### Technothlon Previous Question Papers- Hauts Squad 2015

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**HAUTS SQUAD – 2015**

##### Chess With A Twist

**Question 1:**

Consider a chessboard of dimensions 4xN. What is the value of N among the following such that, you can start with a knight(horse) at some square of the board, proceed by valid moves, visit each square exactly once and can return to the starting point as the 4N+1st square?

[Note: If the knight is at a position as shown gure, a valid knight move would be any square marked ‘X’]

(A) 2 (B) 4 (C) 8 (D) no N exists

**Question 2:**

Consider a chessboard of dimensions 4xN. What is the value of N among the following such that, you can start with a knight(horse) at some square of the board, proceed by valid moves, and can visit each square exactly once, with no requirement that we ever return to the starting square?

(A) 2 (B) 4 (C) 8 (D) No N exists

**Question 3:**

Find out the minimum number of moves required to move from the situation in the left to that in the right if the valid moves for bishop, rook, knight, king and pawn are shown in the gures below respectively.

(A) 9 (B) 18 (C) 16 (D) 8

You and your friend save as much money as possible by participating in the contest and y to Athens.Exploring Athens you and all other teams come to Parthenon, the temple of Athena, who is the goddess of wisdom and intelligence. You also see many images of an owl. You learn from the guide that it is the“Athena Noctua”, also called the Minerva owl or the little owl. The guide also mentions some interesting stories of Athena’s wisdom and cleverness.

The tourist guide at Parthenon takes you to an abandoned tunnel. And here you nd an inscribed stone and a bowl of colored owers. Suddenly you hear a deep and ancient gong ringing as you start reading the Stone Of Instructions. The instructions read as follows:

Every person entering the Temple of Athena will be given a bowl with 15 red owers and 12 yellow owers. Each time the Gong of Time rings, he/she must do one of two things:

1. Exchange: If he has at least 3 red owers in his bowl, then he may exchange 3 red owers for 2 yellow owers.

2. Swap: He may replace each yellow ower in his bowl with a red ower and replace each red ower in his bowl with a yellow ower. That is, if he starts with i red owers and j yellow owers, then after he performs this operation, he will have j red owers and i yellow owers.

[You can leave the tunnel only when you make all the combinations (no. of red, no. of yellow) of owers

that are possible].

**Question 4:**

After how many operations will you have 5 each of red and yellow owers?

(A) 5 (B) 10 (C) 15 (D) None of the above

**Question 5:**

What is the number of combinations you need to make to leave the tunnel?

(A) 27 (B) 52 (C) 54 (D) None of the above

The next day, on your way back from Parthenon, you see an owl injured under a large, old tree. You decide to save the owl and x its wing, so you take it along with you. Later you get to know that owl needs to reach IIT Guwahati, India as soon as possible. Realizing how dangerous it could be for the owl to travel to India, you decide to help the owl reach its destination.

**Question 6:**

You and your friend along with the owl now reach Turkey. Tired by your journey, you decide to go a bit sightseeing in the town of Istanbul. Inside the Hagia Sophia, while you all are walking, you encounter a weird looking signboard, which reads“Lay down the letters,Left it begins,Right it goes,Starting a new line”.

A trapdoor opens below you and you fall onto a gigantic chessboard. Overhead, you see a clock, which is rotating in the anticlockwise direction. Seeing this, you get puzzled.

Realizing that the only exit is an elevator across the board, you rush to it. There you nd a ghost guarding it. When asked about the exit, the ghost replies.The board is all you have, follow lines, stay away from the edges, and watch the clock as it goes.You realize that lift is not the exit and ask the Ghost what is to be done. He then gives you two words”Powder and “Magic” and tells you to enter these words in their secret language following the clues. You decoded “Powder” as “hfvkma” . If this is the right word then what is “Magic” in their secret language?

(A) erfpk

(B) rfpkd

(C) hfvrp

(D) erfpd

Another break comes in your journey when you encounter a long wall in Aleppo. The wall is made of bricks of dierent sizes and if you put all the bricks in order of their sizes, you will get a staircase and be able to cross it. So, you have to sort the bricks such that the smallest is on the top and largest at the bottom. To re-order them, you can hold any number of bricks from the top of pile, lift them, invert the pile you are holding and place the inverted pile back on the rest of the bricks below. The move has been explained in the below diagram:

The above diagram indicates one move or step. By doing a certain number of appropriate moves, you can bring the entire pile of bricks to suit your need.The wall stopping your way looks like the gure shown:

**Question 7:**

What is the minimum no. of moves required?

(A) 10 (B) 9 (C) 8 (D) 12

After crossing the wall, your journey hits another wall (see below), but this time sorted and built of hollow numbered blocks. On reaching the top of it, you see a signboard which says: “From the top block, move down one layer a time to either of the two touching blocks just below the one where you are. At the bottom, you will nd a glider which ies for as much time as the sum of the numbers on the blocks through which you pass”. As you would want the glider to y for as much time as possible, you must choose the path which gives you the maximum sum.

**Question 8:**

Find the path and give the maximum time for which the glider can be own.

(A) 248 (B) 335 (C) 249 (D) 250

**Question 9:**

Can you nd how many paths pass through the block ‘55’ ?

(A) 48 (B) 12 (C) 24 (D) 10

While travelling through the Syrian Desert, you are in desperate need of water. The owl, still bursting with energy(God knows how) comes ying back to you to tell you that she had located a water faucet nearby.Hearing this, you rush to it, to nd a real water cooler. But the problem is, it has seven pieces lying on the ground which have to be tted to form a rectangle in order to have the faucet working.

**Question 10:**

Which of the following statements are correct about the rectangle formed?

(A) The corner four pieces are 1,3,5,7

(B) The corner four pieces are 1,2,5,6

(C) The corner four pieces are 2,4,5,7

(D) Rectangle cannot be formed

Suddenly, the Desert God emerges out of the faucet. He says, “I am pleased with your hard-work, and will reward you with food if you pass my challenge.” He gives you innite number pieces of ve dierent types as shown below with numbers etched on them. He then tells you to use a maximum of 30 of these pieces to make a rectangle such that the sum of the numbers on all of the small squares is maximum. Also, each type should be used at least once.

**Question 11:**

What is maximum sum of all the numbers that is possible?

(A) 137

(B) 136

(C) 127

(D) Rectangle cannot be formed

As you reach Iraq, you come across a poster mentioning a contest which could help you earn more money for your trip. So you decide to participate and below are the questions of the contest.

**Question 12:**

If you want to build a pile of 100 coins (1 and 2 dinars) such that, number of coins between any two ‘2’ dinar

coins is not equal to ve, then what are the maximum number of 2 dinar coins that can be included in the pile?

**Question 13:**

Minimum length of ruler required with 4 marks such that every two marks are at a dierent distance is 6 as

shown in the diagram.

Similarly, what is the minimum length of ruler required with 6 marks such that every two marks are at a different distance?

**Question 14:**

The government of Iraq wants to issue ‘d’ denominations of coins (in whole numbers of dinars(Iraqi currency)) so that by using no more than 3 coins, citizens can pay any amount from 1 dinar to 36 dinar. Find the value of ‘d’ and all the ‘d’ denominations and give the sum of all these ‘d’ denominations?

(Ex: if d=3 and 3 denominations are {4,5,3}, answer= 4+5+3= 12.)

As you travel from Iraq towards IIT Guwahati, after a long time you nd yourselves on a mountain trail at Tehran, Iran. The mountain trail is dicult and slippery. You suddenly get caught in a landslide and fall into a deep hole. Then the stones of 2 kg weight start falling onto you from above.The hole is a 4×4 grid i.e., it is divided into 16 squares. Each square can accommodate only one stone (of any weight). To survive, you need at least one empty square. Owl notices your trouble and gives you the magical power to meld together stones, but only those of the same mass can be mould together. Since the stones are falling very fast, you are only able to push the stones in straight lines (horizontally/vertically).For example, when the arrangement of one of the rows in gure 1 when pushed horizontally to left transforms as shown:

**Question 15:**

If one stone falls every second, what is the maximum time for which you can survive i.e., when no square is empty and all stones become immovable?

(A) (2^16) -1 (B) (2^17) -1

(C) (2^17) – 2 (D) 2^16

**Question 16:**

What is the largest stone that you can make if every second, either 2kg stone or 4 kg stones fall?

(A) 2^17 (B) 2^16

(C) (2^16) -1 (D) (2^17)-2

After a day lled with adventures in Tehran, you now reach Kandahar. You realize that you are falling short of time and should reach Kabul as early as possible but the problem is that you have lost your map and you can’t see anybody nearby. So, you don’t know where to head towards. You suddenly see a Kabuliwala, whom you ask for help.

He says, he will show you the way to Kabul only if you help him get his treasure. Then he takes you to a place which looks similar to a grid, which has some rocks, a xed metal ball gun and a treasure box at the other end which can be opened only when the metal ball hits the 3 buttons. Now help the Kabuliwala arrange those blocks in order to open the treasure box.

**INSTRUCTIONS:-**

1. The initial direction of ball cannot be changed. The metal ball once released only travels in straight lines unless and until it hits a rock.

2. Rocks reect the ball when it hits them.

The place where you go looks something like the above gure. For the arrangement shown in the gure,the metal ball could only hit one button. Make appropriate shifts in the positions of rocks so that the ball can hit all the three buttons.

**Question 17:**

What is the sum of all the positions that have rock (including xed) on them?

(Two digit integer type)

After the kabuliwala leaves, you nd another similar grid. Solve this one to have a treasure for yourself :-

WORMHOLE ROCK :- Acts as a teleport for the ball. How it exactly works is shown in the below 2 gures.

**Question 18:**

What is the sum of all the positions that have rock (including wormholes) on them?

(Two digit integer type)

You then head towards Kabul. After reaching there, you come to know that the King’s descendant found an ancient-Treasure Box. One of the men of the ancient court discovered that the code that unravels the treasure, is hidden in the below puzzles. Help the descendant and get 10% of the treasure as reward.

**PUZZLE 1:**

You are given ten ‘L’ shaped tiles of size 1 through 10. Stack tiles so that each tile is fully supported and also the stack is as high as possible. For example, we can stack L’s of side 1 through 5 so that the stack is 18 units high and stack L’s of side 1 through 6 so that stack is 26 units high as shown in gures below.

[Note: It is not necessary that all the L’s should be present from one to 10]

**Question 19:**

What is the maximum height of the stack formed on stacking tiles of side 1 through 10? And now, the shape of the stack gives you the shape of the key that unlocks the box but you still don’t know the size (in terms of number) of the key, which is hidden in the next puzzle.

**PUZZLE 2:**

A large square can be formed by small integer sided squares so that each square either lies on the bottom

layer or lies on top of one or more larger squares. Some examples are shown below:

**Question 19:**

What is the maximum height of the stack formed on stacking tiles of side 1 through 10?And now, the shape of the stack gives you the shape of the key that unlocks the box but you still don’t know the size (in terms of number) of the key, which is hidden in the next puzzle.

**PUZZLE 2:**

A large square can be formed by small integer sided squares so that each square either lies on the bottom layer or lies on top of one or more larger squares. Some examples are shown below:

**Question 20:**

What is the smallest odd square that can be formed using above method?

You are running out of time. You now reach the outskirts of Nepal but have no idea about the route to Assam. You then ask a person nearby for help. He helps you but through a puzzle.He shows you 12 boxes of dierent sizes, each having two cavities(right and left) in which smaller boxes can be kept. He then numbers each box and places one inside the other randomly without showing you.The nal result is a single box. He says, he will give you the correct map only if you tell which box is in the right cavity of box numbered ‘2’. If you nd out the number of that box you will get the map.Here are your clues:

1. If he unpacks all the boxes and arrange them in a line in the following way,

then the nal numbering of boxes in which the boxes are arranged would be

2. If he unpacks all the boxes and arrange them in a line in the following way,

then the nal numbering of boxes in which the boxes are arranged would be

**Question 21:**

Which box is in the right cavity of ‘2’?

(A) 11 (B) 6 (C) 7 (D) 5

**Question 22:**

If you found out the right box then also tell how many boxes of smallest size did he show you initially?

(A) 2 (B) 3 (C) 4 (D) 5

The unfolding of the map came as a complete surprise. It was not the whole map but just a part of it. The instructions below will tell you how to get the other parts.The rest of the parts are hidden in the houses shown in the gure below. You will get the parts of the map only on delivering the right article(one per house) to all the houses.

(i) An ‘X’ coloured ARTICLE should be delivered to an ‘X’ coloured HOUSE only by using an ‘X’ colouredTRUCK.

(ii) A truck can also carry other coloured articles so that it can place them at any of the CHECKPOINTS from where other truck can later carry it to the destination.

(iii) Trucks can carry any number of articles at a time.

(iv) The main objective is to start all the trucks at the same time, with same speed and deliver the articles without collision of trucks. Also, a truck picks up every article that comes on its way.

(v) Last and the most importantly, the PATHS of the trucks should not overlap at any point other than junctions. At junctions, the paths can though cross each other (but both trucks should not reach that junction at same time which leads to collision). The trucks can move only forward.

**Question 23:**

Draw the nal path of all the trucks and give the number of turns taken by the blue truck.

(A) 8 (B) 7 (C) 6 (D) None of the above

**Question 24:**

Give the total number of articles carried by the red truck on it’s way.

(A) 1 (B) 2 (C) 3 (D) 4

You come to the end of the journey and the owl in your arms utters its wings and ies above your head to face you. A light glow lls the air around it as it speaks:”Athena Noctua I am called. Sent by the goddess of Wisdom herself to help you on this and select the best members on the earth to fulll the ancient prophecy of Athena. The good you’ve collected on your way will decide your fate. Till we meet again! ” and the owl you held so dear to you for so long ies o leaving you waiting for the results.

**2015 HAUTS SQUAD KEY**

CHESS WITH A TWIST

1. D

2. C

3. B

TROUBLE WITH FLOWERS

4. D

5. B

CRACK THE CODE

6. A

PYRAMID WALLS

7. C

8. C

9. C

FRAGMENTS

10. D

11. 137×4 (MARKS TO ALL)

ENIGMATHIC

12. 52

13. 17

14. 40

COALESCE

15. A

16. A

KABULIWALA’S TREASURE

17. 71

18. 45

TILING

19. 56

20. 21

FINAL HUNT

21. B

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