Pi Day

Today is the "Pi DAY". We have dedicated special dates to celebrate various reasons. And March 14 (or in numbers 3-14) is the day dedicated to celebrate the mathematical constant "Pi". The representation of this date consists the first three digits of the mathematical constant "Pi". Thus it is selected as the "Pi" day. It is a day celebrated by mathematics lovers and probably a day hated by maths haters.

Pi Day Google Doodle 

There are sites specially designed to celebrate this day such as http://www.piday.org/. And it contains useful information about the constant "Pi". Alternatively another day is celebrated as the "Pi" approximation day which fall on July 22nd (22/7) which represents the widely used approximation value of "Pi".

The Cake Cutting Problem - Fair Division Problem (Part 1)

As humans we tend to own everything. When dividing something among people we need to own a fair portion, if not a bigger portion. Thus problems regarding dividing has become the source of many conflicts, wars and sometimes crimes. Cake Cutting problem is an interesting thing in mathematics which deals with dividing a cake between several people such that each person is fully satisfied with the piece he gets.The generalized dividing problem is also known as the fair division problem and there are several algorithms to solve this. This problem is very much associated with real life problems.

Basics

Before solving the problem of fair division the word "Fair" should be defined properly. Fair can mean two things. The first condition is everyone should be satisfied, that the piece they got is at least an equal portion of the total resources. That is, if the resource is divided between n people each person should feel that they have received at least 1/n portion. The other condition is that every person should feel that no one has received more than himself. In other words everyone should feel that they received the largest part. Satisfying the first condition is less problematic than the second. Thus sometimes the dividing problem is limited to satisfy only the first condition. Yet the human behavior demands not only the first condition but also the second condition should be satisfied so that everyone is happy about the division of the resource. The problems which mandate the second condition is also known as Envy Free division. 

Also the criteria to decide the value of each portion is important. In the simplest case the value of each piece is based on its size. But in complex (and practical) cases the value of each piece will be different     to each participant. One participant may value a piece of cake with more icing on it while one prefer a  piece with cherries. Even under these conditions the definition of fair will help us to tackle the fair division problem easily.

Simplest Case

The simplest case is the division of a cake between two people. This can be easily solved using the divide and choose method. In this scenario one person divide the cake into two pieces which he thinks that is fair. The other person gets the chance to select a piece he like. The other piece is left to the first person. As the first person initially divided the cake in to two fair parts he gets a fair portion. Although second person do not agree with the division of the cake in to two as fair he gets the chance to select any piece. Thus he will also be satisfied with the piece he gets.

The Case of Three

The next case is the division of a cake between three people. There are several methods to solve this fair division problem for three people. Almost all these methods can be generalized to solve problems containing more than three people.The divide and select approach which was used for two person case can not be extended to solve this.  And there are solutions which seems plausible but proved wrong when thoroughly analyzed.

Let's now analyze one such incorrect approach. Let the three people be named as A,B and C. This solution includes following steps.

• A cuts the cake in to two pieces where he thinks one is 1/3 and the other is 2/3 of the cake.
• B cuts the 2/3 part of the cake in to two fair pieces.
• C chooses any piece he likes.- So C is satisfied
• Then A chooses.
• Finally B chooses.

Now lets analyze the above solution. Since C selects first, clearly C is satisfied. If C has chosen a piece that is cut by B then A can select the initial piece he cut. If C has selected the initial piece cut by A, then A can select one of the pieces cut by B. As A initially divided in to 1/3 and 2/3 parts, at least one piece cut by B should be 1/3. So A is satisfied in any case. The problem is with B. If B thinks the initial cut by A is not fair (ie. The smaller piece is larger than 1/3 of the cake) B is only satisfied with that piece. Thus if that piece is selected by A or C then B is left unsatisfied. Thus the solution given here is not correct.

A Simple Card Trick

This is a simple card trick and it's correctness can be easily shown using the Pigeon-Hole principle.For details about Pigeon-Hole Principle visit http://shashindrasri.blogspot.com/2011/11/pigeon-hole-principle.html.

Card Trick based on Pigeon hole principle from Shashindra Silva

VJD System - Alternative to the Duckworth-Lewis Method

Although the Duckworth-Lewis method is widely used in international and first class level there are many criticisms about this method.One of the main criticism about the DL method is that it gives a higher weight to number of wickets remaining rather than number of overs left. As an example DL par score jumped from 117 to 140 at the end of 20 overs when West Indies lose there 4th wicket in the 20th over at the recently concluded first ODI match between England and West Indies.WI were 1 run ahead before the wicket and suddenly 23 runs behind the target after the wicket.Thus it is very important to a chasing side, not to lose wickets rather than maintaining the required run rate.


Another problem with the DL method is it's inability to handle the power play overs.There is no way to account for the number of power play overs left and number of power play overs played. Also since it uses a G50 score concept where the average score was considered as 250 for an ODI match ,the target scores obtained by DL method was not very justifiable for very low scores and and for very high scores. Thus the effectiveness of the Duckworth-Lewis method is limited to a range of scores distributed around 250. Although this error was corrected for a certain extent in the professional edition still the Duckworth-Lewis method is not that accurate in lower scores and higher scores.

Duckworth-Lewis Curves

Also when multiple interruptions happens the targets given by the Duckworth-Lewis have many anomalies.  The target given by the Duckworth-Lewis method when the interruption happens during an innings break clearly favors the chasing side if the first innings score is high.This is because they have all the ten wickets in hand at the start and the number of overs are reduced. Due to this fact these days captains choose to field when they win the toss if rain is expected during the match.


Also Duckworth-Lewis method was originally designed for ODI matches. Thus when this was used in T20 matches it gives inefficient and unfair results. This may be because the lack of data to construct the resource curves for T20matches. In-fact the unsuitability of the Duckworth-Lewis method in T20 is the main criticism against it.

V. Jayadevan

Thus the need of a new system to be used in rain affected matches have been raised. The main competitor that has challenged the Duckworth-Lewis method is V. Jayadevan method which is used in Indian domestic matches. This is widely known as VJD method.


The designer of this method V. Jayadevan is currently working as an Engineer, Irrigation Department, Kerala. He has spent fifteen years to develop VJD system which is used for calculating revised target due to interruptions in ODI & T20 cricket matches, as an alternative to the existing D/L system.




Previous posts on Duckworth-Lewis method.

1. Duckworth-Lewis Method (part 1)
2. Duckworth-Lewis Method (part 2)
3. Duckworth-Lewis Method (part 3)

The Hawk Eye Technology

Technology has become an integral part of our lives. The use of technology in sports have also been increased. At first it was about broadcasting events worlwide.  But in the modern times technology is used to increase the fare-play in sports. Cricket has been one of the sports that uses technology extensively.  The first instance of using technology in cricket may be the third umpire concept which was first used in a test match in 1992. Although it was just a simple TV replay it helps a lot to take correct decisions on run outs and stumping and sometimes the legality of catches and to decide about boundaries.

But within the last few years more sophisticated technologies have been emerged and they are now been used in the game of cricket. One such technology is the hawk eye technology which is used in UDRS (Umpire Decision Review System).

History

Dr.Paul Hawkins

Hawk-Eye is a ball tracking system which is also used in other sports such as tennis. This technology is controlled by the Hawk-Eye Innovations Ltd which is a part of Sony Europe. Initially this was used as a broadcasting tool in cricket LBW decisions and now has become an important part in the decision making process in cricket matches.

The initial research for the Hawk-Eye technology began in 1999 by Dr.Paul Hawkins at Roke Manor Research Ltd. In the year 2001 channel 4 used this technology in broadcasting the Ashes series. In February 2002 this technology was first used in tennis broadcasting. In 2005 Hawk-Eye was permitted to be used as an officiating aid in tennis. This is mainly used to detect the line-calling decisions. In 2007 MCC world cricket committee announces that the Hawk-Eye will be used in the Decision Review System used in cricket.   In 2011 Hawk-Eye was used in a Cricket World Cup for the first time in history. Also there are plans to use Hawk-Eye in Soccer. This is still at the testing stages of FIFA and if the results are successful, this will be used in 2014 world cup.

Technology

Camera Placement in Tennis

The Hawk-Eye uses six high speed vision processing cameras positioned at different places in the ground along with two broadcast cameras to calculate the trajectory of the ball. Although there are 8 cameras available only 5 cameras are used depending on the side of the wicket that’s been used. In tennis there are around 10 cameras to track the ball. These cameras obtain a 3D visualization of the path of the ball after it’s been bowled. There will be two trajectories for each ball one for releasing the ball by the bowler up to pitching and another one for pitching the ball up to hitting the batsmen. Using this technology the speed of the ball, the swing of the ball from the bowler’s hand to the pitching point, the pitching point, the bounce of the ball, the spin or deviation of the ball can be obtained. Based on the second trajectory the expected path of the ball will be calculated and used to determine whether it will hit the stumps. This is widely used to decide LBW decisions.

Controversies

Camera Placement in Cricket

When technology is used in a sport there will be three very important things to consider. First it should be accurate. Then it should be fast enough so that the momentum of the game is not lost due to technology and finally it should be economically and technically affordable to use in normal games.

Hawk-eye is financially feasible. It is a low cost solution which only needs several cameras and some computers to do the calculation. Since cameras will be always available in an international sporting event the installing cost is very low. Also the time taken to calculate the path of the ball is short but sometimes this may damage the momentum of the game. This is one reason that the number of challenges by the players against the normal decisions taken by umpires or referees is limited. In cricket this is two reviews per team in each inning. Also in tennis the player will get two incorrect challenges per and three challenges in a tie break.

But the problem with the Hawk-eye is its accuracy. Although in tennis they have proved that it has an accuracy of 3.6mm in line calling decisions Hawk-Eye has failed to obtain the trust among players, referees and the general public. Also there are few instances in cricket that the decision given by the hawk-eye system is different than what most of the people expected.

So far they WON all matches,So will they LOSE the next ? ( The Gambler's Fallacy)

Well they won all the test matches against India. And so far all the matches of the CB series. They are not a very strong team as the previous Australian teams who controlled the world cricket arena for more than dozen years. But recently they have managed to win a fair amount of matches continuously.  Although Australia have the home advantage I think all the three teams have a somewhat fair chance of winning a match. The probability of an Australian win may be high but the probability of two consecutive wins will be low. And of course the probability of three continuous wins will be much lower. So Does this mean that Australia have a very high chance of loosing the next game ? or India have a high chance of winning the match on 12th ? or more importantly if Australia win on 12th Sri Lanka have very very high chance of winning the next match against Australia ?

This leads us to a very interesting fallacy in probability named as the Gambler's fallacy or Monte Carlo fallacy. So its time to think about this fallacy leaving cricket for a moment.

The Gambler's Fallacy

                 The basic idea of the gambler's fallacy is based on the perception we have that a result which we have obtained for a lengthy sequence will change in the next event. If we got 5 or more heads in a row while tossing an unbiased coin most of the time we will expect a tail in the next event. Not only we think but also we may incorrectly derive that the probability of a tail is much higher than 0.5 and reaching to 1. Lets check this using an example. Lets assume we have a fair coin so that the probability of a head or a tail equals 0.5. Lets consider the case where we get 5 consecutive heads. Then the probability will be 0.5^5=0.03125 which is a very rare event. So now consider a situation where we have already got 4 consecutive heads. So the big question is what is the probability of getting another head in the next tossing of the coin ? One may solve the problem in the following way.
Since the probability of getting 5 consecutive heads is 0.03125 the probability of not getting a head is 1-0.03125=0.96875. So the probabilities of getting a head and getting a tail is respectively 0.03125 and 0.96875. This reasoning highly favors a tail in the next tossing of the coin and this leads to the famous Gambler's Fallacy.
                            The problem of the above reasoning is that it does not take the independence of each tossing. It may be true that we had 4 consecutive heads so far. But it does not alter the probability of the next event. What has happened so far is simply the past and unlike in some real life scenarios they don't affect the future. There will be no demons or angels from the past events affecting the future when the events are simply independent.
                            So the probability of getting a head in the next tossing is still 0.5.  This confusion is the basic root of the gambler's fallacy. When a same result is observed for a large number of times people tend to think that the opposite will occur in the future. Based on that assumption they will bet on those results. This simple fallacy will lead to many interesting explanations of some events. Also there may be some instances where the gambler's fallacy may not be applied. As an example assumption of independence of events may not be totally correct. 
                             In the scenario at the beginning the events are not independent. There is a very slight dependence. Due to the consecutive wins they  may decide to give an opportunity to a new player or to rest some senior players. Then the Gambler's fallacy will not become a fallacy any more and the other two teams may increase their chance of winning. 

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. 

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)