2.4 OTHER IMPORTANT STRATEGIES 53mind during any investigation. We will also discuss more advanced strategies in later
chapters.
Draw a Picture!
Central to the open-minded attitude of a "creative" problem solver is an awareness
that problems can and should be reformulated in different ways. Often, just translating
something into pictorial form does wonders. For example, the monk problem (Exam
ple 2.1.2 on page 17) had a stunningly creative solution. But what if we just interpreted
the situation with a simple distance-time graph?
Summlt.....,.-------------..."
First Days Path
Base�-------------�
BAM NoonIt's obvious that no matter how the two paths are drawn, they must intersect some
where!
Whenever a problem involves several algebraic variables, it is worth pondering
whether some of them can be interpreted as coordinates. The next example uses both
vectors and lattice points. (See Problem 2.2.17 on page 37 and Problem 2.3. 38 on
page 52 for practice with lattice points.)
Example 2.4. 1 How many ordered pairs of real numbers (s,t) with 0 < s, t < 1 are
there such that both 3s + 7t and 5s + t are integers?
Solution: One may be tempted to interpret (s,t) as a point in the plane, but that
doesn't help much. Another approach is to view (3s + 7t, 5s + t) as a point. For any
s,t we have
(3s+7t,5s+t) = (3, 5)s+ (7, 1)t.
The condition 0 < s,t < 1 means that (3s+ 7t, 5s+t) is the endpoint of a vector that lies
inside the parallelogram with vertices (0, 0), (3, (^5) ), (7, 1) and (3, 5) + (7, 1) = (10,6).
The picture below illustrates the situation when s = (^0) .4, t = 0.7.