A took half of the apples, returned 10. B took one-third of what were left...it went on...Evaluate number of apples in the beginning, but no algebra please.

## The baffling puzzle of how to evaluate number of apples the woman had in the beginning

Three friends Anish, Bibhu and Chandan while returning home came upon a woman with a basket of apples. The friends, in lighter mood, decided to buy the apples. Anish first took half of the apples, but promptly returned ten to the basket. Bibhu took one-third of what were left and again returned two he didn't like. Chandan picked up half of the remaining apples but threw back one that was worm eaten. The woman was left with only 12 in her basket. How many apples the woman had to start with?

Evaluate the elusive number, but **without any Algebra please.**

**Time for you should not be more than 5 minutes.**

See, it should not be a difficult one, though a little baffling. All you have to do is to tell your brain to think in a totally new way, different from the conventional.

**Aside:** *When I told such a puzzle (but a different one) to a senior college student during a little waiting time, she got quite confused and left me for another seat in the waiting hall.*

But you are not just any person, in not just any place. This is the place of inventive problem solving to encourage you to think in new ways.

Have a bit of a relief from the small time given for you to solve. The puzzle must be easy and quick to solveābut only if you think in the right way.

## Let us explain the solution to the puzzle of how to evaluate number of apples the woman had

I cannot use Algebra, which means there is no place here for use of any variable like unknown number of apples X. So, I thought: What do I know for certain? I don't know the target number of apples, but **do I have any information with total certainty?**

This is the way a problem solver should think, homing in on to the first important question.

Yes, of course, i know the number of apples the woman had at the end for certain. It was 12. I got my starting point as well as how I should go ahead.

#### Working backwards from end to the beginning event by event: Taking care of last purchase by Chandan

I realized I can easily reverse the action of Chandan and move one step ahead. At the last stage Chandan took half the remainder after the first to events and returned one.

**Logic analysis says**,

- If I return Chandan, (of course mentally) the one apple he returned to the basket, my number of apples will become 11. More important is: these 11 apples must be half of what was left after the first two purchases by Anish and Bibhu. The number left after first two purchases is double of 11, that is, 22.
**Firm conclusion:**After Anish and Bibhu did their purchase and returning drama, the woman had 22 apples left in the basket.

#### Again work backwards to take care of the second purchase by Bibhu

*Bibhu returned two apples that I must return to him.* My basket (it has become my basket by now) has now 20 apples. Bibhu got one-third of what were left after Anish took his share.

- So, this number of what was left after Anish thankfully finished his finicky purchase must be
**three times of half 20, that is 30**.*Bibhu took one-third of 10 apples left 20 and then returned 2 to make it 22 for Chandan.* **Firm conclusion:**After Anish finished his purchase, 30 apples were left.

#### Third time working backwards taking care of the first purchase gives me the solution to the puzzle of how to evaluate the number of apples the woman had

Anish returned ten apples to the basked. Reversing his action in my mind, I must return those ten apples to him. **My number of apples in the basket becomes 20.**

- This
**must be the half of the total number of apples the woman had in the beginning**(for the simple reason that Anish got half of the number of apples in the basket).

Finally I get my magic **number of apples in the basket the woman had in the beginning:** it is double of 20, that **was 40.** **This is your answer.**

**Aside:** Writing so many lines of over-elaborate and verbose explanation takes time. But the actual calculation while working backwards? It should take max to max three minute's time.

**Don't forget:** Check the answer repeating the three purchases.

*Welcome to the world of inventive problem solving where thinking new is the norm. Enjoy your stay here.*

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