In the following problems, Θ = 20 + units digit of your day of birth. For example, if you
were born on April 1st, then Θ = 20 + 1 = 21. If you were born on March 30th, then Θ = 20
+ 0 = 20.
1. Alok and Bhanu play the following game on arithmetic expressions. Given the
expression
N = (Θ + A)/B + (Θ + C + D)/E
where A, B, C, D and E are variables representing digits (0 to 9), Alok would like to
maximize N while Bhanu would like to minimize it. Towards this end, they take turns in
instantiating the variables. Alok starts and, at each move, proposes a value (digit 0-9)
and Bhanu substitutes the value for a variable of her choice. Assuming both play to their
optimal strategies, what is the value of N at the end of the game? Also find a sequence
of moves (digits by Alok and variables by Bhanu) that would yield this value.
Note: Moves that lead to a divide-by-zero condition are disallowed. A non-optimal
sequence of moves is (5 → B, 6 → C , 3 → D, 2 → E, 0 → A) and the expression
evaluates to Θ/5 + (Θ+9)/2.
2. The mean, unique mode, median and range of 21 positive integers is 21. What is the
largest value that can be in this sequence? Also find such a sequence.
Note: Given a sequence of numbers a(1) ≤ a(2) ≤ ... ≤ a(n),
The median of the sequence is the middlemost value in the sequence if n is
odd and the average of the two middle values if n is even.
The mode is the most occurring value in the sequence
The range is the difference between the largest and the smallest values, i.e.
a(n) - a(1).
For example, the sequence 2, 3, 4, 6, 6, 9 has mean = (2 + 3 + 4 + 6 + 6 + 9)/6 = 5,
median = (4+6)/2 =5, mode = 6, and range = 9 – 2 = 7.
3. A secret message is divided into Θ parts and each part is shared with a different
person. People communicate with each other using two-way phone calls and, in each
communication, share all the information they know until that point. What is the minimum
number of communications required for all Θ of them to know the secret? Find a
sequence of communications that achieves this minimum.
4. An equilateral triangle ABC with sides of length Θ cm is placed inside a square AXYZ
with sides of length 2*Θ cm so that side AB of triangle is along the base of the square
(as shown). The triangle is rotated clockwise about B, then C and so on along the sides
of the square until the points A, B and C return to their original positions. Find the length
of the path (in cm) traversed by point C.
5. A bag contains printed articles of 4 different kinds: periodicals, novels, newspapers
and hardcovers. When 4 articles are drawn from the bag without replacement, the
following events are equally likely:
the selection of 4 periodicals
the selection of 1 novel and 3 periodicals
the selection of 1 newspaper, 1 novel and 2 periodicals and
the selection of 1 article of each kind
What is the smallest number of articles in the bag satisfying these conditions? How
many of these are of each kind?
6. Given a 9 x Θ chessboard, a rook is placed at the lower left corner. Players A and B
take turns moving the rook. A plays first and each turn consists of moving the rook
horizontally to the right or vertically above. The last person to make a move wins the
game. At the completion of the game, the rook will be at the top right corner. For
example, the figure below shows a 3 x 4 chessboard and the sequence of moves that
leads to a win for player A.
Does player A have a winning strategy in the given 9 x Θ chessboard? If so, what is the
strategy? If not, what is player B's winning strategy?
7. A spaceship on an inter-galactic tour has to transfer some cargo from a base camp to
a station 100 light sec away through an asteroid belt. The ship can carry a maximum of
100 kgs of cargo and, as a result of colliding against the asteroids, every 2 light sec of
travel causes it to lose 1 kg of cargo. There are 300 kgs of cargo available at the base
camp. Find the maximum amount of cargo (in kg) that the ship can transfer to the
station? Assume that the spaceship can store the cargo at any intermediate point along
the way and that stored cargo is not depleted by the asteroids.
Prerequisites
Please answer as many questions as you can.
