@[283374:viveknitw]
Ingredients :
Note that in every round 1st, 4th, 7th, 10th..contestant get eliminated.
ie. in every elimination round, (3n+1)th contestant is struck off. 'n' here starts from 0.
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Lets make the dough now :
Start with 150 contestants, read terms, if every (3n+1)th term is to be eliminated, the number of terms eliminated will be 50, since (3.49+1)=148, which is the last term to be struck off.
Initialy number of terms in Round 0) 150
No. of terms left in Round 1) 150 - 50 = 100
Now, for second round we have 100 terms, we again eliminate every (3n+1)th term. Since 100 = 3.33 + 1. The last term will get eliminated in this round. And total number of terms struck off in this round = 33 + 1 = 34, as we start n from 0
So, No. of terms left in Round 2) 100 - 34 = 66
Now for third round we have 66 terms, we again eliminate every (3n+1)th term. Since 66=3.22. This ensures that 66th term will not get eliminated since we are looking for 1 more than the mutliple of 3. So, which term wil get eliminated from the end. The (21.3+1)th or 64th term. And hence the total number of terms eliminated in this round = 21 + 1 = 22
So, No. of terms left in Round 3) 66 - 22 = 44
Continuing in this manner, we obtain the number of terms left after each round as
No. of terms left in Round 4) 44 - (14+1) = 29 ................since 14*3 = 42
No. of terms left in Round 5) 29 - (9+1) = 19 ................since 9*3 = 27
No. of terms left in Round 6) 19 - (6+1) = 12
No. of terms left in Round 7) 12 - (3+1) = 8
No. of terms left in Round 8) 8 - ( 2+1) = 5
No. of terms left in Round 9) 5 - (1+1) = 3
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The Final Recipe :
Now since we know the number of terms eliminated in each round as well the no. of terms remaining in subsequent round we can start with the process.
Let us write down the number tags of each contestant.
In Round 0), we have all 150 terms/contestants
*1 2 3 *4........................140 141 *142 143 144 *145 146 147 *148 149 150
The contestants who are to be eliminated are awarded a * , a consolation prize.
After the terms with * are struck off we proceed to
Round 1)
The terms left in this round are 100, writing them below :
*2 3 5 ............................140 *141 143 144 *146 147 149(99th term) *150(100th term)
Eliminating the terms with *, we proceed to
Round 2)
The terms left in this round should be 66. The last term which is 66th term cannot be eliminated since it is a mutiple of 66=3.22. So the term to be eliminated is 3.(22-1) = 63rd term
*3.........................................140 143 *144(64th term) 147(65th term) 149(66th term)
Round 3) : 44 terms remaining in this round
............................................. 140 143(42nd term) *147(43rd term) 149(44th term)
3.14+1 = 43rd term is the last term to be struck off.
Round 4) : 29 terms remaining in this round
.............................................. 140 *143(28th term) 149(29th term)
3.9 + 1 = 28th term is the last term to be struck off.
Round 5) : 19 terms remaining in this round
............................................... 140(18th term) *149(19th term)
3.6 + 1 = 19th term is the last term to be struck off.
Round 6) : 12 terms remaining in this round
................................................ 140(12th term)
Note that the number of terms is a multiple of 3, so the last term wont get eliminated.
Round 7) : 8 terms remaining in this round and 140 = last ie 8th term.
Note that 3.2+1 = 7th term is the last term to be struck off. So 8th term which is 140 still remains
Round 8) : 5 terms remaining in this round and 140 = last ie 5th term.
Note that 3.1+1 = 4th term is the last term to be struck off. So 5th term which is 140 still remains
Round 9) : 3 terms or contestants remaining in this round
let them be Miss Australia(with number tag X)..................Miss Wales(with number tag Y)............ and Miss China(with tag 140)
As per the strange but interesting rule of the contest, we eliminate 1st contestant and are left with
in Round 10)
Miss Wales(with tag no. Y)............ and Miss China(with tag 140)
Eliminatin again 1st contestant we are left with winner of the competition.
Miss China(with tag no. 140)
So the number tag of winner is 140........Answer
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Garnishing the dish now :
Important thing to note here is that there is no need to bother about the terms in the starting of the series.
After round (2), I stopped writing the numbers in the beginning of the series. I was only interested in the numbers in the last of series.
Since I knew this beforehand I made sure to write atleast last 11 numbers in the series.
How did I come to know this? Did 2,3 dry runs with few numbers and noticed that the winner is always the one who stands towards the end of the series.
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Is this the correct answer?
This method sure required a lot of brainstorming.
Do you know any other method to solve this Qn?
If you do pls share.
ashish0102