Problem Solving and Computing (CSC-103 98S)

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Problem Set 5: Conceptual Problems

Assigned: Tuesday, February 24, 1998
Due: Tuesday, March 3, 1998

Feel free to work on the problems in any order.

1. Glaeser's Dominoes

[From Thinking Mathematically, p. 172]

George Glaeser of Strasbourg put a set of dominoes more or less randomly in a flat tray and took a photograph. The exposure was not correct, and although the numbers can be discerned, the positions of the individual dominoes cannot.

Each domino is a rectangle composed of two adjacent squares, each with a number. In the set there is only one domino with each combination of numbers from the numbers 0 through 6. For example, there is one domino with 2,3, and one with 4,4.

Can you reconstruct the dominoes?

3  6  2  0  0  4  4
6  5  5  1  5  2  3
6  1  1  5  0  6  3
2  2  2  0  0  1  0
2  1  1  4  3  5  5
4  3  6  4  4  2  2
4  5  0  5  3  3  4
1  6  3  0  1  6  6

2. Tetrominioes

[Based loosely on some problems in chapter 2 of Another Fine Math You've Got Me Into ... by Ian Stewart.]

Sam and Michelle Rebelsky have recently purchased a house and need to retile some rectangular and square floors. They visit the local Home Depot and discover that there are many shapes and sizes of tiles. They are particularly attracted to the tiles labeled "tetrominoes".

Here are the basic tetrominoes. Note that each one is made up of four squares.

+--+  +--+--+  +--+--+  +--+     +--+
|  |  |  |  |  |  |  |  |  |     |  |
|--|  +--+--+  +--+--+  +--+--+  +--+--+
|  |  |  |  |  |  |     |  |  |  |  |  |
|--|  +--+--+  +--+     +--+--+  +--+--+
|  |           |  |     |  |        |  |
+--+           +--+     +--+        +--+
|  |
+--+

What is the minimum number of each tetromino you need to completely tile a rectangle? A square? You cannot cut the tetrominoes.

3. Pentominoes

[Based loosely on some problems in chapter 2 of Another Fine Math You've Got Me Into ... by Ian Stewart.]

The Acme Tile company has observed a growing market for oddly shaped tiles, such as

+--+--+        +--+--+--+--+
|  |  |        |  |  |  |  |
+--+--+--+--+  +--+--+--+--+
   |  |  |  |        |  |
   +--+--+--+        +--+

Tiles of this form made up of five squares are called pentominoes. How many different shapes of pentominoes are there? What are they?

4. Positive Divisors

[From the notes of Emily Moore. Slightly modified by Sam Rebelsky]

A divisor of an integer x is an integer y such that y evenly divides x. That is, there is another integer z such that y*z = x.

Which positive integers have an odd number of positive divisors?

5. Cartesian Chase

[Taken from Thinking Mathematically, p. 162. Modified by Sam Rebelsky]

Cartesian chase is a two-player game which is played on a rectangular grid with a a fixed number of rows and columns. Play begins in the bottom left hand square where the first player puts a mark. On each turn, a player may put a mark into a square that is

Can you come up with a strategy for three-player game of cartesian chase?

6. Friday the 13th

[Taken from Thinking Mathematically, p. 160. Modified by Sam Rebelsky]

Noted cartoon character Churchy LaFemme is terrified of Friday the 13th, particularly when it falls on Fridays. (He claims that like many holidays, birthdays, and other special days, Friday the 13th can fall on any day of the week.) He's worried that Friday the 13th will fall on Fridays more often than he can cope with.

How many times will Friday the 13th occur on a Friday this year? What is the maximum number of times Friday the 13th can occur on a Friday in a calendar year, using the standard calendar? The least number of times? Does it matter whether you start in January or another month?


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