## Half a poodle riddle

Once from her latest litter of French poodles, Sonia sold half of the litter and half a poodle to the local pet shop...read on...

### The half a poodle riddle

Sonia breeds French poodles as a hobby. She sold half of her puppies and half a puppy from the latest litter to the local pet shop. Next she sold half of the puppies that were left in the litter and half a puppy to Puppy Palace. She then decided to give the remaining puppy to her friend Kaveri.

How many puppies did the original litter have?

Oh well, that's easy, let's assume $x$ as the number of puppies Sonia started with - you start thinking immediately. knowing this, without any delay I stop you on this track, "No, you can't use algebra. You have to solve the riddle in your head without using $x$ or $y$ or any variable. You have to use your common sense reasoning only."

**Time to solve:** 5 minutes.

### A brief history of the riddle

Years back when teaching Problem Solving I used this fun learning riddle in my class and thoroughly enjoyed the confusion it created.

Though not really difficult, the riddle has an effect of putting up an immediate roadblock to further coherent thought (perhaps because of its mild shock) - how is it possible for Sonia to sell half a puppy!

True.

Later, I was with my son waiting for our turn in a day-long counseling session for selection in M.Sc course in a renowned university. After lunch, all of us waiting were very bored and tired. To enliven the air, I asked the girl candidate sitting by me, "Would you like to solve a little riddle?" Unsuspecting she nodded yes.

After I told her the half a poodle riddle, her eyes turned blank, she mumbled something and a few minutes later discreetly changed her place.

It must have been boredom and tiredness of waiting through those long hours, I am sure to this day.

### Solution to the half a poodle riddle: First stage: Discovery of the mystery of half a poodle

The obvious first question you ask,

How is it possible to have half a poodle?

As a well-grounded problem solver you stick to your common sense and decide firmly,

No, half a poodle is not possible in reality.

With this conviction, you are sure that "half a poodle" must mean something else, but what else?

You realize at this point,

Obviously, "half a poodle" actually meant arithmetic half of a poodle. After all, the riddle is an arithmetic calculation problem, isn't it?

With a clue to riddle you continue reasoning.

There have been two sales and in each half a puppy is added to another number of puppies. The purpose of adding this half a puppy must have been then to make the number of puppies a whole integer.

It makes full sense. Yes, that must be it.

Continuing in the same track you now realize the crucial truth,

Then the

other number to which half a puppy was added must have been an odd numbercreating a half that needed to be made a whole by adding another half.

### Solution to the half a poodle riddle: Final stage: Using the clue of arithmetic half a poodle and odd number of rest of puppies, but where to start calculating?

You are sure now that you have to use the arithmetic half a poodle as well as the other number of puppies at the beginning of each sale as an odd number. But how to start calculating?

You cannot start at the beginning. That is number of poodle puppies you have to find out.

You wonder, where and how to start calculating the number of poodles without using any variable! This is the final hurdle.

Never to be stopped, you take the path supported by your common sense again,

You will evaluate only what you are sure of. You cannot start at the beginning. So you will start at the end.

You sense that with the clue of arithmetic half a poodle, you can calculate number of poodles sold at the second sale without any confusion.

In the second sale, Sonia sold half of rest of her litter after first sale and half a poodle to Puppy Palace.

Without giving much load on your thinking neurons, you take the straight path of trial with 5 numbers of poodles as REST OF HER LITTER after first sale. Half of it is 2 and a half. Sonia added half a puppy to make it 3 poodle puppies and sold it to Puppy Palace leaving, oh no, 2 puppies!

But after the second sale only one puppy is left, not 2. Isn't it!

Finally, by this trial you get the easy way to calculate the **number of puppies Sonia started with in each sale,**

Just double the number of poodle puppies left after the sale and add 1 to get the number of poodle puppies Sonia started with in each sale.

At the beginning of the second sale, Sonia must have had 3 poodle puppies half of which is 1 and a half. She added another half a puppy to make a whole number of 2 puppies and sold them.

This is how, after the second sale only one poodle puppy is left that Sonia gave to Kaveri as gift.

You realize that with complete certainty you have got this calculation of 3 number of poodle puppies left in the litter at the beginning of the second sale. And You are WORKING BACKWARDS.

Taking one more step backwards TO THE BEGINNING, you double the **number of poodles left after the first sale 3** and add 1 to get 7 as the number of poodle puppies Sonia started with.

Unknowingly you have been forced to use the powerful problem solving technique of WORKING BACKWARDS APPROACH to solve the little riddle.

Now you know how to use your common sense and working backwards technique.

In real life problems also unknowingly we use the working backwards approach often whenever at a future date you have a target event, say an exam starting on a fixed future date, or you have decided to take a jungle safari trip next year March!

Automatically you would backtrack from the future date to the present times detailing the tasks to be done starting now. That is working backwards approach in action.

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