Two old friends hadn't seen each other for a long time when they meet
in the streets. 'How are your three sons?' -the first asks. 'Hey, I can't
remember their ages'. The other replies, 'Well, the product of their ages
is 36. Also the sum of their ages equals to the number of windows in that
house over there'. The first one says: 'Listen, I still can't figure out
the ages of your sons'. The second replies 'Sorry, I forgot to tell you
my oldest son has blue eyes.' At which point the first one figures out
the ages of the three sons. Can you do it?

A bag contains a total of N balls with either blue or red colour. If
five balls are randomly chosen from the bag, the probability is precisely
1/2 that all five balls are blue. What's the smallest value of N for which
this is possible? (Hint: Use different number of blue/red balls to get
to the answer?)

A breakfast cereal maker is putting toys in the boxes of cereals as part
of a promotion. There are four different toys to collect. A child desperately
wants to collect the whole set, and his/her parents are concerned about
how many boxes of the same cereal they have to go through before he/she
achieves this. It is assumed that each cereal box is equally probable
to have any one of the toys inside, and that there is only one toy per
box. On average, how many cereal boxes are needed to be bought so the
child has the complete set?

At what time between 3 and 4 o'clock will the hands of a clock be in
a straight line?

A column of troops one km long is moving along a straight road at a uniform
pace. A messenger is sent from the head of the column, delivers a message
at the rear of the column and returns. He also moves at a uniform pace
and arrives back at the head of the column when it has just covered its
own length.
How far did the messenger travelled?

A cyclist practising for a road race found that one complete circuit
of the course took exactly two hours and fifteen minutes. The course at
one part was very hilly. His speed on the level was at the rate of 16
km per hour, uphill it was 12 km per hour and downhill 24 km per hour.
What is the distance round the course?

Four dog dealers purchase a bargefull of dogs. The first buys one airedale,
three spaniels and seven puppies for £14. The second buys one Airedale,
four spaniels and ten puppies for £17. The third buys ten airedales,
fifteen spaniels and twenty-five puppies. The fourth buys one Airedale,
one spaniel and one puppy.
How much did the third and fourth dealers pay?

One morning, Ben is in a hurry as he approaches the escalator at the
tube station, but Fred is in an even bigger hurry. For every 2 steps Ben
took, Fred took 3. Altogether Fred took 25 steps before getting off the
escalator, while Ben took 20. Both of them step on every step.
How many steps does the escalator have showing at any given time?

There are 10 open boxes containg 100 coins each. In 9 of these boxes
the coins are made of gold, and in the other the coins are made of copper.
You are given a large digital balance which can be used once only. Can
you identify the box containg copper coins knowing the weight of both
gold and copper coins?

100 prisoners are given the chance to be set free tomorrow. They are
all told that they will each be given a hat to wear and that they cannot
see the colour of the hat assigned to them. They know that there are 3
possible colours of hat: red, blue, and white. Each prisoner can see everyone
else's colour hat except their own. The hats colours are assigned completely
at random and once the hats are placed on top of each prisoner's head
they cannot communicate with others in any form, or else they are immediately
executed. The prisoners will be called out in random order and they are
to guess the colour of the hat that he/she is wearing. They shout the
colour of the hat so that everyone else can hear. If the prisoner guesses
correctly the colour of his/her hat they are set free immediately, otherwise
executed.
They are given the night to come up with a strategy amongst themselves
to save as many prisoners as possible. What is the best strategy they
can adopt and how many prisoners can they guarantee to save? ( Hint :
Start with the case where there are only 2 possible colour hats).

Alan took out an insurance policy for his son on his son's first birthday.
Under the scheme the first premium was £10 and the premiums increased
£10 each year. At the end of a number of years the son found that
the amount already paid in premiums was equal to the amount which would
still have to paid before the sum insured would be due for payment.
What was the son's age at this period and at what age would the insurance
be due?

Two streets of houses meet at right angles. One is 20m wide, and the
other 10m wide. What is the longest ladder which could be taken round
the corner? The ladder cannot be tipped up.

You are participating in a TV game. You have reached a stage where you
can select one box out of three knowing that only one contains a cash
prize and the others contain nothing. The TV host knows which of the boxes
contains the prize. Having made your choice, the host decides to reveal
the content of one of the two boxes you haven't selected which he knows
is empty. He then offers you the possibility to change your selection.
Should you switch or stick to your choice? does it matter?

Two people live far apart from one another and need to secretly communicate
via exchanged messages. To stop anyone reading the messages they are put
into a box which is pad locked and then sent. Anything sent without it
being in a box and pad locked will be read by the post service and confenscated.
The pad lock has only one key which the person putting the lock owns.
How can you send messages without the post service reading the messages.
Hint: You can use more than one lock on the box.

A line of 100 airline passengers is waiting to board a plane. They each
hold a ticket to one of the 100 seats on that flight. (For convenience,
let's say that the nth passenger in line has a ticket for the seat number
n.)
Unfortunately, the first person in line is crazy, and will ignore the
seat number on their ticket, picking a random seat to occupy (equally
likely). All of the other passengers are quite normal, and will go to
their proper seat unless it is already occupied. If it is occupied, they
will then find a free seat to sit in, at random.
What is the probability that the last (100th) person to board the plane
will sit in their proper seat (#100)?