The technique is to be methodical in constructing the table.
Take care to chose one method and follow it.
If you remember truth tables from logic, same idea.
Example 1.4, page 15 (?):
Make an orderly list
How many different total scores could you make if you hit the
dartboard shown with three darts?
(Three nested circles. Scores 10 for the innermost, 5 for the middle,
and 1 for the outer ring.)
Understand the problem
Three darts hit the dartboard and each scores a 1, 5, or 10. The
total score is the sum of the scores for the three darts. There could
be three 1s, two 1s and a 5, one 5 and two 10s, and so on. The fact
that we are told to find the total score when throwing three darts is
just a way of asking what sums can be made using three numbers,
each of which is either 1, 5, or 10.
Devise a plan
If we just write down sums hit or miss, we will almost surely overlook
some of the possibilities. Using an orderly scheme instead, we can
make sure that we obtain all possible scores. Let’s make such a list.
We first list the score if we have three 1s. then two 1s and one 5,
then two 1s and no 5s, and so on. In this way, we can be sure that
no score is missed.
Carry out the plan
Number of 1s
Number of 5s
Number of 10s
The possible total scores are listed.
Here the key to the solution was in being very systematic. We were
careful first to obtain all possible scored with three 1s, then two 1s,
then no 1s. With two 1s there could be either a 5 or a 10 as
shown. For one 1 the only possibilities are two 5s and no 10s, one
5 and one 10, or no 5s and two 10s. Constructing the table in
this orderly way makes it clear that we have not missed any
My additional notes:
Note the similarity with numbers. Consider listing all numbers that fit the following
Use only the digits 0, 1, 2, and 3.
Have at most three digits.
Then drop those numbers whose digits do not add to three.
That’s another way to construct a table like this: Consider a larger table
where it is easier to be systematic, then remove numbers that do not fit the
Groups are fine, turn in your own work. Homework is due in or before class on
Write out (briefly) your approach to each problem.
Problem set 1.1:
Problems 10, 13 (Hint: find a way to make a smaller table)
Construct an example like 1.3 where there is no solution. Explain what lead
you to its construction. (Hint:
Problem set 1.2:
Problems 5, 7, 8
Consider solving Example 1.3 with a table starting at a=1, b=1, c=1. How long
would the table be if you step through the choices? How many entries would
you check if you bisect the choices? (Hints:
Don’t make the lists, but rather count the steps in the method for
making the list.
Or just go ahead and use a program or spreadsheet to build the lists.)
Note that you may email homework. However, I don’t use MicrosoftTM products
(e.g. Word), and software packages are notoriously finicky about translating
If you’re typing it (which I advise just for practice in whatever tools you use), you
likely want to turn in a printout. If you do want to email your submission, please
produce a PDF or PostScript document.