DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

1.4 Another Application of the Sum Principle
◦30. US coins are all marked with the year in which they were made. How
many coins do you need to guarantee that on at least two of them,
the date has the same last digit? (The phrase “to guarantee that on
at least two of them,...” means that you can find two coins with the
same last digit. You might be able to find three with that last digit,
or you might be able to find one pair with the last digit 1 and one pair
with the last digit 9, or any combination of equal last digits, as long
as there is at least one pair with the same last digit.)
There are many ways to explain your answer to Problem 30. For example,
you can separate the coins into stacks or blocks according to the last
digit of their date. That is, you can put all the coins with a given last digit
in a stack together (putting no other coins in that stack), and repeat this
process until all coins have been placed in a stack. Using the terminology
introduced earlier, this gives a partition of your set of coins into blocks of
coins with the same last digit. If no two coins have the same last digit, then
each block has at most one coin. Since there are only ten digits, there are at
most ten non-empty blocks and by the Sum Principle there can be at most
ten coins. Note that if there were only ten coins, it would be possible to
have all different last digits, but with eleven coins some block must have at
least two coins in order for the sum of the sizes of at most ten blocks to be
11. This is one explanation of why eleven coins are needed in Problem 30.
This type of situation arises often in combinatorial situations, and so rather
than always using the Sum Principle to explain your reasoning, you can use
another principle which is a variant of the Sum Principle.
The Pigeonhole Principle
If a set with more than n elements is partitioned into n blocks, then at
least one block has more than one element.
The Pigeonhole Principle gets its name from the idea of a grid of little
boxes that might be used to sort mail or as mailboxes for a group of people
in an office. The boxes in such grids are sometimes called pigeonholes in
analogy with the stacks of boxes used to house homing pigeons back when
homing pigeons were used to carry messages. People will sometimes state
this principle in a more colorful way as “if more than n pigeons are put into
n pigeonholes, then some pigeonhole contains more than one pigeon.”
31. Prove the Pigeonhole Principle follows from the Sum Principle.
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