We’ve already seen this in building tables, guessing, etc.
If the result must be even, don’t list odd numbers.
In the diagram where we filled in circles along the sides of a triangle
to equal integers in the vertices, we never needed to guess larger than
the smallest integer.
Generally, use whatever rules apply to reduce your problem.
In an extreme case, only one result may be left standing.
Classical problem structure. Provide lists of items along with properties
linking the lists. Then derive the only possible matching between the lists.
Zillions of logic puzzles available on-line or in puzzle magazines.
Each of Bill, John, Fred, and Jim are married to one of Judy, Gretchen,
Margie, and Loretta.
Judy’s husband’s name does not begin with J.
Margie’s husband’s name has the same letter twice.
The name of Loretta’s husband has three letters.
General strategy: Form a table to track possibilities.
Fill in what is known from each rule (noted (1), (2), (3) below).
Cross out possiblities that cannot occur (similar notation).
What possibilities remains? Judy and Fred
That leaves one spot left, John and Gretchen
Judy
Gretchen
Margie
Loretta
Bill
—(2)
—(2)
Yes(2)
—(2)
John
No(1)
Yes
—(2)
—(3)
Fred
Yes
—(2)
—(3)
Jim
No(1)
—(3)
—(2)
Yes(3)
For an example that does not work, see problem 8 in set 1.4 (page 49). Whoever
wrote that has no idea what is in a decent banana split or double-dip cone.
If there are more pigeons than pigeonholes, then at least one holeholds more than one pigeon.
Thought to have been presented first by Dirichlet in 1834 as the “shelf principle”.
Useful for proving or demonstrating a fact through counting.
Assume there are 14 black socks and 6 white socks in the drawer. Without
looking, how many socks must you retrieve to have two of the same
color?
How might you consider solving this? Write out all possibilities foreach number of socks you retrieve. Yuck.
Here there are two “holes”, black and white.
You need draw three socks to guarantee two of the same color!
Frighteningly powerful principle.
Ignoring baldness, there are at least two people in the tri-cities area with
the same number of hairs on their head.
There are about 150 thousand hairs in a typical head of hair.
According to the 2000 census, there are 480 091 people in the
tri-cities area.
If we assume no one has over 480 000 hairs, and each “hair count”
is a category / pigeonhole, then there must be two people with
the same number of hairs.
A “lossless” file compression system must expand some files.
Compressed files are pigeonholes.
There are fewer possible files of smaller size.
Hence if all files can compress to smaller sizes, there will be two
files that compress to the same file.
The pigeonhole principle and its variations are an indispensable tool of mathematics!
There is a wonderful description and exploration of different phrasings from Edgar
Dijkstra:
Note that 12 wants the number that guarantees two people have
the same birthday.
Which of these statements demonstrate inductive reasoning, and which
demonstrate deductive reasoning? Justify.
It has rained for the past week. It will rain tomorrow.
All men are mortal. Socrates is a man. Therefore, Socrates is mortal.
Satellite-based network access does not function through heavy rain.
It is raining heavily. I cannot upload the notes right now.
The next number after 3, 8, 13, 18, and 23 is 28.
Note that you may email homework. However, I don’t use MicrosoftTM products
(e.g. Word), and software packages are notoriously finicky about translating
mathematics.
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.