# Puzzling Adventures: River Run--How to Make the Most of Going Upstream

In this puzzle, figure out how fast each leg of your trip should be to make it as fast as possible

Image: Cloe Liane Shasha

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If you ever swim or paddle upstream, you will notice two things. First, a river's speed varies a lot. Second, those variations should cause you to pull harder when you hit rapidly flowing water. If you don't, you will simply make no progress. This puzzle replaces your muscles with a motor, but still asks you to figure out how to trade off energy for time.

Here are the facts:

•  You want to go 72 kilometers (km) upriver.
•  The first 24 km has a downstream speed of 7 kilometers per hour (kmh).
•  The next 18 km has a downstream speed of 2 kmh.
•  The last 30 km has a downstream speed of 0 kmh (the river becomes a lake).

You have an electric motor with three settings that can push the boat forward at a water speed of:

•  5 kmh using 1 kilowatt (kW) of power
•  10 kmh using 3 kW
•  15 kmh using 5 kW

Recall that land speed = water speed - downstream speed.
So, for example, if your water speed upstream is 15 kmh but the river has a downstream speed of 2 kmh, then your land speed is 13 kmh.

Warm-up:
Suppose you went full speed on all legs of the voyage. How long would the journey take and how much energy would you expend?

Solution to Warm-Up

Here now are the challenges for you.

1. What is the least energy you could use to make the entire trip, assuming you were in absolutely no rush? How would you do it?

Hint: On a lake, you would use the slowest speed, but this may not hold on all parts of the trip.

2. Suppose you have a battery that holds 30 kWh. How could you arrange to arrive as quickly as possible without consuming more than 30 kWh?

Dennis Shasha is at the Courant Institute of Mathematical Sciences, New York University. His most recent puzzle book, Puzzles for Programmers and Pros, was published in 2007 by John Wiley and Sons/Wrox.

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1. 1. hector 01:21 AM 9/4/08

There is a better solution to the minimum time part:
First 24 km at top speed, 3 h, 15 kwh,
next 18 km at medium speed, 2.25 h, 6.75 kwh,
next 30 km at slowest speed, 6 h, 6kwh.
Total time 11.25 h, total energy 27.75 kwh.

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