and C.H. Gibson and Aubrey Faulkner barely did so

Cover structures by base

In recent weeks I have been rereading the 2000s file ‘Generalizing Sierpiński Numbers to Base b’, written by a team from the University of Tennessee at Martin.

Although Sierpiński numbers to base 2 are well-known in studies of prime numbers because Proth primes — those of the form k*2ⁿ+1 — occur frequently as possible factors of binomial numbers like Mersenne and Fermat numbers, similar numbers for other bases were not studied until the 1990s. Moreover, although a major project exists to verify the smallest Sierpiński number for all bases up to 1030, bases as small as 71 have not been started yet.

What was really interesting to me re-reading ‘Generalizing Sierpiński Numbers to Base b’ (my off-line .pdf copy is under a slightly different title but has the same text) this summer was the discussion of various covering set periods. The UT Martin team noted that different bases have vastly different minimal periods for a covering set of primes to repeat. For bases that are 2 or one fewer than a power of 2, this period is relatively long, since for any b of the form 2ⁿ-1, b²-1 has no “primitive divisor” — that is, no prime divisor that does not divide a smaller number of the form bⁿ-1. This is because:

1. b²-1 = (b+1)(b-1)
2. b+1 is a power of 2
3. both b+1 and b-1 divide by 2
4. thus, b²-1  lacks a prime divisor for these bases
5. Bang’s Theorem states that the only other such case is 2⁶-1, which I will not discuss further

The absence of a factor with period 2 means that for bases of the form 2ⁿ-1 covering sets must be built from prime numbers with longer periods. For other numbers, there will always be a cover with period 12 or shorter, but for base 3, there can be no cover repeating more frequently than every 48 terms, as was established by Yannick Saouter in 1995.

In the following table, I will indicate the presence or absence for bases from 2 to 175 of covers with the following periods:

• 2
• 3
• 4
• 6
• 8
• 9
• 10
• 12
• 15

Covers with periods 5, 7, 11 and 13, as was noted by the UT Martin team, do not occur for any base so small as 175. 14-covers have not been investigated, although such a cover would be expected to involve eight primes, one with period 2 and seven with periods 7 or 14.

8-, 9-, 10- and 15-covers were not discussed in the UT Martin study, although I have long known of the 8-cover {11, 73, 101, 137} in base 10. The smallest base for which an 8-cover provides the smallest Sierpiński number is 168, while I know of no base where a 9-cover, 10-cover or 15-cover provide the smallest Sierpiński or Riesel number. Nevertheless,  my re-read made me feel these were worthy of study.

For all bases:

• red means the base lacks a cover with that period
• light green means a non-primitive cover
• dark green means a primitive cover that cannot be reduced

Presence or absence of N-cover for Bases from 2 to 175
N-cover	2	3	4	6	8	9	10	12	15
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175