7 problem solving questions. Help me answer it.

[mkerala]

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.

[hypervalent_iodine]

Since this is homework, you'll have to show us what working you've done and specific points where you need help or simple clarification. We aren't here to do your work for you; that part is entirely up to you.

[mkerala]

Since this is homework, you'll have to show us what working you've done and specific points where you need help or simple clarification. We aren't here to do your work for you; that part is entirely up to you.

 

1. didnt understand the question properly

 

2. 7, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 28, 28.

 

3. (Θ-1) times for the first person to know everything. as he communicates with the last person he also tells everything he knows to that person so the remaining (Θ-2) people should call once atleast to know everything which adds upto --- (Θ-1)+(Θ-2)

 

4. while moving around a fixed point the side will become the radius for the arc. it takes three rounds for c to come tothe original position and on calculating the whole thing it comes to 2*22/7*Θ*10/3

 

5. a minimum of 7 articles total. 4-periodicals, 1-novel, 1-newspaper, 1-hardcover

 

6. player A can win easily as Θ > 20. he has to make sure that he moves to the right end in more than 9 steps

 

7. 150kgs (i assumed empty ship has no wt loss)

 

Are these answer correct?

Edited March 13, 2012 by mkerala

[imatfaal]

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.

 

To explain what I think this means

1. Θ is a whole number between 20 and 29

2. Your equation looks like this

 

[math] N= \frac{\Theta +A}{B} + \frac{\Theta +C +D}{E}[/math]

3. alok (a) wants to maximise the end result of that equation, bhanu (b) wants to minimize it

4. each round of the game a chooses a number between 0 and 9

5. and b chooses which of the 5 variables it will replace

eg.

Round 1. a - I choose 2. b - I choose to make that replace C

 

[math] N= \frac{\Theta +A}{B} + \frac{\Theta +2 +D}{E}[/math]

Round 2. a - I choose 9. b - I choose to make that replace B

 

[math] N= \frac{\Theta +A}{9} + \frac{\Theta +2 +D}{E}[/math]

and so on for 5 rounds to replace A,B,C,D, and E

6. The equation is enumerated. As a and b want different results each will/might have a strategy for maximising or minimizing the value of N

7. I presume the optimal solution exists - but I would have to think about ti for a while

 

2. Your answer is incorrect. FYG I can get considerable higher

 

7, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 28, 28.

 

why not change the 2nd number from 21 to 19, then you could add one to the 1st and the last numbers with out changing any of your averages or range

 

8, 19, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 28, 29. Carry on doing this sort of swapping value.

 

Remember

1. Mode is the most common. Median is the 11th. Mean is the sum divided by 21. Range is the biggest less the smallest.

2. Use a spreadsheet - unless that is forbidden. They will have functions to provide the averages - just to make it a bit easier.

3. You need to take away from the middle to weight the top end - however there is no good reason to have that 28 in second last place

 

[imatfaal]

question 3 - got the same although i would explain it differently; its easier for me thinking of person A calling everybody once, then calling everybody again apart from the last person.

 

question 4

 

I get a different answer. For each lap of the triangle (of which you need three) you get 3 steps of [imath] \frac{2\pi}{3} [/imath] or 120degrees , 3 steps of [imath] \frac{\pi}{6}[/imath] or 30degrees, and two times the corner is the pivot. For each lap you get [imath] 3*\frac{2\pi}{3} + 3*\frac{\pi}{6} + 3*0 = \frac{5\pi}{2} [/imath]. In degrees that's 450degrees. And doesnt match your sums

[imatfaal]

6. player A can win easily as Θ > 20. he has to make sure that he moves to the right end in more than 9 steps

 

But how would he do that? What stops B from doing what he wants - he can move any number of spaces to right!

 

There is a complete strategy that guarantees a win for one of the sides. Draw out a 5x4 board. Work from the goal backwards - mark squares as "if I leave the rook here I must lose" or "if I leave the rook here I must win". The first square is the top-right, thats a "must win" (ie its the goal). what happens if you leave it on the rightmost column but not in the top square ? or in top row but not right hand square? Carry on backwards - you will soon notice the pattern of "must win" squares

 

 

7. 150kgs (i assumed empty ship has no wt loss)

Nope. That's what you get if you make 3 simple voyages from base camp to station. Try thinking in terms of percentage or ratio of cargo lost per mile - rather than just tonnes of cargo lost per mile.

 

You lose 1 tonne per mile if the ship is full (1/100) and 1 tonnes per mile when it is half empty (1/50). You want the ship to travel with as the lowest loss ratio - so you want it to travel full more of hte time. Note there is no penalty for setting up intermediate stations.