The 12 marbles appear to be identical. In fact, 11 of them are identical, and one is of a different weight. Your task is to identify the unusual marble and discard it. You are allowed to use the scales three times if you wish, but no more.

Note that the unusual marble may be heavier or lighter than the others. You are asked to both identify it and determine whether it is heavy or light.

52. You are a bug sitting in one corner of a cubic room. You wish to walk (no flying) to the extreme opposite corner (the one farthest from you). Describe the shortest path that you can walk.

53. A mythical city contains 100,000 married couples but no children. Each family wishes to “continue the male line”, but they do not wish to over-populate. So, each family has one baby per annum until the arrival of the first boy. For example, if (at some future date) a family has five children, then it must be either that they are all girls, and another child is planned, or that there are four girls and one boy, and no more children are planned. Assume that children are equally likely to be born male or female.

54. There are 100 light bulbs lined up in a row in a long room. Each bulb has its own switch and is currently switched off. The room has an entry door and an exit door. There are 100 people lined up outside the entry door. Each bulb is numbered consecutively from 1 to 100. So is each person.

Person No. 1 enters the room, switches on every bulb, and exits. Person No. 2 enters and flips the switch on every second bulb (turning off bulbs 2, 4, 6, …). Person No. 3 enters and flips the switch on every third bulb (changing the state on bulbs 3, 6, 9, …). This continues until all 100 people have passed through the room.

What is the final state of bulb No. 64? And how many of the light bulbs are illuminated after the 100th person has passed through the room?

55. You and I are to play a competitive game. We shall take it in turns to call out integers. The first person to call out “50” wins. The rules are as follows:

• The player who starts must call out an integer between 1 and 10, inclusive;

• A new number called out must exceed the most recent number called by at least one and by no more than 10.

Do you want to go first, and if so, what is your strategy?

56. You are standing at the centre of a circular field of radius R. The field has a low wire fence around it. Attached to the wire fence (and restricted to running around the perimeter) is a large, sharp-fanged, hungry dog. You can run at speed v, while the dog can run four times as fast. What is your running strategy to escape the field?

57. Why is that if p is a prime number bigger than 3, then p2-1 is always divisible by 24 with no remainder?

58. You have a chessboard (8×8) plus a big box of dominoes (each 2×1). I use a marker pen to put an “X” in the squares at coordinates (1, 1) and (8, 8) - a pair of diagonally opposing corners. Is it possible to cover the remaining 62 squares using the dominoes without any of them sticking out over the edge of the board and without any of them overlapping? You cannot let the dominoes stand on their ends.

59. How many consecutive zeros are there at the end of 100! (100 factorial). How would your solution change if there problem were in base 5? How about in Binary?

60. You're a farmer. You're going to a market to buy some animals. On the market there are 3 types of animals for sale. You can buy:

Horses for £10 each, goats for £1 each and ducks, you get 8 of these per bunch and each bunch costs £1.

The aim is to acquire 100 animals at the cost of £100, what is the combination of horses, goats and duck that allows you to do this? (you must buy at least one of each.)

61. A man has built three houses. Nearby there are gas water and electric plants. The man wishes to connect all three houses to each of the gas, water and electricity supplies.

Unfortunately the pipes and cables must not cross each other. How would you connect each of the 3 houses to each of the gas, water and electricity supplies?

62. How many squares are there on a chessboard? Can you extend your technique to calculate the number of rectangles on a chessboard?

63. There is a round table and 12 chairs around it. 12 coffee mugs are there on the table in front of chairs. Three persons can sit to any random position and drink the coffee in front of him. Another person then can fill one coffee mug. These three persons can move to two chairs either left or right and can drink the coffee again. How the other person will make sure that six mugs are always filled.

64. There is an array of “n” elements and there are “1” to “n-1” numbers are stored. So which number occurred double time?

65. There are three tires, each of which can go for some distance, let x, y, and z. How long you can go with these tires?

66. There is a number which ends with the “6”, like “

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