Spanning Test history

Although the question of how many players it takes to ensure that every Test match in history featured at least one of them ostensibly concerns cricket, it soon became clear that finding the answer, and proving that that was indeed the minimum number required, would require rather a lot of mathematics. This was how we eventually reached a conclusion...

The simplest case, considering only the 43 Tests played in 2010, can be solved by trial and error. England and India played 14 Tests each during the year and never played each other, so if each of them had at least one ever-present player, we're most of the way there already. Fortunately they both did: England had Cook, Pietersen, Prior, Swann and Trott, and India had Sehwag and Tendulkar.  Of the remaining 15 matches, all featured at least one of Pakistan, West Indies or New Zealand, so if each of those had an ever-present player (in the Tests not already covered - not necessarily all those which their team played over the year) then we have a set of five spanning all 43 Tests. Pakistan had one (Umar Gul), West Indies four (Bravo, Chanderpaul, Gayle, Nash) and New Zealand eight (Guptill, B McCullum, McIntosh, Martin, Southee, Taylor, Vettori, Watling), so any set containing one player from each of those lists (of which there are 320 possibilities) is a set of five players, at least one of whom played in every Test in 2010.

To prove that five players are necessary, consider a spanning set of teams rather than of players - since no-one played Tests for more than one team in 2010, if it requires at least five teams to span every Test during the year then it certainly requires at least five players. There were four series during the year which had no teams in common (Australia vs Pakistan, South Africa vs England, Bangladesh vs India and Sri Lanka vs West Indies), so if any set of four teams spans all Tests in 2010, it must consist of one team from each of these pairs and no others. Since New Zealand aren't included, Bangladesh must be since the two played each other during the year; therefore the set can't also include India. It can't have both Australia and South Africa because that would miss the England vs Pakistan series, nor can it contain Pakistan and South Africa because that would miss the Ashes. Therefore it can't contain South Africa at all - but if it doesn't contain either India or South Africa then it would miss the two series they played against each other. This is a contradiction, from which we conclude that the conjectured set of four teams does not exist - so five is the minimum numbers of teams required such that at least one of them was involved in every Test in 2010, and hence also the minimum number of players.

Clearly such an approach is not feasible when considering a larger set; some algorithm must be found which can produce a list of players from a set of any number of matches without trial and error. One candidate for such an algorithm is:

1. Let S be the set of all Test matches under consideration.

2. Let X be the player who has played most matches in S (if two or more players have played an equal highest number, pick any of them).

3. Add X to the list.

4. Remove from S all the Test matches played by X.

5. If S is empty, output the list and stop. If S is not empty, go to step 2.

In non-mathematical terms: start by picking the player with most Tests (for the period since 1st January 2000 only, Ricky Ponting; for the entirety of Test history, Sachin Tendulkar), then at each subsequent stage pick the player who has played most Tests which did not also feature any of the players already on the list. Stop when you run out of matches, ie when the list of players you've already picked span all the Tests in the set. After causing Statsguru to crash numerous times, this algorithm produced a list of 54 players to span all of Test history. The exact lists depend on the choices made at the points where two or more remaining players had an equally high number of matches; for example, a few steps from the end one finds that Michael Clarke is the only player to feature in the recent Australia vs India series who is already in the set, meaning that one player who appeared in the three matches which Clarke missed is required to complete the set. 14 players featured in all three matches, so any of these can be chosen in order to include those three matches in the set. 51 players are enough to span all but three Tests, but unfortunately the remaining three matches are spread across Test history (South Africa vs England in 1892, New Zealand vs South Africa in 1932 and India vs Pakistan in 2007) that three further players are required to include them; clearly any of the 22 players can be chosen for each match. Ishant Sharma played in the 2007 match, so until the end of 2014 only 53 players would have been required; his dropping from the final Test of the series necessitated a 54th.

Here is one example of a set of 54 players who between them have appeared in all Tests played to date, listed in the order the algorithm adds them to the set:

SR Tendulkar, SR Waugh, JH Kallis, SM Gavaskar, DPMD Jayawardene, MC Cowdrey, S Chanderpaul, IVA Richards, WR Hammond, Javed Miandad, GS Sobers, IR Bell, SE Gregory, RW Marsh, SP Fleming, TG Evans, FE Woolley, JG Wright, PR Umrigar, Inzamam-ul-Haq, JHB Waite, BE Congdon, JM Blackham, WM Lawry, AJ Stewart, BB McCullum, A Ranatunga, MJ Clarke, RR Lindwall, JH Sinclair, Mohammad Ashraful, DI Gower, WAS Oldfield, Intikhab Alam, Azhar Ali, W Rhodes, KF Barrington, A Flower, CL Walcott, J Briggs, H Masakadza, Hanif Mohammad, AI Kallicharran, BC Lara, CK Nayudu, AD Nourse, V Kohli, RG Nadkarni, HW Taylor, SK Warne, MA Butcher, WL Murdoch, B Mitchell, R Dravid

The above algorithm is guaranteed to find a set of players who between them have played in every Test in the set in question, but not necessarily the smallest set. 

A proof that the algorithm runs in polynomial (more specifically, quadratic) time relative to the number of Tests in the set - returning to the initial algorithm:

1. Let S be the set of all Test matches played to date.

2. Let X be the player who has played most matches in S (if two or more players have played an equal highest number, pick any of them).

3. Add X to the list.

4. Remove from S all the Test matches played by X.

5. If S is empty, output the list and stop. If S is not empty, go to step 2.

The key factors here are how long steps 2-4 take, and how many times you have to do them. Let's break it down a bit:

2. Let P be the set of all Test players, and denote a player in P by Pi. Let n(Pi) = 0 for every i.

a) For each Test match Sj in S, add 1 to n(Pi) iff Pi played in Sj.

b) Let X be the player Pi for whom n(Pi) is greatest.

Step 2.a) involves |S| operations (one check for each match in S) and thus is carried out in |S| time. Step 2. b) involves |P| operations (checking the value of n(Pi) for each Pi in P); the number of players is at most 22 times the number of matches, so |P|=<22|S| and step 2.b) is therefore also carried out in |S| time. So the whole of step 2 is also carried out in |S| time. Step 3 only involves one operation, while step 4 involves at most |S| (since the number of matches being removed from S is at most |S|). Therefore each iteration of steps 2-4 is carried out in |S| time.

Since each iteration of steps 2-4 adds one player to the list, and at most |S| players can be added to it (obviously |S| players are sufficient to span |S| matches), the maximum number of iterations required is |S|. Therefore the algorithm is guaranteed to give the optimum solution in at most |S|^2 time (that is, quadratic time with respect to |S|)

Michael Jones, Manish Yadav, Arnold D'Souza

Chains of players

Chains linking each country's first Test to its most recent (as at the end of 2014) i.e. the first named player appeared in the country's inaugural Test, each player in the chain appeared in at least one match alongside the next player, and the last played in its most recent match:

Australia (1877) - 10

JM Blackham - SE Gregory - C Kelleway - DG Bradman - RN Harvey - RB Simpson - GM Wood - SR Waugh - RT Ponting - MJ Clarke

England (1877) - 10

G Ulyett - WG Grace - W Rhodes - LEG Ames - DCS Compton - MC Cowdrey - JH Edrich - GA Gooch - AJ Stewart - JM Anderson

West Indies (1928) - 7

LN Constantine - GA Headley - CL Walcott - GS Sobers - CH Lloyd - DL Haynes - S Chanderpaul

New Zealand (1930) - 8

ML Page - J Cowie - B Sutcliffe - RO Collinge - RJ Hadlee - MD Crowe - CL Cairns - BB McCullum

India (1932) - 7

CK Nayudu - L Amarnath - VL Manjrekar - EAS Prasanna - Kapil Dev - SR Tendulkar - Virat Kohli

Pakistan (1952) - 4

Hanif Mohammad - Younis Ahmed - Wasim Akram - Taufeeq Umar

Sri Lanka (1982) - 2

A Ranatunga - DPMD Jayawardene

Zimbabwe (1992) - 2

GW Flower - Hamilton Masakadza

Bangladesh (2000) - 2

Mohammad Rafique - Shakib Al Hasan

No continuous chain exists for South Africa, since no-one played for them both before and after the apartheid ban. One chain of 8 connects their first Test (1889) with their last before the ban (1970), and another of 2 connects their first after readmission (1992) with their most recent:

FW Smith - JH Sinclair - AW Nourse - HW Taylor - B Mitchell - AD Nourse - DJ McGlew - PM Pollock

AA Donald - JH Kallis

Most of the chains are not unique; one or more other players could be substituted for some of those in the chain to give another of the same length, although certain players must be included in order to minimise the length. Some countries' chains are very close to needing one link fewer: Garry Sobers played his first Test later in the same series in which George Headley played his last, and although Chris Cairns made his debut before Richard Hadlee retired, they never played together - so in each case the chain requires one extra link to cover a relatively short time gap. On the other hand, some countries' chains fit together neatly, with minimal overlap between successive players' careers: WG Grace's last Test was Wilfred Rhodes's first, and between them they span 50 years.

Some players are the last of their 'era', so when they retire their country will need one more link in its chain: Chanderpaul, Sangakara and Jayawardene are all the last player still active to have played with the previous one in the chain. Sri Lanka and Zimbabwe are nearing that point: Rangana Herath also played with Ranatunga, so when either he or Jayawardene retires, the other will become the last of their 'era'. Australia have just reached this position: until Ponting's retirement they only required a chain of 9, but now they have no active player left who played alongside Steve Waugh, so an extra link is needed. For the time being the new link could be any of a number of players, and it remains to be seen which of them will keep the chain at that length for the longest - probably not Clarke, as his career overlapped Ponting's by several years and another player who made his debut shortly before Ponting retired is likely to play on longer.

Including players from different countries, the first Test can be linked to the present day by a chain of only eight players (thanks to Charles Davis for this one):

JM Blackham - SE Gregory - FE Woolley - FR Brown - TW Graveney - DL Underwood - A Ranatunga – KC Sangakara

The physics of six byes

Cricket abounds with tales of a ball going for six byes, with Charles Kortright, Roy Gilchrist and Jeff Thomson among those to which the feat has been attributed. Leaving aside for the moment the fact that the laws of cricket state that a ball only scores six runs if it clears the boundary on the full after having been struck by the bat, just how likely is it that a ball can bounce in the middle of the pitch and then clear the boundary without bouncing again? Clearly such a ball must be extremely fast - but how fast?

Assume that the ball bounces approximately in the middle of the pitch. Most international grounds have a distance of around 80m from there to the boundary in any direction, so let's say that the ball travels exactly 80m before landing. Even the fastest bouncer does not pass very far over the batsman's head - say a height of 2.5m, 10m after pitching. The path of a projectile in a gravitational field is a parabola, and identifying three points on it is enough to determine its equation. The general equation of a parabola is y = ax^2 + bx + c and the three known points on it are (0, 0), (10, 2.5) and (80, 0), so the coefficients a, b and c must satisfy the simultaneous equations

0 = 0a + 0b + c

2.5 = 100a + 10b + c

0 = 6400a + 80b + c

Clearly the first equation implies that c = 0, so the second and third can be reduced to 2.5 = 100a + 10b and 0 = 6400a + 80b. Multiplying the second equation by 8 gives 20 = 800a + 80b, and subtracting this from the third -20 = 5600a, so a = -1/280, and substituting this value into either of the equations gives b = 2/7. Thus the height y of the ball at a horizontal distance x from its point of pitching is given by the equation y = 2x/7 - x^2/280. Differentiating this gives the equation of the tangent to the graph - the direction in which the ball is travelling at a given moment: dy/dx = 2/7 - x/140. At x = 0 this has the value 2/7, the direction at the moment of pitching; this is equivalent to an angle of about 16 degrees from the horizontal.

A parabola is symmetrical, so the highest point in the ball's trajectory is reached at half the horizontal distance between pitching and landing: x = 40m, y = 80/7 - 1600/280 = 40/7m. This information enables us to use the equations of motion:

s(horizontal) = 40

u(horizontal) = 7x/sqrt53

v(horizontal) = 7x/sqrt53

a(horizontal) = 0

t = ?

s(vertical) = 40/7

u(vertical) = 2x/sqrt53

v(vertical) = 0

a(vertical) = -9.8

t = ?

where s = the distance travelled in that direction between the ball pitching and reaching its highest point, u the component of the initial velocity in that direction, v the component of the velocity when the ball reaches its highest point, a the acceleration in that direction, t the time taken for it to reach that point (which is independent of the direction), and x the magnitude of the initial velocity (including both components. The horizontal acceleration is zero if air resistance is disregarded (and thus the horizontal speed is constant); the vertical acceleration is that caused by the Earth's gravitational field (9.8m/s^2), and the ball has an (instantaneous) vertical velocity of zero at its highest point. The horizontal and vertical components of the initial velocity as functions of its overall magnitude are given by applying Pythagoras's Theorem to the known direction of the trajectory at that point.

The equation s = ut + at^2/2 applied to the horizontal motion gives 40 = (7x/sqrt53)t + 0t^2/2, so 40 = (7x/sqrt53)t and thus 40/t = 7x/sqrt53.

The equation v = u + at applied to the vertical motion gives 0 = 2x/sqrt53 - 9.8t, so 9.8t = 2x/sqrt53. Multiplying this equation by 7/2 gives 34.3t = 7x/sqrt53.

Putting the resulting two equations together gives the result 40/t = 34.3t, so t^2 = 40/34.3 and t = 1.08 seconds. Then 9.8t = 2x/sqrt53, so 9.8(1.08) = 2x/sqrt53 and x = 10.6(sqrt53)/2 = 38.6m/s. This is equal to 86.8mph, which is fairly fast in itself - but this is the speed at which the ball must be travelling after pitching. The next factor which needs to be taken into consideration is the coefficient of restitution - the ratio of the ball's speed after pitching to its speed before. Even on the bounciest of pitches this will be no more than 0.7 (on most pitches probably less), which would mean that in order to achieve a speed of 86.8mph after pitching, the ball would have to be released at a minimum of 124mph - a quarter as much again as the highest recorded speed of any ball bowled in an international match (and that's even if you believe the speed guns). The actual figure would be more, because the above calculation does not take into account the effect of air resistance

In conclusion: 'six byes' (even though they wouldn't be counted as such) might just about be possible at a very small club ground - although even if the boundary is 60m from the centre of the pitch rather than 80m, applying the same method shows that the ball would still have to be bowled at a minimum of 105mph. It is certainly impossible at an international-sized ground.

Manish Yadav/Michael Jones

Most runs off an over in the second innings, the fastest wicket-taking ball, and sixes off de Villiers

Is it true that 18 is the most runs scored off an over in the second innings of a T20I? - Ramesh Parajuli Avi

18 isn't even the highest this week: Chris Gayle hit 22 off Kyle Abbott (0, 4, 4, 4, 4, 6) in West Indies' record chase at Johannesburg. The overall record in the second innings is 30, by Aaron Finch (4, 1) and Glenn Maxwell (4, 6, 6, 5 no balls, 4) off Bilawal Bhatti at Mirpur in the 2014 World T20. - Adee Amjad

What is the fastest ball ever to take a wicket? - Martin Ructions Rusis

Any answers to questions involving bowling speeds come with the caveat that data on the subject is far from complete, but the ball with which Shoaib Akhtar blasted out Stephen Fleming in Pakistan's crushing victory at Lahore in 2002 is a plausible candidate. The Cricinfo commentator reported that the ball was "measured at 157kph on the broadcaster's gun and 153kph on the ground sponsor's gun"; the fact that two guns are capable of giving results differing by 4kph illustrates the margin of error involved, so no delivery can ever be definitively declared the fastest, but Shoaib's effort seems to have as good a claim as any. - Michael Jones

Did anyone ever hit Fanie de Villiers for six in an international match? - Malay Shah


In Tests, there are three sixes recorded off de Villiers, all in 1994: by Craig McDermott at Johannesburg, Mike Atherton at Leeds and Shane Thomson at Durban. Atherton's was one of only four sixes in his Test career.

Keith Miller took the most wickets in Tests without ever being hit for six (170); Maurice Tate bowled the most balls. - Charles Davis

No home Tests in a year, doubles in most countries, players wearing glasses, and the number of first-class matches

India didn't play a home Test in 2014. When was the last calendar year this happened? - Sivakumar Venkatesan

The last year in which India didn't host a Test was 1992, when they played three in Australia, one in Zimbabwe (the hosts' inaugural Test) and three in South Africa, but none at home. It was the third year in four they had done so: after hosting nine Tests in 1987 and four in 1988, they only played a solitary home match (against Sri Lanka at Chandigarh in November 1990) in the next four years, before a three match series against England in 1993. - Ramesh Parajuli Avi

Kumar Sangakkara has now scored Test double centuries in six different countries (Sri Lanka, Pakistan, Bangladesh, Zimbabwe, UAE and New Zealand). Has anyone else done this? - Anusha Ranaweera

No, Sangakkara is the first to reach this milestone, beating Brian Lara's previous mark of five (West Indies, Australia, Sri Lanka, South Africa and Pakistan). Younis Khan also currently has five (Pakistan, India, Bangladesh, UAE and Zimbabwe), and still has the chance to add to his collection. - Michael Jones

Did Harry Bromfield, who played 9 Tests for South Africa between 1961 and 1965, wear glasses on the field? - Ashru Mitra

Yes, he did - the 1965 series featured five bespectacled players (Geoff Boycott, MJK Smith, Eddie Barlow, Peter van der Merwe and Bromfield). - Mike Leach

The current Test match between New Zealand and Sri Lanka is number 2155 in the chronological ordering. How many first-class matches have been played? - Richard Mackey

Cricket Archive numbers all its first-class matches. Right now there are 56,047 matches in its database, from 1772 to the end of 2014; new matches are added on an almost daily basis. - Charles Davis

Most centuries in first 100 Tests, and most away runs in 2014

Who scored most centuries in their first 100 Tests? - Kazim Krunchoo

There's a three-way tie at the top here: Sunil Gavaskar, Sachin Tendulkar and Matthew Hayden all scored 30 centuries in their first 100 matches. Ricky Ponting and Kumar Sangakkara follow with 28, Michael Clarke 26, Graeme Smith and Alastair Cook 25. Don Bradman scored 29 centuries, and only played 52 matches; Garry Sobers made 26 in 93 matches. Younis Khan currently has 28 in 96, so if he continues his recent good form he may equal or even break the record. - Michael Jones

Who scored most Test and ODI runs away from home in 2014? - Sivakumar Venkatesan

Counting away matches only (i.e. the home of the opposition), Murali Vijay led the way in Tests with 852 runs, and Hashim Amla in ODIs with 800. Including all matches outside the player's own country (and excluding Pakistani batsmen, since all their "home" matches are played in a neutral country), Kumar Sangakkara's 943 was the most in Tests, and Amla retains the top spot in ODIs with 892. - Martin Briggs

Penultimate Tests at a new ground

Has there been any other player whose penultimate Test was the first at a new ground, as Kumar Sangakkara's was at the Hagley Oval? - Manjula Aravinda

This happened rather a lot in the early days of Test cricket, when several new grounds were used within a few years, and many players appeared in only a handful of matches. The first Test of all, at Melbourne, was the penultimate one for ten of the players who appeared in it (Tom Armitage, Harry Charlwood, Andrew Greenwood, Allen Hill, John Hodges, Harry Jupp, Tom Kendall, James Lillywhite junior, James Southerton and Nat Thomson). Over the following years, the first Test at the Oval was the penultimate one for George Alexander; the first at Old Trafford for Alfred Lucas; the first at Lord's for Alfred Lyttelton and Lord Harris; the first at Port Elizabeth for Monty Bowden, Charles Coventry, Basil Grieve, Philip Hutchinson, Arthur Ochse, Albert Rose-Innes and Bernard Tancred; the first at Cape Town for William Milton, Fred Smith and Charles Vintcent; and the first at Johannesburg for Frank Hearne. All three of Smith's Tests were the first at the ground in question (Port Elizabeth, Cape Town and Johannesburg), as were both of Alexander's two (The Oval and Adelaide).

More recently, Imran Khan's penultimate Test was the only one played at Gujranwala. - Michael Jones/Aslam Siddiqui/Anusha Ranaweera