Directions for Questions 1
and 2:
Thus, if n is even, then n/2 players
move on to the next round while if n is odd, then (n + 1)/2 players move on to
the next round. The process is continued till the final round, which obviously
is played between two players. The winner in the final round is the champion of
the tournament.
Solution:
Choose (1) if Q can be answered
from A alone but not from B alone.
Choose (2) if Q can be answered
from B alone but not from A alone.
Choose (3) if Q can be answered
from A alone as well as from B alone.
Choose (4) if Q can be answered
from A and B together but not from any of them alone.
Choose (5) if Q cannot be
answered even from A and B together.
In a single elimination
tournament, any player is eliminated with a single loss. The tournament is
played in multiple rounds subject to the following rules:
(a) If the number of players, say n, in any round
is even, then the players are grouped in to n/2 pairs. The players in each pair play a match
against each other and the winner moves on to the next round.
(b) If the number of players, say n, in any round
is odd, then one of them is given a bye, that is, he automatically moves on to
the next round. The remaining (n − 1) players are grouped into (n −
1)/2 pairs. The players in each pair play a match against each other and the
winners move on to the next round. No player gets more than one bye in the
entire tournament.
1.
Q: What is the number of matches played by the champion?
A: The entry list for the tournament
consists of 83 players.
B:
The champion received one bye.
2. Q: If the number of players, say
n, in the first round was between 65 and 128, then what is the exact value of n?
A: Exactly one player received a bye
in the entire tournament.
B: One player
received a bye while moving on to the fourth round from third roundSolution:
Q1)
To find the
number of matches played by the champion:
StatementA
specifies that the entries are 83.
64 < 83
< 128 => 2^{6} < 83 < 2^{7} => the number of
rounds = 7
no. of players

RoundI

RoundII

RoundIII

RoundIV

RoundV

RoundVI

RoundVII

83

41+1

21

10+1

5+1

3

1+1

1

(Observe
that ‘bye’s are indicated in colour)
If the
champion has not received any bye, then he has to play 7 matches,
But if he has
received a bye, then he has to play 6 matches. Hence it is not possible to decide
it from the statement A alone and statement B is also required to answer the
question.
Answer (4)
Q2)
The numbers
when repeatedly divided by 2 – always leaving remainder zero until it ultimately
becomes 1 are the powers of 2:
2^{0},2^{1},2^{2},2^{3},2^{4},2^{5}....
ie.,
1,2,4,8,16,32,64,.....
The numbers
when repeatedly divided by 2 – always leaving remainder zero but exactly one time leaving 1 until it ultimately
becomes 1 are in the form of (2^{n}1)2^{k}, where n > 1, k ≥
0 and n,k being positive integers:
3,7,15,31,63,.....
and 3*2 = 6, 3*4 = 12, 3*8 = 24,.....
and 7*2 = 14, 7*4 = 28, 7*8 = 56,.....
and so on...
Now this set
of numbers is to be considered here for this problem as it is mentioned that “Exactly one player received a
bye in the entire tournament”
As “the
number should be between 65 and 128”, the possibilities are:
3*32 = 96; 7*16 = 112; 15*8
= 120; 31*4 = 124; 63*2 = 126; 127*1
= 127
Now we check
up case by case for the occurrence of ‘bye’s:
no.
of players

RoundI

RoundII

RoundIII

RoundIV

RoundV

RoundVI

RoundVII


96

48

24

12

6

3

1+1

1

one
player received bye while moving on to 7th round from 6th round

112

56

28

14

7

3+1

2

1

one
player received bye while moving on to 6th round from 5th round

120

60

30

15

7+1

4

2

1

one
player received bye while moving on to 5th round from 4th round

124

62

31

15+1

8

4

2

1

one
player received bye while moving on to 4th round from 3rd round

126

63

31+1

16

8

4

2

1

one
player received bye while moving on to 3rd round from 2nd round

127

63+1

32

16

8

4

2

1

one
player received bye while moving on to 2nd round from 1st round

(Observe
that ‘bye’s are indicated in colour)
Now it is
evident that, occurrence of onetime bye can happen in six cases. Hence it is
not possible to decide it from the statementA alone and statementB is also
required to answer the question.
Answer (4)
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