A devotee visits three temples each with 100 steps. He puts 1 rupee coin on each step going up, coming down and offers half he has to deity. Read on...
The devotee and three temples riddle
A devotee visits three temples. Each temple has 100 steps. The devotee puts 1 rupee coin on each step while going up and while descending.
In each of the temples, he always offers to the deity half of his coins.
After the visit to the last temple, all his coins got exhausted. How many coins he had in the beginning?
No algebra allowed.
Time to solve: 10 minutes.
Solution to the devotee and three temples riddle
Where to start is the big question. What do you know for sure?
The start is unknown alright. But the end? What happens at the end? The money put on the 100 steps is known with complete certainty. No confusion there.
Can I mentally collect the coins left by the devotee on the steps while he descended the steps of the third temple? You consider the choice.
Sure you can.
So you collect the Rs.100 from the exit steps of the third and last temple and reach the deity of the temple.
You are retracing the steps of the devotee, but backwards. From end to beginning.
The schematic of the three temples shown with coin deposit points broadly labeled.
Nobody stops you from retracing the path followed by the devotee and while moving back to front, you will collect mentally the coins deposited by the devotee. Again, you can very well do that.
At the first stage backwards, you collect Rs.100 on the steps and reach the deity at point J. Because of the scheme followed by the devotee, the number of coins with you must be the same as offered by the devotee to the third deity.
He deposited half of his coins to the deity and rest half was left with him, isn't it?
So you collect the Rs.100, mentally of course, from the third deity and cross over to the point H. At this point you have Rs.200.
When you descend the steps to point G, you collect another Rs.100, increasing your collection to Rs.300. No confusion at all.
Now you know the last steps to the solution.
Retracing the path backwards at the second temple, total collection Rs.400 from G to F on the exit steps, passing the second deity to the point E, total collection Rs.800 and descending the steps of second temple to point D, total collection Rs.900.
At the first temple, going up the steps to point C total collection Rs.1000, crossing the deity to point B total collection Rs.2000 and finally descending the entry steps of the first temple, collect Rs.100 more. Total collection is Rs.2100.
You have reached the starting point of the devotee. And you have exactly the number of coins the devotee had at the start.
To be sure, you may play act the role of the devotee going from point A to the end point K to check whether finally at K you have exhausted what you had in the beginning.
If you can hit upon the crucial idea of going back to front, solution takes less than 3 minutes.
End note
This is the Working backwards approach, a powerful problem-solving technique. We use it so often even in our daily life problem solving, such as making the study schedule, when the exam date has just been announced.
Any other example? What do you do when you know the time you have to reach your destination! You backtrack and fix the time you have to start the journey.
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