SOLUTION: Hi there, <p> We were browsing through a folder of our maths teachers, for any challenging maths questions. We arrived at one that had been originally in a popular maths exam, b

Algebra ->  Rectangles -> SOLUTION: Hi there, <p> We were browsing through a folder of our maths teachers, for any challenging maths questions. We arrived at one that had been originally in a popular maths exam, b      Log On


   

Question 134685: Hi there,


We were browsing through a folder of our maths teachers, for any challenging maths questions. We arrived at one that had been originally in a popular maths exam, but had no idea how to figure it out in a time-efficient manner. Still, we haven't found the answer and would appreciate any help or an explanation of how you arrived at your answer:


"We say that squares can be packed together to form a 'jagged rectangle' if they fill a rectangular box in the way shown in the diagram. The diagram shows a 6 by 4 jagged rectangle and it contains 39 squares of the same size. In a 9 by 7 jagged rectangle, how many such squares would there be?"


It's quite hard to do the diagram here with such limited keyboard buttons, but if you reply in a confidential email to the given email, we would like to send you an image of this so called "jagged rectangle." Please help us.


Thank you!


Jasmine Zerlina
Brett Hagan
Melissa Salvagage
Jordan Kemp
Don Farlane
Answer by Earlsdon(6294) About Me  (Show Source):

You can put this solution on YOUR website!
While the picture would be helpful, one can easily imagine how it would look!
Let's first look at the given example of 39 squares in a 6 by 4 jagged rectangle.
First, each individual square would be tilted at an angle of 45 degrees, so that one of its corners would be in the upper-most position.
Now, lay 6 such squares side by side so that each square touches a adjacent square at a corner.
Now, beneath that row of 6 squares, place 5 squares so that they fit in the triangular spaces between two adjcent squares of the first row of squares.
Continue this arrangement until you have 4 rows of squares in 6 columns.
The pattern is then is 1 row of 6 squares, then 1 row of 5 squares, then 1 row of 6 squares, in an alternating fashion.
You can see that the total number of squares is the sum of 4 rows of 6 squares plus 3 rows of 5 squares, or 4*6+3*5 = 24+15 = 39 squares.
Now, applying the same reasoning to the larger jagged rectangle, you can see that there would be alternating rows of 9 squares and 8 squares.
There would be 7 rows of 9 squares plus 6 rows of 8 squares for a total of 7*9 + 6*8 = 63 + 48 = 111 squares.
I suppose one could generalise this for a jagged rectangle of m by n squares.
Total number of squares = m*n+(m-1)(n-1) = 2mn-m-n+1