MATC Mathematics Club
Problem
Set Volume 5 Page
The Puzzle Corner of the MATC Mathematics Club
Madison Area Technical College
Madison, Wisconsin
Volume 5, Fall 2003 Problem Sets 

Problem Set #3, Volume #5 ( Click here for the solutions ).  
1. A magic square is a square in which numbers are placed in a grid
as so that the row sums, column sums, and diagonal sums are all equal to the same number. The table below shows a magic square
where all the sums are 15.
Show that for any 3 x 3 magic square the center value must always be equal to 1/3 of the sum. Click here to download a Microsoft Excel spreadsheet that has been programmed to aid in calculating Magic Squares. Click here to see a web page describing a connection between chess and magic squares. 

2. Show that for any 3 x 3 magic square the only possible sum is 15 when each of the digits 1  9 is used to fill the square.  
3. An antimagic square is a square in which numbers are placed in a grid
as so that every row sum, column sum, and diagonal sum is different. The table below shows an
example of an antimagic square.
Show that for any 3 x 3 antimagic square that the sum of the row sums is equal to the sum of the column sums. Why is this a trivial result? 

Problem Set #2, Volume #5 ( Click here for the solutions ).  
1. A circle O_{1}(with
radius r_{1}) is externally tangent to a line l and two
circles O_{1}(with radius r_{1}) and
O_{2}(with radius r_{2}) as shown in the illustration below.
Show that the radii are related by the following equation: This problem is an example of Japanese geometry known as San Gaku. 

2. The top two diagrams below show how to divide a larger shape into
four smaller, equal shapes. Can you divide the bottom shape into five equal shapes?


3. Use mathematical
induction to prove the following two sums:
Sadly, the formulas for sums of other powers of natural numbers are not so nice...


Problem Set #1, Volume #5 ( Click here for the solutions )  
1. The circles O_{1}(with radius r_{1}) and
O_{2}(with radius r_{2}) are externally tangent to each other and to
the line l at the points A and B as shown in the illustration below.
Find a formula that relates the distance AB to r_{1} and r_{2}. This problem is an example of Japanese geometry known as San Gaku. 

2. Each X in the division problem
below stands for different digits (0 through 9; some are repeated, some may
not appear at all).
Find the unique combination of digits that fills in each X.
This problem was contributed by P. L. Chessin to the April 1954 issue of The American Mathematical Monthly. 

3. An unlimited supply of gasoline is available at one edge of a desert 800 miles wide,
but there is no source on the desert itself. A truck can carry enough gasoline to go 500 miles (this will be called one ("load"),
and it can build up its own refueling stations at any spot along the way. These caches may be any size, and it is assumed there
is no evaporation loss.
What is the minimum amount (in loads) of gasoline the truck will require in order to cross the desert? Is there a limit to the width of the desert the truck can cross? This problem is taken from Martin Gardner's My Best Mathematical and Logic Puzzles. 
Join the MATC
If you are interested, email Jeganathan "Sri" Sriskandarajah ( jsriskandara@madisoncollege.edu ) or contact him in room 211G. Watch this page and the student bulletin for further announcements. 
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