When the big hand of a clock stood at 12, an ant started walking anti-clockwise from mark 6 along the path of the clock hand. After meeting...Read on...

## The Ant on a Clock Face Riddle

When the big hand of an old clock stood at 12, an ant started walking anti-clockwise from mark 6 along the path of the clock hand. Just after meeting the big clock hand, the ant started walking in the opposite direction and continued walking at the same constant speed as before. It met the big clock hand for the second time exactly 45 minutes later.

For how long did the ant walk?

*Recommended time to solve:* 30 minutes.

*With basic arithmetic and some idea on what happens when two people walk together, this confusing riddle can be solved.*

### Solution to the Ant Walk on a Clock Face Riddle

First I imagined the situation of the ant meeting the big clock hand on the clock face. Mental visualization is easy, but a picture should make it easier to understand.

The big hand started from point A and the ant started from point B and met at point P somewhere between 6 and 12. Where they met and how long it took to meet both are unknown.

#### Analysis of the first lap of the ant's journey

With both time of walk and distance of walk unknown, what are known with certainty? Some things are known, though possibly ignored at first.

- The ant and the clock hand started moving at the same time.
- The two moved for the same duration till they met, and,
- Together, the two covered exactly half of the clock face length walking towards each other.

So, I was certain:

At whatever speeds the two moved, both of them took the same time to cover exactly half the clock face distance TOGETHER.

*The 'together' idea is the most important,* I realized. *During the second half of the journey also, the two moved together for the same duration*, even though in the same direction. Isn't it?

#### Analysis of the second lap of the ant's walk

I know the distance covered together in the first lap but don't know the duration of movement.

In the second lap though, I know for certain,

- Both the ant and the clock hand moved for the same duration along the clock face for exactly 45 minutes.
- Both of them started moving at the same time.
- The big hand of the clock moved exactly 3/4th of the whole clock face distance in 45 minutes.

I need to know only how much clock face distance the ant covered in 45 minutes.

#### The distance the ant covered in its second lap of walk

In its second lap of walk, the ant walked in front of the clock hand in the same direction of the clock hand. This implies,

The two could meet only if the ant made a full circle of the clock face from its first meeting point P and in addition, walked 3/4th of the clock face length (the distance covered by the clock hand in 45 minutes) to catch up the big clock hand at the second meeting point Q (as in the figure).

In the race with the big clock hand,

In the second lap then, the ant covered (1 + 3/4) = 7/4th of the whole clock face distance.

#### Final solution to the ant walk riddle

In the second lap,

- Together the two covered the distance of (7/4 + 3/4) =2.5 times the whole clock face distance.
- They covered this total distance together in 45 minutes.

The two covered 2.5 times the clock face distance in 45 minutes together. So,

They must have covered together in the first lap, half the clock face distance, that is, one-fifth the distance covered in second lap in one-fifth of 45 minutes, that is, in 9 minutes.

Voila! I got at last the elusive first lap walk time of the ant.

Total walk time of the ant was (45 + 9) = 54 minutes.

#### Concepts used elaborated (you may skip)

The riddle falls in the category of Time and Distance arithmetic, though common sense concept that is enough to solve the riddle is:

If two objects move for the same duration at different but constant speeds, the total distance covered by the two TOGETHER will take exactly the duration for which each moved.

It is also true,

The direction of movement of the two objects may be altogether different. Even, their starting time and point may also be different. Only their speeds must be constant and duration of movement must be same.

These are *extension of the concepts on two objects moving for the same duration at constant speeds.*

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