The Art and Craft of Problem Solving

2.4 OTHER IMPORTANT STRATEGIES 59

bystander tells her that her hat fell into the river as she originally dived. The swimmer

continues downstream at the same rate of speed, catching up with the hat at another

bridge exactly I mile downstream from the first one. What is the speed of the current

in miles per hour?

Solution: It is certainly possible to solve this in the ordinary way, by letting x

equal the current and y equal the speed of the swimmer, etc. But what if we look at

things from the th e hat's point of view? The hat does not think that it moves. From its

point of view, the swimmer abandoned it, and then swam away for an hour at a certain

speed (namely, the speed of the current plus the speed of the swimmer). Then the

swimmer turned around and headed back, going at exactly the same speed. Therefore,

the swimmer retrieves the hat in exactly one hour after turning around. The whole

adventure thus took two hours, during which the hat traveled one mile downstream.

So the speed of the current is! miles per hour. _

For another example of the power of a "natural " point of view, see the "Four Bugs"

problem (Example 3.1. 6 on page 65). This classic problem combines a clever point of

view with the fundamental tactic of symmetry.

Problems and Exercises

2.4.7 Pat works in the city and lives in the suburbs

with Sal. Every afternoon, Pat gets on a train that ar­

rives at the suburban station at exactly 5PM. Sal leaves

the house before 5 and drives at a constant speed so as

to arrive at the train station at exactly 5PM to pick up

Pat. The route that Sal drives never changes.

One day, this routine is interrupted, because there

is a power failure at work. Pat gets to leave early, and

catches a train which arrives at the suburban station

at 4PM. Instead of phoning Sal to ask for an earlier

pickup, Pat decides to get a little exercise, and begins

walking home along the route that Sal drives, know­

ing that eventually Sal will intercept Pat, and then will

make a V-tum, and they will head home together in

the car. This is indeed what happens, and Pat ends up

arriving at home 10 minutes earlier than on a normal

day. Assuming that Pat's walking speed is constant,

that the V-tum takes no time, and that Sal's driving

speed is constant, for how many minutes did Pat walk?

2.4.8 Prove, without algebra, that the sum of the first

n positive odd integers is n^2 •

2.4.9 Two towns, A and B, are connected by a road.

At sunrise, Pat begins biking from A to B along this

road, while simultaneously Dana begins biking from B

to A. Each person bikes at a constant speed, and they

cross paths at noon. Pat reaches B at 5PM while Dana

reaches A at II: 15PM. When was sunrise?

2.4.10 A bug is crawling on the coordinate plane from

(7, II) to ( -17, -3). The bug travels at constant speed

one unit per second everywhere but quadrant II (neg­

ative x-and positive y-coordinates), where it travels at

� unit per second. What path should the bug take to

complete its journey in minimal time? Generalize!

2.4.11 What is the first time after 12 o'clock at which

the hour and minute hands meet? This is an amus­

ing and moderately hard algebra exercise, well worth

doing if you never did it before. However, this prob­

lem can be solved in a few seconds in your head if

you avoid messy algebra and just consider the "natu­

ral" point of view. Go for it!

2.4.12 Sonia walks up an escalator which is going up.

When she walks at one step per second, it takes her

20 steps to get to the top. If she walks at two steps

per second, it takes her 32 steps to get to the top. She

never skips over any steps. How many steps does the

escalator have?

2.4.13 The triangular numbers are the sums of con­

secutive integers, starting with I. The first few trian­

gular numbers are

I , 1+2=3, 1+2+3=6, 1+2+3+4= 10 .....