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WI vs SL Ist ODI: Caribbeans lost to Lankans due to Mathematical shortcomings in D/L target score method
Sunday was a day of clouds and rain as far as cricket is concerned. More than one match were interrupted due to rain, the overs were shortened and the Duckworth Lewis method was employed to compute and set the target scores for the chasing team. But when analyzed in the light of mathematics, those D/L based target scores were incorrect.

In this article, I bring forward the mathematical shortcomings present, and which can be traced even by you, in those yesterday's target scores.

Cricket's natural mechanism simplified for you

Batsman's data means his runs scored + balls faced value.

On the basis of repeated observations and extensive research, it has been found that both, the score data, S, and the batsman's data, W, have the same remainder once all nines have been removed. This is a case of congruence, the Score is always congruent to the batsman's data module 9.

In mathematical terms, this is written S = W (mod 9).

A simple example, the batsman's data is 34 and the total score is 241. When we remove/cast out all nines from both the numbers, we get-

34 - 27 = 7 and 241- 234 = 7

Natural Integer Partition In Actual Score

Date   Match Score  Integerpartition 
 11 June 1975 Pakistan Vs West Indies  266  270 -4
 27 November 1979  Australia v West Indies   193 18013 
 25 June 1983  India v West Indies  183 180 3 
 11 February 1984  Australia v West Indies  222 180 42
 18 April 1986  India v Pakistan  245 270 -25
 13 April 1984  India v Pakistan  188 180 8
 17 November 1985  India v Pakistan  203 270-67 
 7 April 1986  New Zealand v Pakistan  214 180 34

Always congruent

Pick up any match from the past, or any live match, observe the scoring details of the first and second wicket batsman. Simply add his balls faced figures to his runs. And compare it with the first innings score. Their remainders will be equal (or minus version) in more than 80 per cent matches. This is the main mathematics of cricket.

The defense wall

This is the ideal state the first team should strive to achieve. When equal, the target (or total score, score for wickets) act as a strong defense wall, which the chasing team can't break. Even that's the original purpose a target score, it is a challenge for the chasing team.

Congruence leads to victory

When the remainder values of the batsman's data and the Team 1 score are equal, it creates winning chance for Team 1. When not equal, or when there is a 9 in the data, the chasing team i.e. Team 2 wins.

Ideal Target score formula

Whichever method is used to arrive at the fair value of the target score, it must be slightly adjusted to make its remainder equal to the first wicket batsman's data (his runs + balls). Then only it will be consistent with the logic/purpose of the target score which is to put up a challenge for the chasing team.

Sunday's Matches D/L target scores

West Indies vs Sri Lanka match D/L target 163

West Indies were weak side, still they managed to put a score 159 runs at Colombo. Simple target was 160 runs. Most importantly, the score had a natural congruence with its first out batsman's data-

Charles out scoring 1 run in 2 balls, batsman's data = 1 + 2= 3. Remainder when divided by 9= 3.

Score remainder= 159- 153= 6 = -3.

There was a correct equivalence between batsman's data and the Team 1 score, there was hardly any need to modify it. On the other hand, the remainder value of the DL method based target 163 is 1. Clearly, equivalence/congruence was absent. Hence, the target score is mathematically incorrect [Statistically, it might have been correct, I am never challenging it].

Pakistan women vs West Indies women D/L target 78

                      D/L target Score: Lack of Congruence
 Date Match  Batsman's Data   Digit Root  1st Ining Score Digit Root 
 Jan 1, 1997 Zimbabwe Vs England  14 5 200 2
 Apr 12  Bangladesh Vs Kenya  17 8 242 8
 Oct 11 South Australia vs Queensland  1 1 280 1
 Oct 12 Tasmania Vs NSW  61 7 218 2
 Oct 20 Tasmania Vs Victoria  95 5 320 5
 Nov 1 KwaZulu-Natal Vs South Western Districts  58 4 238 4
  Sri Lanka v West Indies  3 3 159 6
  Cape Cobras Vs Lions  38 2 125/8 7
  Pakistan women Vs West Indies women  18 9 88 7


"There was lack of congruence between the original target 89 as well as the first/second wicket batsman's data, the defense wall was absent, team 2 Pakistan were to win. But the D/L target made a blunder of a match- it resulted in a tie, and finally a win for WI at St George's. Alas, captain Sana Mir had to face one more defeat for such a low actual target 89.

Cape Cobras v Lions D/L target 124

Similar mathematical shortcoming was witnessed in the Sunday's South African domestic T20 competition, Ram Slam 2015. Even the actual score 125/8= 7 was the additive inverse of the Levi's data= 38= 2. So was the D/L target 124=7. But, as explained earlier, an ideal target score must not be additive inverse or any other number, but exact equivalent of the remainder of batsman's data. It is just the matter of adding/deducting 1 to 9 to the target derived by D/L (any other) method.

D/L history

When we check the history, the D/L target score in the past matches, the plight is the same. There is lack of congruence as detailed in the table.

D/L resource table itself non congruence

The resource table provides percentage values, how can we expect the target score to be integer. But the cricket score are integers. Clearly, the method has failed to trap the natural linear integer mechanism of cricket scores.

Suppose, the current score is X runs (for z wickets). The resource table would provide a specific percentage which denotes resource left. When we attempt to calculate the 100 percent of the same (which means total potential score), it is never equal to actual match total score.

Any equation should work both ways, output to input, input to output. The D/L table lacks this basic test. And further, it never generates integers, never congruent to live scorecard's current status or even batsman's data.

The reason is simple, the natural score mechanism seems to be 'addition/subtraction' based, but the DL method is multiplying/dividing and what not. The mechanism of cricket is not getting trapped.

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