The Hour Hand and the Minute Hand

             We all know that the hour hand and the minute hand on a clock travel at different speeds .However, there are certain occasion when they are exactly opposite each other. Can you give a simple formula for calculating the times of these occasions?

Answer:   Here is the formula that gives the minutes past twelve to which the hour hand points when the minute hand is exactly thirty minutes ahead.

Minutes Past twelve Y= 30/11[(n-1) 2+1]

Where n is the next hour _

Let’s take the case of at what time between 4 and 5 will the hands be opposite each other? (n=5).

      Y= 3/11 × 9=270/11 + 246/11.               

I.e. the hour hand will be 24 6/11 minutes past 4. The formula may be derived from the following: If X is distance moved by the minute hand y is the distance moved by hour hand.

         Then X-Y = 30

First time the hands move round X= 12 Y

Second time the hands move round X=12 Y- 5                   

Third time the hands move round X= 12 Y-10 etc

The Bus Number

    While visiting a small town in the United States, I lost my overcoat in a bus. When I reported the matter to the bus company I was asked the number of the bus. Though I did not remember the exact number I did remember that the bus number had a certain peculiarity about it.  The number plate showed the bus number was a perfect square and also if the plate was turned upside down. The number would still be a perfect square – of course it was not?

   I came to know from the bus company they had only five hundred buses numbered from 1 to 500. From this I was able to deduce the bus number. Can you tell what the number?

Answer :  By experiment we find that the only numbers that can be turned upside down and still read as a number are 0, 1, 6, 8 and .

The numbers 0, 1 and 8remain 0, 1 8 when turned over, but 6 becomes 9 and 9 becomes 6. Therefore the possible numbers on the bus were 9, 16, 81, 100, 169 or 196. However the number 196 is the only number which becomes a perfect square when turned over because 961 is the perfect square of 31

Therefore 190 is the correct answer.  

Bicycle Thieves

     A friend of mine runs a bicycle shop and he narrated to me this following story. 

  A man, who looked like a tourist, came to his shop one day and bought a bicycle from him for Rs.350. The cost price of the bicycle was Rs. 300 So my friend was happy that he had made a profit of Rs. 50 on the sale. However, at the time of settling the bill, the tourist offered to pay in travelers’ cheques as he had no cash money with him. My friend hesitated. He had no arrangement with the banks to encash travelers’ cheques. But he remembered that the shopkeeper next door has such a provision, and so he took the cheques to his friend next door and got cash from him.

  The travelers’ cheques were all of Rs. 100 each and so he had taken four cheques from the tourist totaling to Rs. 400. On encasing them my friend paid back the tourist the balance of Rs. 50

  The tourist happily climbed the bicycle and pedaled away whistling a tune. 

   However, the next morning my friend’s neighbor, who had taken the travelers cheques to the bank called on him and returned the cheques to the bank, called on him and returned the cheques which had proved valueless and demanded the refund of his money. My friend quietly refunded the money to his neighbor and tried to trace the tourist who had given him the worthless cheques and taken away his bicycle, but the tourist could not be found. 

  How much did my friend lose altogether in this unfortunate transaction?

Answer:  One can think of different answers for this question but yet the correct answer is very simple. All we have to consider is that the shop owner could not have      possibly lost more than the tourist actually stole.

               The tourist got away with the bicycle which cost the shop owner Rs. 300 and the Rs. 50 ‘change’ and therefore he made off with Rs.350. And this is the exact amount of the shopkeeper’s loss.

The Digits and Square numbers

   All the nine digits are arranged here so as’ to  form four square numbers:

9,   81,   324,   576

How would you put them together so as to form a single smallest possible square number and a single largest possible square number?

Answer:  the lowest square number I can think I can think, of containing all the nine digits once and only once, is 139854276, the square of 11826, and the highest square number under the same conditions is 923187456 the square of 30384.

Over the Golden Gate

       While in San Francisco some time back, I hired a car to drive over the golden gate bridge. I started in the afternoon when there was no traffic rush. So I could drive at a speed of 40 miles an hour. While returning, however. I got caught in the traffic rush and I could only manage to drive at a speed of 25 miles an hour.

What was my average speed for the round trip?

Answer:  No, the answer is not 32½ miles an hour, though this figure is the obvious answer! However, this represents the average of the 2 speeds and not the average speed for the whole trip.

         If the time is equal to the distance divided by the average speed, then the time for the trip starting from San Francisco equals S/40 and the time for the return trip is ˢ/25 which gives us a total time of s/40+s/25, which equals 13 s/200 which equals 2S times 200/135, which equals 400s/13s or 30- 10/13 miles an hour.