## Back of the train brainteaser

By David Shaw, 14 July 2021

**Difficulty: Tricky**

Ben is driving his train slowly. He realises he’s left an important package in the last car of the train. He needs to hop out of the train, walk all the way to the back to retrieve it, and then walk to the front of the train again.

The train is 3 kilometres (km) long and it’s going at 1 km per hour. Ben can walk at 2 km per hour.

Ben stops the train to retrieve the package, as he knows it’s the safest option. But he wonders if it would have been faster to let the train keep rolling. Or, perhaps it would take the same time either way.

Can you work out the fastest option?

**Scroll down or click for a hint, or the answer!**

## Brainteaser hint

To calculate how fast Ben gets to the back of the train, add Ben’s speed and the speed of the train together.

To calculate how fast Ben returns to the front of the train, subtract the speed of the train from Ben’s speed.

## Brainteaser answer

It’s faster for Ben to stop the train.

When the train is stopped:

- Ben walks 3 km to get to the back, and 3 km to return to the front. That’s 6 km all up. At 2 km per hour, it takes 3 hours.

If the train was rolling:

- The trip to the back would be faster. In an hour, Ben can walk 2 km and the train goes 1 km – getting him all the way to the end of the train.
- On the way back, the train would be moving away from Ben. Every hour he’d walk 2 km, but the train would travel 1 km away from him, so he’d only make it 1 km down the train. At that speed it would take him 3 hours to return to the front.
- That means the trip would take 1 + 3 = 4 hours.

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23 July, 2021 at 3:48 pm

I think you may need to publish a correction to the answer regarding Ben and the train.

If the train is moving, Ben’s speed relative to the train is 4km/hr so it will take him 3/4 hrs = 45minutes to get to the back. The return trip is at a relative speed of 3-1 = 2km/hr so it will take him 3/2 = 1.5 hours. The total trip time is thus 2hr 15min.

If the train is stopped, he travels 6km at 3km/hr = 2hrs, so it is 15 minutes faster to stop the train.

23 July, 2021 at 4:19 pm

Hi Alan,

I think you may have misread the question – Ben walks at 2 kilometres per hour, not 3.

Cheers,

David

25 July, 2021 at 5:23 pm

Hi David,

Just another thought to complicate the answer.

In calculating the answer you did not consider how late the train would be to its final destination.

If the train is stopped it will be 3 hours behind schedule.

If the train continues to roll it will still be on time to its destination.