## 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 gave 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 think immediately. Knowing this, with no delay I stop you on this track, "No, you can't use algebra. Solve the riddle in your head without using $x$, $y$ or any variable. 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 riddle in my class and thoroughly enjoyed the confusion it created.

Though not truly hard, 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 MSc course in a renowned university. After lunch, all of us waiting got exhausted. 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.

Possibly because of boredom and tiredness of waiting through those long hours!

### Solution to 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 resolve,

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 stick to the only practical alternative,

"Half a poodle" must mean arithmetic half-of-a-poodle. After all, the riddle is an arithmetic calculation problem, isn't it?

Time to go deeper.

There were two sales, and in each, half a puppy added to another number of puppies. The purpose of adding this half a puppy must have been to make the number of puppies a whole integer. No other possibility.

On the same track, now realize,

The

other number, to half of which half a puppy was added, must be an odd number.Half of the number resulted in a free-standing half that needed to be made a whole integer by adding another one half. For example, half of 3 is 1 whole plus one half. Add one half and it will be 2.

### Solution to half a poodle riddle: Final stage: Use the clues, but where to start?

You are sure now that you have to use ideas of the arithmetic half-a-poodle and the number of puppies at the beginning of each sale an odd number. But how to start calculate?

Starting figure is what you need to find. No place to start, using no variable. You wonder where and how to start calculations using no variable! This is the last hurdle.

Undaunted, you take the path supported by your common sense again,

You will calculate only what you are sure of. As calculations from the start not possible,

you will start calculations from the end.

With the clue of arithmetic half-a-poodle, you know you can calculate the **number of poodles sold at the second sale** with no confusion.

In the second sale, Sonia sold half of rest of her litter after first sale and half a poodle to Puppy Palace. Rest 1 puppy she gave to Kaveri.

Hold on to this **1 puppy ultimately left.** You know it for sure and this is your point of start.

Anything else you are sure of?

Sonia added half a puppy to HALF OF THE PUPPIES BEFORE THE SECOND SALE. Now you know she did it to make the number a whole integer.

You now know this figure of ONE AND A HALF should be equal to half of the puppies before the second sale. So double this figure and get 3 as the number of puppies BEFORE THE SECOND SALE.

While working back,

You must take the

key actionopposite to what Sonia had taken.Sonia had halved. So you will double.

*3 is the number of poodles at the START OF SECOND SALE and AFTER THE FIRST SALE.*

*Add again half a poodle to 3 and get 3 and a half.*

This is **half of the number of poodles at the start of the first sale.**

Double 3 and a half and get 7 as the number of poodles that Sonia started with.

**7 is your answer.**

You have learnt how to use your common sense reasoning and *working backwards technique.*

*In real-life problems, unknowingly we use the working backwards approach often whenever you set a target event at a future date, say an exam to start on a fixed date, or a jungle safari trip planned next year March!*

Automatically, you will backtrack from the future date to the present times detailing what to do, starting now. This is—**working backwards approach in action.**

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