TCS Ignite Open Lab

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TCS Ignite Open LabThe Craft of Problem Solving

I hear and I forget; I see and I remember; I do and I understand.

- ConfuciusChinese philosopher (551 – 479 BC)

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Cover Page:

Lifting of Govardhan HillThe hand drawn image on the cover depicts the lifting of Govardhan hill by Krishna as

found in Krishna Mandapa in Mammalapuram near Chennai. Trainees here at Ignite are

called to a life of humility, patience and calmness to bring out their fuller self and

overcome obstacles.

Krishna and the Govardhan Hill © by J.P. Ignite, TCS

The ArtistJ Prabhakar, known as J.P., is a Chennai based artist who specializes in pen-and-ink

line drawings. With no formal education in art, J.P. is entirely self-schooled. Although

his work is focused exclusively on the sacred arts and his themes tend to be temples,

monuments and sculptures, he innovates constantly in terms of technique, technology,

materials, form and content. His pursuit of excellence is a constant source of inspiration.

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The TCS Ignite Open Lab – The Craft of Problem Solving

The Challenge

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. You can solve any three of the seven given problems.

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.

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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

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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

Scope and Effort

Total expected effort is about 1 person day. This means that one person will be able to

solve these problems in a day.

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Prerequisites

Any final year science graduate registered for the Open Challenge can attempt this

challenge. The challenge only requires knowledge of basic mathematics and a general

mathematical thinking ability.

Submission Process

You will need to submit a document containing the solution to the problems as well as

the steps you have followed to arrive at the solution. The submissions can be in txt, .doc

or .pdf format

Evaluation

You will be evaluated based on the following parameters.

S.No. Parameter

1. Problem comprehension

2. Correctness of solution

3. Elegance of your solution