What are some ways to make a problem 'simpler'? One is
to consider a smaller case. For example, suppose we're trying
to solve this problem:
A good first step might be to ask the same question with
a smaller number of days:
What day of the week is the 3,824th day after Wednesday?
Well, 7 days past is clearly another Wednesday, so we just
have to worry about (9 - 7), or 2 days. We can count those
off: Thursday, Friday.
What day of the week is the 9th day after Wednesday?
We haven't actually solved the problem yet, but we've made
progress. If we can find the Wednesday closest to the day
we're looking for, we'll be able to count off from there,
as we did in the smaller case.
Another way to make a problem 'simpler' is to choose nice,
round numbers - so nice that you can almost work the solution
out in your head. For example, given a problem like
we might change the numbers to
A computer has a hard drive that stores approximately
8.4 x 10^7 bytes. A high density 5.25 inch floppy disk holds
approximately 1.3 x 10^6 bytes. How many floppies are needed
to back up the hard drive?
In this form, it's clear that we can get the answer (10 floppies)
by dividing the size of the hard drive (1000 bytes) by the size
of one floppy (100 bytes). Now that we know this, we're ready
to deal with numbers of any size or complexity.
A hard drive holds approximately 1000 bytes.
A floppy holds approximately 100 bytes.
How many floppies are needed to back up the hard drive?
Specific kinds of problems often have specific ways that they
can be made 'simpler', e.g.,
Q: At midnight, a jet left Seattle Washington for St. Louis, Missouri,
2100 miles away flying at 500 mph. One hour later a high-speed jet
left St. Louis for Seattle at 700 mph. At what time did they pass
Q': At midnight, the first plane leaves Seattle, and by 1 am, it's 500
miles along the way there. Which means that it's
2100 - 500 = 1600
miles away from St. Louis. So now we have a simpler problem:
One plane leaves a spot 1600 miles from St. Louis at 1 am,
traveling at 500 mph. At the same time, a second plane leaves
St. Louis traveling towards that spot at 700 mph. With each
passing hour, how much closer do they get to each other?