Assorted Brain Teasers by Kundan Pangtey - HTML preview

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Imagine  a  contest  of  beauty  pageant.  Participants  come  onto stage  one  by  one  in  front  of  Judges  for  each  event.  The participants  and  Judges  do  not  know  each  other.  The  Judges are asked to award points on a scale from 1 to 10 (no fractions) to each candidate.

If  you are made one of  the judge and is asked to choose one out  of  following  choices  for  the  first  participant  in  first  event which one would you choose ?




The contests are generally conducted to select a winner from among a  group  of  contestants.  In  a  contest,  it  is  not  important  how  much marks  or  points  (absolute)  an  individual  gets,  rather  the  whole process  is  to  make  a  proper  sorting  from  among  a  chaotic  group through a comparative analysis. Sorting cannot be done in isolation, it  can  only  be  done  if  there  are  more  than  one  candidates  put together and compared together. If the winner candidate gets 5 point on a scale of 1 to 10, it does not mean that she or he is not worth 10, it merely indicates that all other candidates have points less than 5.

When 1st  candidate comes onto stage, judges may find it difficult to award a point as they would not know whether she is best or worst among all candidates. Judges would normally rely on their own past experience  to  make  an  imaginary  benchmark   or  standard  and attempt a comparison with that. Therefore the whole process would be  highly  influenced  by  an  individual’s  experience  which  would amount to biased judgment, especially during initial stage.

If 1st  candidate is awarded 1 point, for example, it would mean that an  assumption  has  been  made  that  all  remaining  candidates  are better than her and would get either one or more than 1 point. Now if all remaining participants turn out to be worst than 1st  candidate then there would be a big dilemma as no numbers are available less than one  hence  all  shall  end  up  having  1  point  each.  Similarly  if  1st candidate  is given  10  points  same  situation  would  arise  in  reverse order

Let us compare beauty contest (crude comparison) to a “ball game” where participants are replaced by balls of various sizes chaotically placed in a cluster. These balls are required to be sorted out based on their sizes.



The balls are ten in numbers and named as A, B, C, D, E, F, G, H, J & K and judges award points on the basis of their sizes on a scale of 1 to 10 (no fractions). The smallest one gets 1 and largest one gets 10 points. The sizes of balls are not known to judges prior to contest.



The name and diameter of balls are shown in a table below:


Let us imagine that D is the first one paraded in front of judges.

• If D is awarded 1 or 2 (minimum marks), the result would be correct up to 99%.

•     However, if D is awarded 10 or 9 marks (maximum) the result will be 99% wrong.

•     In case D is awarded 5 marks (mid point of  scale), at least half of the remaining candidates would get correct markings.

•     Suppose  E  is  the first  candidate  paraded  in  front  of  judges and is awarded 1 or 2 marks (minimum), the result of contest would   be   50%   correct   and   if   awarded   10   or   9   marks (maximum)  still  result  would  be  50%  correct,  however,  if  E gets 5 point (middle), the result would be 100% correct.

Awarding  of  marks  at  extreme  end  is  a  total  gamble  and  there  is every danger  of  going  wrong  by 99%  and  candidates  may end  up getting incorrect marks.

To  be  on  safe  side  the  correct  way  would  be  to  choose  the  mid- point, i.e.5 so that a fair amount of points are available on either side and a dead end is avoided.

Following conclusion can be drawn from the analysis;

(a) If points are picked up from extreme ends of a scale (1, 2 or 9, 10) there is a danger of going 100% wrong or right by 0%. So the probability of choosing points correctly would always vary from 0%  to 100%.

(b) However,  if  points  are  chosen  from  middle  of  the  scale  (5) the  answer  would  always  be  50%  to  100%  right  and  will never  be  100%  wrong.  The  probability  of  choosing  points correctly would always vary from 50% to 100% and will never be less than 50%.

(c)  Therefore the correct way would be to choose the mid-point for  initial  participants  so  that  the  danger  of  going  wrong  by 100%  is  avoided  and  a  fair  amount  of  points  are  made available on either side for awarding later participants and a dead end is averted.