Cynthia Lanius

Rectangle Pattern Challenges-

Preliminary Questions

1. Examine Stage No. 1 and Stage No. 2. Then examine Stage No. 2 and Stage No. 3. Describe what you have to do to Stage (N - 1) to create Stage N.

Answer Add a row of squares all around the rectangle. Make the corner squares blue, the side squares red, and the top and bottom squares, green. This process, building one stage from the previous one is called recursive.

2. Observe the designs looking for patterns. Use the symmetry of the design to ease your counting. Organize your information into a table like the one below.

Stage No.     1         2         3     ...     n
No. Blue Squares 6 10 14   4n+2
No. Red Squares 2 8 18   2n2
No. Green Squares 4 12 24   2n2+2n
Total No. Squares 12 30 56   4n2+6n+2

Thought Questions

1. Which color of squares is growing at the slowest rate? At the fastest rate?

Answer Blue is growing the slowest (at a constant rate - increasing by 4 each time). Red is next, then green is fastest.

2. How many squares of each color will be in the 8th stage of the design? In the 0th?

Answer When n = 8, Blue = 4(8) + 2 = 34, Red = 2(8)2 = 128, green = 2(8)2 + 2(8) = 144. When n = 0, Blue =2, Green and Red both = 0.

3. Will the design ever use 42 blue squares? Will it ever use 102 red squares? Will it ever use 870 squares in all? If so, state the stage number for each answer.

Answer The design will use 42 blue squares if when you make 4n + 2 = 42 and solve, you get a whole number. Solving gives N = 10, so yes. The design will use 102 red squares if 2n2 = 102 gives us a whole number for n. It doesn't, so no. The design will use 870 squares in all if 4n2 + 6n + 2 = 870 has a whole number solution. It does - 14. So yes when n = 14, the design will use 870 squares in all.

4. What are the dimensions (length and width) of the rectangular designs? Write these in terms of n? Show that when you multiply the length and width you get the total number of squares in each rectanglular design. Be sure to do this for n.

Answer When n = 1, L = 4 W = 3
When n = 2, L = 6 W = 5
When n = 3, L = 8 W = 7
So for n, L = 2n + 2 and W = 2n + 1
(2n + 2)(2n + 1) = 4n2 + 6n + 2, the total number of squares.