WESTERN AUSTRALIAN JUNIOR OLYMPIAD MATHEMATICS 2006

Team Question                                                              45 minutes

Consider a garden table made of 15 square tiles in a 5 x 3 arrangement. The table has a straight crack along a diagonal Seven of the individual tiles are broken.

Now consider a 6 x 4 rectangle. This time eights tiles are broken.

A.      How many tiles get broken when an 8 x 6 tables is cracked a long a diagonal?

Solution: 12 tiles get broken.

B.      Give the dimensions of two different rectangular tables that get nine tiles broken when   they are cracked along a diagonal.

Note:  Remember that a square is also rectangle. Also, note that for this and subsequent question, an 8 x 6 table, for example, is considered the same as a 6 x 8 table.

      Solution: The possible dimensions are: 9 x 1, 7 x 3, 9 x 3, and 9 x 9.

C.      How many different rectangular tables can you find that get ten tiles broken when they are cracked along a diagonal? Write down their dimensions.

Solution: The possible dimensions are: 10 x 1, 10 x 2, 10 x 5, 10 x 10, 9 x 2, 8 x 3, 7 x 4, and 6 x 5.

D.      Try some squares tables. Describe what happens:

Solution: The number of broken tiles is the same as the length of the side of the square

E.      What happens when the shorter dimensions of the table is 1?

Solution:  Every tile is broken, i.e. the number of tiles is broken is the length of the rectangle.

F.      For what sort of dimensions does the crack go through corners tiles inside the rectangle?

Solution: For dimensions that have a common factor larger (larger than 1).

G.     How many tiles are cracked when the diagonal does not go through any corner of a tile inside the rectangle? Explain your reasoning.

Solution:  If the diagonal does not go through any corner of a tile inside the rectangle, a   tile is cracked when and only when the diagonal enters a new column or enters a new row.

               Say the table has m rows and n columns. The first tile broken is in the first row and the first column. There are m-1 further rows and n-1 further columns. Therefore the number of tiles cracked is 1 + (m -1) + (n – 1) = n + m-1.