Merry Christmas and a Happy New Year !

It's Christmas.Though now it has become a commercial event, there is something special in every Christmas for me. In the early years of my life it was about the Santa and the presents I received. But the time has passed and now entering the period of my life where I have to give presents to others rather than expecting .Now its the time you see scary Santas in shops trying to sell various items.It's the time where you cant travel in a road even in the off peak hours.It's the time you see discounts everywhere.Anyway beside all the changes still its a time to rejoice.It's a time to enjoy with the family and friends.

This carol "Mistletoe and Wine" is a popular single by Cliff Richard.Written by Jeremy Paul, Leslie Stewart and Keith Strachan, this was originally performed as part of the musical Scraps which was based on Hans Christian Andersen's fairytale at the Orange Tree Theatre, Richmond, London in 1976.

The child is a King, the Caroller sing,
The old is past, there's a new beginning.
Dreams of Santa, dreams of snow,
Fingers numb, faces aglow.

Christmas time, Mistletoe and Wine
Children singing Christian rhyme
With logs on the fire and gifts on the tree
A time to rejoice in the good that we see

A time for living, a time for believing
A time for trusting, not deceiving,
Love and laughter and joy ever after
Ours for the taking, just follow the master.

Christmas time, Mistletoe and Wine
Children singing Christian rhyme
With logs on the fire and gifts in the tree
A time to rejoice in the good that we see

Silent night, holy night

Its a time for giving, a time for getting,
A time for forgiving and for forgetting.
Christmas is love, Christmas is peace,
A time for hating and fighting to cease.

Christmas time, Mistletoe and Wine
Children singing Christian rhyme
With logs on the fire and gifts on the tree
A time to rejoice in the good that we see.

Christmas time, Mistletoe and Wine
Children singing Christian rhyme
With logs on the fire and gifts on the tree
A time to rejoice in the good that we see.

Merry Christmas  and a Happy New Year !

Duckworth-Lewis Method (part 3)

1. Duckworth-Lewis Method (part 1)
2. Duckworth-Lewis Method (part 2)

After giving an introduction of Duckworth-Lewis method and some examples of Duckworth-Lewis method in the previous posts this post focuses on the  Duckworth-Lewis method and some of the previously used methods in rain affected games.

Before the invention of this method there have been several other methods used at the international level to decide the result of a rain affected match.One of them was the run rate based system. The main disadvantage of this method was that the number of wickets fallen was not considered. Thus in this system if a team has scored 100 runs for the lost of 9 wickets the position of the team is considered higher than scoring 90 runs for the loss of 1 wicket.

Eng v SA World Cup 1992
Situation at end of match SA need 22 runs.www.patrickeagar.com

Also another method that was used is the Highest Scoring Overs method which compares the maximum runs scored by team1 in any set of overs equal to the number of completed overs received by team 2 against the team 2 in those completed overs. So if team 2 received 31.4 overs their score after 31 overs is compared to the highest scoring 31 overs of team 1's innings. So this method becomes very unfair as the maidens bowled by the second team is not considered. So eventually this method is more biased towards the first team. The most controversial usage of this method arises in the 1992 world cup semi final between South Africa and England. Play was halted when South Africa had scored 231/6 from 42.5 overs against England's 252/6 from 45 overs. When play was able to resume there was time for South Africa to receive only one more ball, 43 overs in total, so their target was revised to 252 by discarding the two maiden overs in the England innings, one of which yielded one extra. So the number of bowls available was reduced by 12 but the target remained unchanged. Due to the use of this method an apparently attainable target suddenly became an impossible one. 


There are many other methods used in a rain affected match. All of these methods either uses the run rate , highest scoring overs or pre-calculated curves (as used in Duckworth-Lewis method) with slight variations. Yet Duckworth-Lewis method is regarded the best method so far used. But a slightly different method named as VJD system introduced by V.Jayadevan (an engineer from Thrissur in Kerala) has challenged the Duckwoth-Lewis method recently. 

THEN THEY CAME FOR ME..........

First they came for the communists,

and I didn't speak out because I wasn't a communist.

Then they came for the trade unionists,

and I didn't speak out because I wasn't a trade unionist.

Then they came for the Jews,

and I didn't speak out because I wasn't a Jew.

Then they came for me

and there was no one left to speak out for me.

 A famous statement attributed to pastor Martin Niemöller (1892–1984) 

How Secured is Your Security Question ????

You may have seen different kinds of questions such as "What is your library card no ?" , "what is your mothers birth town ?", while registering for an online account.These questions are used in password recovery processes and additional sign in verifications. Normally we all care about our passwords and worry about the security of the password.

But do we really care about these security questions ? It seems most of us don't care much about these.Most of the time we give our real details as the answers to theses questions and simply forget about the security question.Even I was unaware of security questions selected for my email address until i decided to write this post.The most surprising thing is that I can't come up with the correct answer for those questions.I think this is common between most of us.We really DON'T CARE about those.

But in reality this is a great security risk.Even if you have a very good password that can not be guessed easily,you may be vulnerable to security threats.It is like locking the front door while leaving the backdoor open.The confidentiality of the answers for a security question will be far less than a password.If some one asks you a password you probably wont tell.But if some one asks you the birth town of your mother you may answer it.Also most of the details that are asked in security questions can be found publicly or can easily be guessed.That is what happened with the email account of Sarah Palin the vice president candidate during 2008 elections.

A good security question should have a answer which can be easily memorized, which is not publicly available and which does not change overtime. Further the possible number of answers for the question should be very large so that it can not be guessed. There are some websites which offers good security questions. But it is worth to note that the goodness of the question depends on the expected answer as well.

Pigeon-Hole Principle

It's amazing how a very simple and obvious thing in life become a mathematical principle and helps to prove many complex results. Pigeon-Hole principle is one such concept. At first it seems as a very obvious fact and one may wonder whether it deserves such a place in mathematics. But this principle becomes very useful in many occasions to prove that there exists some answer for a problem.

Duckworth-Lewis Method (part 2)

(For an introduction of Duckworth-Lewis method visit the part 1)

Below, are two examples given for the betterment in understanding the Duckworth-Lewis method used in different scenarios.A sample of reduced D/L table is given below.

	Wickets lost
Overs left	0	2	5	7	9
50	100.0	83.8	49.5	26.5	7.6
40	90.3	77.6	48.3	26.4	7.6
30	77.1	68.2	45.7	26.2	7.6
25	68.7	61.8	43.4	25.9	7.6
20	58.9	54.0	40.0	25.2	7.6
10	34.1	32.5	27.5	20.6	7.5
5	18.4	17.9	16.4	14.0	7.0

Ex. 01: Team A scored 263 runs within their allotted 50 overs and the match was interrupted when team B had scored 132 for the loss of 2 wickets in 30 overs.

Case1: If match is not resumed.

The percentage of resources lost by Team B = 54% (wickets = 2 and overs left = 20)

The percentage of resources available to Team B = 100-54 = 46%

The percentage of Team A’s resources = 100% (no interruptions occurred during 1st innings) Revised target for 30 overs = 263*46/100 = 120.98

Since Team B has scored 132, they will be declared as the winners.

Case 2: If match is reduced to a 40 overs.

The percentage of resources at the time of interruption for Team B = 54%

The percentage of resources when match resumed (10 overs left, 2 wickets down) = 32.5% Percentage loss of resources for Team B = 54-32.5 = 21.5%

The percentage of resources available to Team B=100-21.5=78.5%

Revised target in 40 overs = 263*78.5/100 = 206.45 = 207(to win)

Ex2: Team A scored 132/2 in 30 overs and the match was interrupted. The match resumed as a 30 over match.

The percentage of resources at the time of interruption for Team A = 54%

The percentage of resources used by Team A = 100-54 = 46%

The percentage of resources available for Team B = 77.1% (30 overs 10 wickets)

Since Team 2 has more resources, their “revised target” must be raised upwards. The G50 value comes into play at this moment. The additional number of runs that has to be added to the target is calculated as a percentage of G50 value with respect to the additional resource percentage available for Team B.

The additional resource percentage available for team B = 77.1-46 = 31.1%

Additional runs that should be added to the target = 225*31.1/100 = 69.975 = 70 (225 is the G50 value)
Therefore Team B should score 202 runs to win the game within 30 overs.

So this shows how simple D/L method is used during matches.The professional version of this D/L method is used in ODI's.

The Duckworth-Lewis Method (part 1)

History

Though they may not have played cricket at international or first class level, they were surely the match winners in more than 80 One Day Internationals (ODI) including a world cup final. They were neither umpires nor match referees as well. Though many won’t realize the relationship between mathematics and cricket, they both strived to describe the gentlemen’s game in a very simple yet realistic mathematical sense. Over the past few years they have become the most influential duo in limited overs cricket. They both are none other than Frank Duckworth and Tony Lewis, the statisticians who introduced the “Duckworth Lewis Method” more popularly known as The D/L method within the realm of cricket and among cricket enthusiasts.
The D/L method was successfully experimented within the year of 1997 by the International Cricket Council (ICC), the ECB (England & Wales Cricket Board) and the Zimbabwe Cricket Union (ZCU). It was first used in international cricket during the second game of the 1996/1997 Zimbabwe versus England One Day International, in which Zimbabwe won by 7 runs. Having seen its potential, the D/L method was formally adopted in 2001 by the ICC and henceforth was the standard method to calculate target scores for rain interrupted one-day matches.

Basis

Unlike previous methods used for rain affected games, The D/L method considers both the number of wickets lost and overs remaining. The said two are considered as the available resources for a particular team. A table is formulated capturing most of the variations pertaining to one-day cricket encounters based on the outcomes of a detailed analysis on hundreds of match data. Based on the depth of the mathematics used, The D/L method was split into a Professional and Standard Edition. The Standard Edition preserves the use of a single table and the adaptation of simple calculations (suitable for any one-day cricket match at any level), where else The Professional Edition utilizes a statistical modeling that is substantially sophisticated and requires the use of powerful computing. The Professional Edition had been put into use for all One Day International matches since early 2004.
This article is based on the standard edition of The D/L method. One major consideration in this Edition is the use of the G50 value which is defined as the average score obtainable in a 50 overs match. Normally, this value is taken as 225, but The D/L method states that a freedom to choose any other value is viable given that all parties involved are aware of that change from the very beginning. This greatly exemplifies the flexibility of The D/L method.

	Wickets lost
Overs left	0	2	5	7	9
50	100.0	83.8	49.5	26.5	7.6
40	90.3	77.6	48.3	26.4	7.6
30	77.1	68.2	45.7	26.2	7.6
25	68.7	61.8	43.4	25.9	7.6
20	58.9	54.0	40.0	25.2	7.6
10	34.1	32.5	27.5	20.6	7.5
5	18.4	17.9	16.4	14.0	7.0

A sample of a reduced D/L table is shown here. The resource percentage (that is, the No. of wickets and overs remaining) is indicated as shown. At the very commencement of an innings, each team has 100% of its resources available (that is, 50 overs and 10 wickets in hand). If at the end of 30 overs a team is at a loss of 5 wickets, its available resources reduces to a frail 40% percent.

Usage

The following procedure is used to recalculate a target score when an interruption occurs.
1. For the innings of each team.
(a) Note the initial resource percentage from the table (that is, at the start of an innings).
(b) Using the table, calculate the resource percentage lost due to each interruption.
(c) Henceforth determine the available resource percentage.

2. If Team 2 has fewer resources available than Team 1, the ratio of the available resources for the two teams is calculated and Team 2's “revised target” is obtained by scaling down Team 1's score by this ratio.

3. If Team 2 has more resources available than Team 1, the amount by which Team 2's resource percentage exceeds Team 1's is calculated and this excess is worked out as a percentage of The G50 value to determine the extra runs required to add on to Team 1's score to give Team 2's “revised target”.
(some examples will be calculated in the next post)

Calculate Squares of Numbers Instantly

The following method may help to calculate squares of numbers from 21-120 easily .Only some basic multiplications have to be done and some basic square values have to be remembered in order to use this method.

The base of this method is the well known algebraic identity

a^2-b^2=(a-b)(a+b)

This identity can be rearranged to the following form.

a^2=(a-b)(a+b)+b^2

So when 'a' is given we should select 'b' such that the terms b^2 and (a-b)(a+b) are easily calculable.The 'b' value can be selected so that the square of 'b' is a known number.Most of us know the squares of numbers up to 12 and some times even up to 25.To use this method I may suggest to remember square values up to 20.One can use this method only remembering square values up to 5 but knowing more values will ease the other calculation part in the equation.

The next part is the calculation of (a-b)(a+b).We can choose 'b' such that either (a-b) or (a+b) is divisible by 10.Then the multiplication becomes more simple and can be done mentally.Since there are several values of 'b' that gives a factor divisible by 10 the trick is to select 'b' such that either (a-b) or (a+b) gives the multiplier of 10 nearest to the value of 'a'.

To get an idea about the method let me show some examples.

• 38

We'll select b as 2 (since this will give us a+b=40).Thus the calculations we have to do is 40*36 which is effectively 36*4.(this multiplication may done in two separate steps 30*4+6*4,which are very easy).then the only thing we have to do is adding 4 (2*2) to this value to obtain the square of 38.

So 38*38=30*40+6*40+4=1200+240+4=1444.

One may also try b=8 if they are more comfortable with multiplication with 3 rather than 4.

• 74

We'll select b as 4.The calculations we have to do is 70*78 which is effectively 78*7.(this multiplication may done in two separate steps 70*7+8*7,which are very easy).To obtain the real value 16 should be added to the answer.

So 74*74=70*70+8*70+16=4900+560+16=5476.

One may also try b=6 if they are more comfortable with multiplication with 8 rather than 7.

• 87

In this case we may select b=3 but than we have to multiply 84*90.But if we select b=13 we only have to multiply 100*74 and add 13*13 (169) to the result.The calculations have become very easy.As the number of square values we remember the calculations will become very simple.

As an example if we remember square of 26 (which is 676) then we may select b=26 in the previous example and the multiplications will become very simple.

I think this method is very useful depending on the values you select for b and the ability of using basic multiplications.Although in the beginning I mentioned about numbers up to 120 this can be extended to more numbers depending on your ability to multiply mentally.To the better use of this method the following points may be important.

• If the number is near 50 or 100 select 'b' such that (a-b) or (a+b) is equal to 50 or 100.
• Remember more square values.May be up to 25.(This will help to find squares of higher numbers with out much calculations).If we remember squares up to 25 we may calculate up to 200 with out much effort.
• The multiplication part can be done in two steps as given in the example.
• The square of a number ending with 5 can easily be calculated by adding 25 to the value obtained by multiplication of multipliers of 10 that are lower and higher than the required value.(35*35=30*40+25)