An experiment — factoring a pentatrigesimal “century”

Today, following up from work a few years ago on pentatrigesimal numbers — a number system using all numbers and letters except O [because “O” have I have always viewed as a number because of its extreme similarity to zero] — I experimented with a thirteen-decimal-digit pentatrigesimal century with two goals:

1. to see whether an extremely prime-dense group of 100 numbers would influence the prime density of a cluster as large 1,225 numbers
• 1,225 numbers is, of course, the length of a sequence of numbers whose pentatrigesimal representation differs only in the last two digits
2. to see how well I could convert factors calculated in decimal to pentratrigesimal

The 1,225 numbers chosen have decimal representations ranging from 1,277,956,496,425 to 1,277,956,497,649. This is based on the pentatrigesimal representation of the numbers of the decimal century from 1,277,956,497,400 to 1,277,956,497,499, which is the seventh century with seventeen prime numbers, but the first beyond the reach of prime tables like those at Compoasso, Walter Fendt, and Mark Knows Nothing.

Pentatrigesimal Notation and Factors for All Numbers Not Divisible by 2, 3, or 5 Between 1,277,956,496,425 and 1,277,956,497,649:

In the list below:

1. the bold blue numbers are part of the zero-prime decimal century from 1,277,956,497,000 to 1,277,956,497,099
2. the bold red numbers are part of the seventeen-prime century noted above
3. italicised numbers have pentatrigesimal representations ending in 7, E, L, or T. Any number with pentatrigesimal representations ending in those digits are divisible by 7
1. a number is indeed divisible by 7 if and only if its pentatrigesimal representation ends in:
1. 0
2. 7
3. E
4. L, or
5. T

• JV6W5I04 = 9R6 × 21B5CP
• JV6W5I06 = 3Z × AEH × GTBC
• JV6W5I0C = E9 × ED × 5091521
• JV6W5I0G = W × MEWH3YW
• JV6W5I0I = B × 34 × 5I × 7X × G9M
• JV6W5I0M = 22N × 9JW0V4
• JV6W5I0P = DC × 1H3KLS2
• JV6W5I0T = 7 × D × 12 × 1X × 3S4GH
• JV6W5I0Z = N × 1P × 7R × HI × 4LZ
• JV6W5I11 = JI × 10LVMU2
• JV6W5I17 = 7 × 12X × 2LNT2Y
• JV6W5I1B = AN × 1V85CQ2
• JV6W5I1D is prime
• JV6W5I1H is prime
• JV6W5I1J = D × 1IGNXQLN
• JV6W5I1N = 6D × R1 × 46Q6R
• JV6W5I1U = 18 × 3RI × 4AKQG
• JV6W5I1W is prime
• JV6W5I22 is prime
• JV6W5I26 is prime
• JV6W5I28 = H × J × W × 2F1SIR
• JV6W5I2C is prime
• JV6W5I2E = 7 × B × 2D × 3T8U74
• JV6W5I2I = U × 32 × 30G × 2L31
• JV6W5I2P = 2W × KJ × S2 × F5Y
• JV6W5I2R = 23 × 9IAZ42X
• JV6W5I2X = 12 × 16 × 29 × 73PYZ
• JV6W5I31 = B × D × 1I × 16P × 2PC1
• JV6W5I33 = 3M × 5GKMCTP
• JV6W5I37 = 7 × H × 4S × 3YW × ASZ
• JV6W5I39 = 1NN × BUTIY8
• JV6W5I3D is prime
• JV6W5I3J is prime
• JV6W5I3L = 7 × N × 2J × 38 × 7I × 2FR
• JV6W5I3S = D × 3N76 × EKYZ
• JV6W5I3W = 1R × M3 × I24CH
• JV6W5I3Y = 43Z × 4UP × ZV3
• JV6W5I42 is prime
• JV6W5I44 is prime
• JV6W5I48 is prime
• JV6W5I4E = 7 × J × 1C × 2N4 × 17GB
• JV6W5I4G = 2Y × 23P × 3779N
• JV6W5I4M = 1X × 18GZ × 8CAJ
• JV6W5I4R = Q6 × SLMMSG
• JV6W5I4T = 7 × 2UAZFSKP
• JV6W5I4X = B × N × 2TY × Z0PY
• JV6W5I4Z is prime
• JV6W5I53 = PH × 42H × 6Z3C
• JV6W5I59 = D × D × 16 × 3HWXPR
• JV6W5I5B = 49 × 6YG × NGNU
• JV6W5I5H = J × J × LD9 × 35C3
• JV6W5I5L = 7 × 7 × 2N08 × 5BVD
• JV6W5I5N = 6N × 2ZF0631
• JV6W5I5S = W × MEWH3Z2
• JV6W5I5U = U × 4H × 5C16KY
• JV6W5I5Y = 85Z × 2F2Z62
• JV6W5I64 = H × 15WA2DES
• JV6W5I66 = B × 1T6ZMS3R
• JV6W5I6C is prime
• JV6W5I6G = 54 × 3VXM4E4
• JV6W5I6I is prime
• JV6W5I6M = 5P × 3H9GQRY
• JV6W5I6P is prime
• JV6W5I6T = 7 × B × 60J × 1HIK1
• JV6W5I6Z is prime
• JV6W5I71 = 61 × 2ZJ × 13L8P
• JV6W5I77 = 7 × 1ER × 1ZVRUM
• JV6W5I7B = 29X × 8PGLP8
• JV6W5I7D = 1R × 7FY × 1IHJR
• JV6W5I7H = D × U × Y2Z × 1Y9P
• JV6W5I7J = N × W × 1VC × I9RZ
• JV6W5I7N = J × 1FZY × Q3W4
• JV6W5I7U = FG × 19Z5B14
• JV6W5I7W is prime
• JV6W5I82 = B × H × 3X × Y8FXY
• JV6W5I86 = 56 × S26 × 4YV6
• JV6W5I88 = D × 1AM × 160PHD
• JV6W5I8C is prime
• JV6W5I8E = 7 × 7 × 92 × 1JTX5I
• JV6W5I8I is prime
• JV6W5I8P = B × 2J × 4Y × 50ZKC
• JV6W5I8R = J × CIJ × 2X77W
• JV6W5I8X is prime
• JV6W5I91 = H × 15WA2DEY
• JV6W5I93 = 1I × D438106
• JV6W5I97 = 7 × 18 × 8W × 93EAS
• JV6W5I99 = 12 × 2GW × 7JWRC
• JV6W5I9D = 126 × 1VZ × 9X9S
• JV6W5I9J = 1FP9 × DPZG6
• JV6W5I9L = 7 × 2UAZFSLD
• JV6W5I9S = 1C × ESPH8KB
• JV6W5I9W is prime
• JV6W5I9Y = B × 16 × 6W × 171 × 6ID
• JV6W5IA2 = 38 × 65BEX29
• JV6W5IA4 = 37H8 × 66AFI
• JV6W5IA8 = SM × CE9 × 20ZB
• JV6W5IAE = 7 × YNY × 2Y6S9
• JV6W5IAG = D × 1A8 × 16DDWJ
• JV6W5IAM = AH × EJ × 4JJMU
• JV6W5IAR = 5G × 706 × I6I1
• JV6W5IAT = 7 × U × 3EV5D0R
• JV6W5IAX = J × 6U × 193 × 48W4
• JV6W5IAZ = H × 21 × 81 × BC × 7R6
• JV6W5IB3 = W × 1P × 23 × 6D8G4
• JV6W5IB9 is prime
• JV6W5IBB = I469 × 13CV9
• JV6W5IBH = N × RIN × 14VHD
• JV6W5IBL = 7 × 19S × 27MC3J
• JV6W5IBN = 18 × G5V0XZG
• JV6W5IBS is prime
• JV6W5IBU = B × 1E8 × 19XMZN
• JV6W5IBY = D × H × 29 × 1DS84C
• JV6W5IC4 = 1I × 9M × 1CNT8U
• JV6W5IC6 = 1P4 × BRL9BJ
• JV6W5ICC = BQGJ × 1P9Q8
• JV6W5ICG = B × B × U × 1C × 7C × 28B
• JV6W5ICI is prime
• JV6W5ICM = 9U9 × 20NS9I
• JV6W5ICP = D × 1IGNXQMI
• JV6W5ICT = 7 × N × 4B4IY5I
• JV6W5ICZ = 34 × 6D7ZQIR
• JV6W5ID1 is prime
• JV6W5ID7 = 7 × 8JC × BLERN
• JV6W5IDB = 9G × 23HV1MG
• JV6W5IDD = 1VFH × ALT79
• JV6W5IDH = 12 × 76 × 2LPH9G
• JV6W5IDJ = S8 × 4S4 × 5C5H
• JV6W5IDN = 9X × 3682871749
• JV6W5IDU = 49 × 4B × 2047VX4
• JV6W5IDW = H × H × 2E6RWNJ
• JV6W5IE2 is prime
• JV6W5IE6 = D × J × JR × 4ZMAN
• JV6W5IE8 = 2W × 24M3 × 37ZB
• JV6W5IEC = B × 1R × IH × 1YMZW
• JV6W5IEE = 7 × 6D × LM × Q7UC
• JV6W5IEI = Q8 × 5NH × 4V2D
• JV6W5IEP is prime
• JV6W5IER = 32 × C1 × IWNPD
• JV6W5IEX = D × D × 43Z57CI
• JV6W5IF1 = 21 × 9SPKSD1
• JV6W5IF3 is prime
• JV6W5IF7 = 7 × 2UAZFSM6
• JV6W5IF9 = J × 23 × 2S × A9 × LKZ
• JV6W5IFD is prime
• JV6W5IFJ is prime
• JV6W5IFL = 7 × B × 12 × 3AR × 2KDP
• JV6W5IFS = U × 1BS × HXV7U
• JV6W5IFW = 3X × IU × 9F53S
• JV6W5IFY = 9Z × 1B3 × 1HY6P
• JV6W5IG2 = 3Z × 501QQAY
• JV6W5IG4 = 1X × 1T8 × 5R0YU
• JV6W5IG8 = B × 2Y × E1 × 1IKA6
• JV6W5IGE = 7 × D × W × 5A6 × 1M24
• JV6W5IGG = 29 × 7R × 508 × 7Y3
• JV6W5IGM = 2D × AEN × T4T3
• JV6W5IGR = N × V7WSM3S
• JV6W5IGT = 7 × 7 × 7 × H × 461QYI
• JV6W5IGX = 3R × 1Y9 × 2Q8FD
• JV6W5IGZ = 16 × 1G2 × BLTQX
• JV6W5IH3 = 20H × 9V73UU
• JV6W5IH9 = EW × 6PW × 6YJU
• JV6W5IHB = BPR × 1PDEJR
• JV6W5IHH = B × DX × 4IZ1ER
• JV6W5IHL = 7 × 3M × SCY6U4
• JV6W5IHN = 7P × X9 × 2T51Y
• JV6W5IHS = H × 15WA2DFG
• JV6W5IHU = 1C × 1P × 1R × 1R × 2W3N
• JV6W5IHY = 6J × 1171 × 365BX
• JV6W5II4 = B × 1T6ZMS4U
• JV6W5II6 = W × 1I × 39D × 41LP
• JV6W5IIC = 1093 × 1169219119
• JV6W5IIG = 7901 × 161746171
• JV6W5III = J × 56 × 72M7N2
• JV6W5IIM = D × 7I × CKS × JS9
• JV6W5IIP = B4 × KS × 30DW8
• JV6W5IIT = 7 × YX × 2ZW × Z72
• JV6W5IIZ = 6N × 1IN1 × 1Y43
• JV6W5IJ1 = 18 × AP3 × 1HXZJ
• JV6W5IJ7 = 7 × 34 × 2H4 × CTF6
• JV6W5IJB = 16 × GYG4E3W
• JV6W5IJD = B × B × D × N × NIV82
• JV6W5IJH = 1EX × DXGK06
• JV6W5IJJ = QKUU × S5LR
• JV6W5IJN = IGB × 12MMRI
• JV6W5IJU = 12 × 20S × 9A7VW
• JV6W5IJW is prime
• JV6W5IK2 = D6 × 1R2 × V8Y6
• JV6W5IK6 is prime
• JV6W5IK8 = BG × 1U3 × Y4V6
• JV6W5IKC = 3U1 × 56IZMC
• JV6W5IKE = 7 × A3 × 9EC × 11LX
• JV6W5IKI = 1C × 1UU × 7ZHFR
• JV6W5IKP = H × J × N × N × 17J × 43I
• JV6W5IKR = U × 13WP × LJTB
• JV6W5IKX = RRB × QZIXM
• JV6W5IL1 = K9 × 8MW × 3YSZ
• JV6W5IL3 is prime
• JV6W5IL7 = 7 × 1I × 1P × 4N × 8CDR
• JV6W5IL9 = B × 1T6ZMS54
• JV6W5ILD = 2D × 8D5F8N1
• JV6W5ILJ is prime
• JV6W5ILL = 7 × D × 1MZ × 1QU × 2MW
• JV6W5ILS = J × GZJ × 25DLC
• JV6W5ILW = B × 7M4 × 89UAJ
• JV6W5ILY = 12 × J8 × 52G × 6R3
• JV6W5IM2 = IW × 32G × BZMC
• JV6W5IM4 = 2Y × 793 × XIXB
• JV6W5IM8 = IUW1 × 11VJ8
• JV6W5IME = 7 × 7 × U × 6QI × 2J28
• JV6W5IMG = 4B × 4L4UGKB
• JV6W5IMM = H × AN × 2C9 × 1M53
• JV6W5IMR = 2HZ × 7WKKZ9
• JV6W5IMT = 7 × 2L2 × 1362QM
• JV6W5IMX is prime
• JV6W5IMZ = 38 × 65BEX2D
• JV6W5IN3 = D × 1IGNXQNB
• JV6W5IN9 is prime
• JV6W5INB = N × EY × 20SW7Z
• JV6W5INH = W × MEWH3ZM
• JV6W5INL = 7 × H × 23 × 2T15X3
• JV6W5INN = 1FY × DMM316
• JV6W5INS = B × 1X × 4H × 7CKZ8
• JV6W5INU = D × 3IX × F3N7J
• JV6W5INY = J × 18 × 2J × 11H × B88
• JV6W5IP4 = 10C × JNFUQC
• JV6W5IP6 = CU × 1J6PBMZ
• JV6W5IPC = 2TJ × 6ZN × 10BB
• JV6W5IPG = I7C × 1366XD
• JV6W5IPI = 4CW × 4J59SD
• JV6W5IPM = N × V7WSM44
• JV6W5IPP is prime
• JV6W5IPT = 7 × 1R × 2QZ × KSC6
• JV6W5IPZ = H4 × 15LQJ0R
• JV6W5IQ1 = B × J × 3BEQ53Z
• JV6W5IQ7 = 7 × 7 × E6JX8YD
• JV6W5IQB = D × 1IGNXQNH
• JV6W5IQD = A87Z × 1XXAM
• JV6W5IQH = 29 × BP × RCH4X
• JV6W5IQJ = H × W6 × 1E6 × XNX
• JV6W5IQN = B × 4N × H6 × 9QTW
• JV6W5IQU = 8S × 2Q4 × U5RY
• JV6W5IQW = 1C × RJ × 1EI × DSJ
• JV6W5IR2 = D × 1IGNXQNJ
• JV6W5IR6 = 12 × ISLMUU3
• JV6W5IR8 = 5M × 3IHXCPU
• JV6W5IRC = 16 × 2S × 3PZ × 1MN9
• JV6W5IRE = 7 × 18 × LG × 3RV0P
• JV6W5IRI = H × 92 × 92 × PD × Q2
• JV6W5IRP = 2TM9 × 71NXR
• JV6W5IRR is prime
• JV6W5IRX = B × 54 × CCHX7N
• JV6W5IS1 = W × 81 × 2SRQ1R
• JV6W5IS3 is prime
• JV6W5IS7 = 7 × J × 21 × 21 × 19G09
• JV6W5IS9 = N × 1I × JYLMVG
• JV6W5ISD = U × DC × 1N9 × 12S9
• JV6W5ISJ = B × D × KY × 84CNR
• JV6W5ISL = 7 × 1X × AFG × 4YWJ
• JV6W5ISS = 23 × 1A3J × 7DN6
• JV6W5ISW is prime
• JV6W5ISY = 1P × 8D × 1E97MP
• JV6W5IT2 is prime
• JV6W5IT4 is prime
• JV6W5IT8 is prime
• JV6W5ITE = 7 × PJ × 2PN × 1HCW
• JV6W5ITG = H × 4Y × 771 × 157G
• JV6W5ITM is prime
• JV6W5ITR is prime
• JV6W5ITT = 7 × B × W × BZ × UT2G
• JV6W5ITX is prime
• JV6W5ITZ is prime
• JV6W5IU3 = ED × 27Z × LPZP
• JV6W5IU9 is prime
• JV6W5IUB is prime
• JV6W5IUH is prime
• JV6W5IUL = 7 × 2UAZFSP8
• JV6W5IUN is prime
• JV6W5IUS = D × 1V6 × TQ68Z
• JV6W5IUU = 66H × 6AN × HTZ
• JV6W5IUY is prime
• JV6W5IV4 is prime
• JV6W5IV6 is prime
• JV6W5IVC is prime
• JV6W5IVG = J × 8Y × 11N × 3WQG
• JV6W5IVI = D × 4N × G89 × PRD
• JV6W5IVM is prime
• JV6W5IVP = B × U × 22J × 11T14
• JV6W5IVT = 7 × 7 × 83 × 1REG3U
• JV6W5IVZ = 31H × 6IHGE2
• JV6W5IW1 = 16 × 6J × 2KPLX4
• JV6W5IW7 = 7 × N × 4B4IY5M
• JV6W5IWB = B × 18 × 1P × 1AR × NC8
• JV6W5IWD = H × 9M × 1DD × 32JJ
• JV6W5IWH is prime
• JV6W5IWJ = J × 2W × 3X × 1JH × 22U
• JV6W5IWN = 34 × FX × E0XZ1
• JV6W5IWU = 4S × 7652434117
• JV6W5IWW = 2S × 3R × 1X0NTY
• JV6W5IX2 = G3 × GY × 2J9RN
• JV6W5IX6 = 7X × 2HUEXIY
• JV6W5IX8 = 9G × 7DB × 9YL8
• JV6W5IXC = H × U × W × 1KQASZ
• JV6W5IXE = 7 × C72 × 84VRB
• JV6W5IXI = N × 12 × 436 × 6ZMD
• JV6W5IXP is prime
• JV6W5IXR = D × 4H × BX8WKW
• JV6W5IXX = UI × NJECZU
• JV6W5IY1 = 7P × 2KFVN2J
• JV6W5IY3 = 32 × F0R × F4UU
• JV6W5IY7 = 7 × B × B × VNC × XSI
• JV6W5IY9 is prime
• JV6W5IYD = 16 × 3Z1H × 49DJ
• JV6W5IYJ is prime
• JV6W5IYL = 7 × 7 × 5I × 1IZ × 1NDX
• JV6W5IYS = 18 × G5V0XZZ
• JV6W5IYW = 2Y × 1XC × 3HS66
• JV6W5IYY = 1C × 17H × C6GYC
• JV6W5IZ2 is prime
• JV6W5IZ4 = W × 2J × 7ER × 16KR
• JV6W5IZ8 = D × D × 1RR × 2BLIH
• JV6W5IZE = 7 × 1KDN × 1SRE9
• JV6W5IZG = B × 1T6ZMS6B
• JV6W5IZM = 12 × 2KW × 788P6
• JV6W5IZR is prime
• JV6W5IZT = 7 × J × CJ × EKHHJ
• JV6W5IZX is prime
• JV6W5IZZ = D × 15Y × 1AQ0AW

Analysis:

If we count the primes in the “century” [actually, of course, a sequence of one thousand two hundred and twenty-five integers, or twelve and a quarter times as many as a conventional decimal century], we find that there are a total of sixty-three primes.

According to the prime number theorem, there should be an average gap of approximately 27.876 between primes over these numbers. This would mean that there should be about 43.944 primes over the pentatrigesimal “century”, which is approximately nineteen fewer than the number actually observed.

This does imply that the extremely prime-rich decimal century from 1,277,956,497,400 to 1,277,956,497,499 has effects that extend to a full pentatrigesimal “century” more than twelve times as large.

My worst-ever case of absent-mindedness!

 

The “soup” at right resulted from me adding a cupful of milk aiming to thicken my Irish stew without thinking!

Today, after a terrible, erratic sleep following a long trip out to Milleara Mall — where I found some very good value fruit with which I helped make a quince pie (the first made in the house all year) — I agreed with my mother to make an Irish stew. It was fitting given the shift from a long run of warm and sunny days to rain this morning.

 Last night — a very warm morning which hardly got below the temperature of a comfortable afternoon — was terrible for my sleep. This was not wholly my fault. I was repeatedly waking up to hydrate myself after the strenuous travels of yesterday had produced a bad headache. At the same time, my mother and brother have both been suffering a really bad cough, and the noise from that cough helped prevent me from settling down to sleep. The warmth meant that I was unable to get the correct temperature for sleep: it was too warm with the blanket and too cool without it, while I was not bothered to take it off and sleep with just the sheet as I normally would during very warm summer weather.

During the afternoon when I cooked the quinces and the Irish stew, this did not seem to be a problem. However, with a break in the weather around 4 P.M., I went out for a walk around to the Royal Melbourne hospital, and then was asked to thicken the Irish stew. This is when things went really awry. I rushed to get the stew thickened. However, I did not think about how to do it, thinking all I had to do was to make an unmeasured mix of milk and flour! Having recently only made chicken stews thickened via noodles, I could only think to thicken via mixing milk with flour. I did this immediately without asking for any help, but I put so much milk that when I showed my mother what I had done, I was chastised severely. I immediately showed the coffee mug (not in the photograph above) and when inquired as to why the stew was so runny, said that I had put into the stew one whole cup of milk! My mother was seriously upset, and both my mother and brother said that I should not thicken Irish stews because I do not know how to do it. I became  angry, saying that at the very least I could learn how to thicken properly.

Although I hoped that I could evaporate the milk, I know that would never be allowed! Hence I accepted removing some of the fluid as a soup — not bad to eat as I know from past experience. I was soon told that when thickening stews with flour and milk:

1. I must aim for a thickness comparable to honey or even slightly thicker than honey
2. I must add the milk to the flour and never add the flour to the milk.

The real problem for me is remembering this when I next make an Irish stew or some other type!

I any case, despite me being really upset, the stew and the quince tart did provide a nice dinner. My real hope is that I can learn how to thicken stews again!

Competitive Balance in County Cricket: Part I — By Win Percent in Finished Games

For many years now, I have wanted to measure competitive balance in English county cricket between the first official County Championship in 1890 and the last single-division Championship in 1999.

There are however many problems with measuring competitive balance in county cricket with the techniques used for baseball, basketball, gridiron, ice hockey, rugby, [Australian rules] football and even soccer leagues. These include:

1. county cricket games often — in some seasons more often than not — can end without any result
2. over the history of the County Championship, methods of point scoring have varied immensely
1. since 1968, in fact, a large part of scoring has been “bonus points” not directly related to the result of a match
2. the result is that a consistent measure of balance may not be accurate for all eras of county cricket
3. up until World War II, counties usually played schedules of substantially varying length, which makes calculating the idealised standard deviation by conventional means impossible

For these reasons, I plan to analyse competitive balance in county cricket by multiple criteria to see how they compare and how competitive balance has changed over time in the sport.

This first post will analyse competitive balance based purely upon percentage of wins in finished matches. This was the original method of calculating the Champion County, but was discarded because it was seen as encouraging unattractive cricket. Nonetheless, thinking about this idea over many years I have always thought of analysing the observed standard deviation of win percentage in finished matches as the default method by which within-season competitive balance in county cricket might be estimated.

In order to gain at least some idea of the idealised standard deviation of winning percentage in finished games, I have calculated the arithmetic mean of the number of finished games in each season, and based the idealised standard deviation on the ordinary formula of (sqrt(n))/2n. I have then calculated a competitive balance index as the ordinary formula of (ASD-ISD)/(MSD-ISD), where:

MSD = sqrt((N+1)/(12(N-1)))

with N equalling the number of teams in the competition. I have also included skewness and kurtosis for more details on the actual shape of the distribution. Extreme values have been shaded as I previously did in my 2024 study of the “Revolution of 1959”. I have also provided all-series averages for every season of single-division official County championship cricket.

CBI for Each Single-Division County Championship Season, 1890 to 1999:

Season	CBI	SKEW	KURT
1890	0.462358	-0.960141	-0.054772
1891	0.331451	+0.229151	+0.170827
1892	0.842595	-0.210308	-1.337585
1893	0.239742	+0.168580	-0.539230
1894	0.618669	+0.176070	-0.782851
1895	0.327522	+0.307711	-0.623449
1896	0.450441	+0.452382	-1.143677
1897	0.774982	+0.049695	-1.104113
1898	0.750282	+0.411154	-1.371064
1899	0.468059	+0.436129	-1.087182
1900	0.784729	-0.012455	-0.383119
1901	0.571871	-0.119775	+0.485723
1902	0.317136	+0.446170	+0.238533
1903	0.550588	-0.429954	-0.294312
1904	0.575301	+0.493857	-0.213935
1905	0.732449	-0.140498	-1.168476
1906	0.784290	+0.241416	-1.432568
1907	0.761771	+0.046237	-0.682596
1908	0.635791	+0.542957	+0.016871
1909	0.697139	-0.043934	-1.021497
1910	0.621227	-0.364317	-0.144691
1911	0.774390	-0.538290	-1.187153
1912	0.913412	+0.081078	-1.385633
1913	0.564809	+0.335478	-0.908739
1914	0.796326	-0.222787	-1.080853
1919	0.456512	+0.216030	-1.152839
1920	0.794512	-0.380103	-0.857362
1921	0.709633	+0.240165	-1.242194
1922	0.839410	-0.026496	-1.020781
1923	0.835631	0.336090	-1.018051
1924	0.839410	-0.120697	-1.175160
1925	0.858227	+0.271994	-1.036225
1926	0.791844	+0.469658	-1.004929
1927	0.667121	-0.257271	-0.864866
1928	0.878656	+0.073174	-0.473664
1929	0.666758	-0.310386	-0.995948
1930	0.690781	+0.662334	-0.771007
1931	0.454629	+0.085538	-0.322792
1932	0.793709	+0.245013	-1.078973
1933	0.733655	0.011935	-1.166727
1934	0.573070	-0.166672	-1.118039
1935	0.572791	-0.067031	-0.122990
1936	0.508035	-0.395538	-0.508390
1937	0.657703	-0.532268	-0.112059
1938	0.474863	-0.208981	+1.354882
1939	0.523154	-0.687685	+0.030232
1946	0.519580	+0.543093	-0.433125
1947	0.445312	+0.434828	+0.002714
1948	0.373594	+0.137454	-1.469418
1949	0.367723	+0.279477	-0.318211
1950	0.465111	+0.634463	-0.357330
1951	0.505072	-0.065867	-0.496338
1952	0.508765	+0.484745	-0.211689
1953	0.362378	-0.676085	-0.018059
1954	0.522281	-0.256696	-1.183995
1955	0.409663	+0.266860	-0.459407
1956	0.373128	-0.090026	-0.838410
1957	0.495334	+0.373986	-0.129682
1958	0.161731	-0.570909	-0.314868
1959	0.141932	-0.737029	-0.058184
1960	0.388917	-0.502500	-0.743616
1961	0.299882	-0.241965	-0.244088
1962	0.497791	-0.362190	-0.683229
1963	0.323291	-0.403720	-0.159527
1964	0.569966	-0.003793	-1.185038
1965	0.242032	-0.021824	-1.123755
1966	0.196613	+0.002542	-0.898257
1967	0.337281	-0.660471	+0.625506
1968	0.215358	-0.549002	-0.357759
1969	0.484943	+0.583272	+0.128426
1970	0.072694	+0.721005	+0.298446
1971	0.193056	+0.223475	-1.189572
1972	0.521367	+0.177093	-0.235752
1973	0.520214	+0.108313	-0.004182
1974	0.623300	+0.225952	-0.517476
1975	0.541885	+0.173766	-0.741075
1976	0.139265	-0.147423	-1.202972
1977	0.217728	-0.506065	+0.113360
1978	0.446960	+0.975866	-0.540541
1979	0.427807	-0.728434	+0.028456
1980	0.107447	-0.616210	+0.674108
1981	0.361980	+0.183950	-1.009637
1982	0.462704	-0.551719	+0.520057
1983	0.630372	-0.121651	-1.771580
1984	0.497179	-0.408316	-0.375583
1985	0.307034	-0.082135	-0.762481
1986	0.004652	+0.188331	-0.508590
1987	0.521703	-0.022970	-1.246956
1988	0.163785	-0.567549	+0.368864
1989	0.239509	+0.967162	-0.077891
1990	0.334461	+0.777225	-0.488989
1991	0.052972	-0.127006	-1.252023
1992	-0.097526	-0.367632	+1.399614
1993	0.238570	+0.230888	+0.209866
1994	0.179998	+0.525857	+0.763104
1995	0.462702	+0.681663	-0.761303
1996	0.636679	-0.283181	-0.378578
1997	0.337709	-0.472894	-0.460810
1998	0.556307	+0.653322	+0.184129
1999	0.180620	+0.947094	+3.056189
Average	0.48758245	+0.02190829	-0.4853056

Graph of Competitive Balance with 5- and 15-Year Means:

Competitive Balance Index by Win Percent in Finished Games in the County Cricket Championship, 1890 to 1999

Graph of Competitive Balance Index, Skewness and Kurtosis:

Competitive Balance Index by Win Percent in Finished Games, alongside Skewness and Kurtosis of Win Percent in Finished Games, in the County Cricket Championship, 1890 to 1999

Conclusions:

The first graph above clearly shows an improvement in competitive balance in the County Championship since the middle 1930s. In fact, as a fifteen-season running mean, the (ASD-ISD)/(MSD-ISD) index as defined above fell from around 0.75 for the fifteen seasons centred upon 1922 [1911 to 1929] to less than 0.27 for the fifteen seasons centred upon 1987 [1980 to 1994]. This constitutes a dramatic contrast to [Australian rules] football, soccer and basketball leagues, which have seen no improvement in competitive imbalance over the past century-and-a-quarter.

There are several possible explanations for the dramatic improvement, which are not mutually exclusive:

1. standardised professional squads after the 1930s meant that no team relied on low-quality amateur players as many counties before 1930 substantially or largely did
• the exceptions or partial exceptions were:
1. Lancashire, Nottinghamshire, Surrey and Yorkshire — and to a smaller extent Kent, Sussex and Warwickshire
• these counties received sufficient industrial patronage to afford large professional staffs so would only play amateurs who could compete with their best professionals
2. Kent (again), Middlesex, Essex and Hampshire
• these counties possessed a substantial number of high-quality amateurs associated with business in London but able to devote full or partial summers to county cricket
2. standardised mass production and improved coaching of pace and seam bowlers produced more uniform quality amongst counties after the middle 1930s
• this made bowling much more consistently economical in runs conceded and also much cheaper to develop
• of course this greater uniformity at the cost of eliminating the possibility of financially self-supporting first-class cricket, which requires overwhelming predominance of spin alongside the most limited pace and seam
3. increasing breadth of search for players, which began in earnest in the 1930s and 1940s (e.g. Jack Walsh), should have reduced variance in performance
• in this context, the introduction of “special registration” for England-eligible players after World War II should have further improved competitive balance as players were no longer held by teams without need

The skewness data does not suggest a great deal of interest from cursory examination.

The kurtosis data suggests increasing (less negative) kurtosis over time, which is highly consistent with the theories of standardisation noted in 1) and 2) above.

The data suggest that standardisation has either been more radical or more effective (or both) in county cricket than in most other team sports, where similar standardisation has either had no effect on comptitive balance or even, as in [Australian rules] football, lowered it as the talent pool becomes more and more limited. Why a focus on tall, fit pace bowlers that developed after the 1948 Ashes series — and can be traced back earlier — should not lower the talent pool as a similar focus in [Australian rules] football since the 1980s has does deserve discussion as I cannot see a definitive and obvious answer.

Military spending in the US by regional mean rank

Ever since a quarter of a century ago when I read Trotskyist magazines arguing that if all military spending were scrapped and the rich properly taxed there would be far more than enough money to solve all the social problems facing the world today, I have taken an interest in US military spending, in part because I found a lot of information from a late-1970s textbook at Latrobe University.

A Six-Way Division of the US

Since reading the late James Löwen’s Sundown Towns, I have come to propose a six-way division of US states according to the map below:

A six-region map of the United States. An alternative divide of the two western regions is also possible, as well as splitting the into three or four; however, the four more easterly regions are relatively clear even for seven- and eight-way divisions

Comparing Regions’ Military Spending

In the following brief analysis, I hope to look at how these six regions compare in terms of military spending. My first experience with variation in US military spending by region occurred from reading R.W. DeGrasse’s Military Expansion, Economic Decline: Impact of Military Spending on United States Economic Performance as a Melbourne University student exploring the Latrobe library over two decades ago. DeGrasse developed a system of dividing US states slightly different from the conventional one, but one which I have come to see as the most logical available. However, the Southern and Western regions of DeGrasse are sufficiently varied that I have split them into two, as shown in the map above. Löwen’s work from a quarter-century after DeGrasse further supports such a division.

In the table below, I have assessed and compared the ranks of each state in defence spending from, firstly, the latest available state-by-state report (from 2024), and secondly, from DeGrasse’s 1981 work. Comparing DeGrasse’s work with the latest available can, of course, show continuities and changes in how states and regions compare.

US Military Spending By State and Region — 2024 and 1981

2024 Rank	State	1981 Rank
1	Virginia	4
2	Hawaii	3
3	Connecticut	5
4	District of Columbia	1
5	Alaska	2
6	Maryland	13
7	Kentucky	32
8	Alabama	22
9	Maine	23
10	Mississippi	7
11	New Mexico	9
12	Arizona	14
13	Oklahoma	17
14	Colorado	21
15	Texas	16
16	Utah	6
17	Rhode Island	29
18	Missouri	8
19	Massachusetts	15
20	Florida	24
21	South Carolina	19
22	Washington	12
23	Pennsylvania	44
24	Indiana	35
25	North Carolina	27
26	Georgia	18
27	Kansas	25
28	South Dakota	33
29	California	10
30	New Hampshire	20
31	Nevada	31
32	New York	37
33	Vermont	30
34	Wyoming	36
35	New Jersey	40
36	North Dakota	26
37	Michigan	48
38	Louisiana	11
39	Ohio	41
40	Nebraska	39
41	Montana	45
42	Illinois	50
43	Iowa	47
44	Wisconsin	46
45	Delaware	28
46	Idaho	42
47	Arkansas	34
48	Tennessee	38
49	West Virginia	49
50	Minnesota	43
51	Oregon	51
17.00	Southwest Average	13.40
18.63	Lowland (Plantation) South Average	16.50
21.33	Northeast Average	23.75
28.14	Upland (Nonplantation) South average	27.71
30.00	Northwest Average	28.16
39.86	Midwest Average	44.57

Results and Conclusion:

What the table shows is that:

1. with a few exceptions, the states of high and low relative military spending in 1981 and 2024 are the same:
1. seven of the ten states with lowest defence dependence are the same in both years
2. six of the ten states with highest defence dependence are the same in both years
2. military spending in the US is heavily concentrated in the lowland South and the Southwest, and to a lesser degree in the Northeast
3. almost no state in the Midwest (only Indiana in 2024) is outside the bottom third in defence spending
4. eight of twelve states in the Northwest are also consistently below average in military dependence
1. this is more significant than it looks before the four “defence dependent” states in the Northwest, alongside Kansas — the next most dependent in the region — form a periphery either:
1. substantially part of the Southwest at a local level (Colorado, Utah) or
2. peripheral to the Northwest region (Alaska, Washington)
2. we can perhaps then define a “core Northwest” consisting of Idaho, Montana, Nebraska, North Dakota, Oregon, South Dakota and Wyoming
1. states within this “core Northwest” are consistently and uniformly amongst the least defence-dependent in the United States
5. three of the seven states in the Upland South are also consistently amongst the least defence-dependent

In Sundown Towns, Löwen demonstrated that racial segregation has tended to be less in military towns than almost anywhere else in the United States. Löwen also showed that virtually the entire “core Northwest” has historically absolutely excluded blacks, with the exception of a small number of the largest cities. The Midwest was almost as rigid. Löwen’s observation stands in complete agreement with the observation that military spending is concentrated in the lowland South and Southwest, where blacks (and, often, other nonwhites) were valued as sources of labour rather than feared as competitors for land. In the Northeast, whilst blacks were not valued, they were feared less than in the Midwest, Northwest or upland South because land was less abundant.

The military — who saw and sees the value in cheap black labour as soldiers — preferred and prefers thus to locate in areas not virulently intolerant of blacks’ presence. Of course, these areas not intolerant of (or valuing) blacks’ presence are also strategically logical as they are closest to the US’ most dangerous foreign threats from the lower classes of Latin America, but the history noted by Löwen no doubt plays a comparable role.

How I got “pet room Archie’s”!

This evening looking at my unfortunate — if at times hilarious as when “Dick Tyldesley” was corrected to a body part — history of tampering with AutoCorrect, I asked AI about why on earth “petromonarchies” could and would be corrected to the ludicrous “pet room Archie’s”!

AI: That is a perfect example of a "mondegreen" error—where the software tries to turn a complex technical term into several simpler, common words [Mondegreen and phonetic errors in software]. 

It’s easy to see how the algorithm got there:

1. Petro- becomes Pet room 
2. -monarchies (which sounds like “mon-ark-ees”) gets mangled into the name Archie’s [Common AutoCorrect linguistic error].

While hilarious, it highlights why AutoCorrect struggles with academic or geopolitical terms; it’s programmed to prioritise everyday language over niche vocabulary [1]. If you use that term often, you can usually “teach” your phone to stop by adding petromonarchies to your Personal Dictionary or Text Replacement settings [4].

AI: Do you have any other ridiculous corrections saved in your phone's history?

I did not answer on that one, but I do not have any other ridiculous AutoCorrect corrections since I stopped trying to tamper with AutoCorrect out of outrage at toilet wall graffiti [and in fact toilet wall graffiti as I have seen it today is not nearly so horrible as I recall in the Richard Berry Building or in Monash’s Hargrave–Andrew Library].