A lattice is a neatly arranged array of dots covering the plane. One lattice is called the triangular lattice. A piece of this is shown below. Three connected dots in close proximity form a unit triangle, which also happens to be equilateral.Click here to explore the triangular geobord in the GSP file.

Problem1

Form a large equilateral triangle 100 (horizontal) layers tall by connecting dots. By connecting dots, you can form a lot of unit triangles inside the large one. How many?

Answer...

Consuider the first some triangle. A 1-layer triangle has 1 unit triangle inside the triangle. A 2-layer triangle has 4 unit triangles inside triangle. A 3-layer triangle has 9 unit triangles inside triangle. We can find the regularity here.

		number of the unit triangle
	1 layer triangle	1
	2 layer triangle	4
	3 layer triangle	9
	100 layer triangle	10000

Problem 2

On the triangular lattice form a large equilateral triangle 5 layers tall ( as in problem 1). By conecting dots in this triangle you can parallelograms of various sizes, each side of which is parallel to a side of the big triangle. How many sizes of parallelograms can you form? How many parallelograms of all sizes can you form in the large triangle?

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Another lattice is the square latice, a piece of which is shown below. Connect four dots i close proximity and you get a unit square. Click here to explore the square geoboard in the GSP file.

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Problem 3

Connect the dots to form a large square 100 layers tall. In this large square you can connect dots to form a lot of little unit square. How many? You can also form square of other sizes in th elarge square. (Assume the bases of these squares are horizontal, that is, parallel to bottom of page.) What are the sizes you can form? How many squares of all sizes (horizontal bases) can you form?

Answer...

Let's consider some square. A 1-layer suqare has 1 unit square inside the square. A 2-layer square has 5 unit squares inside square. A 3-layer square has 14 unit square inside square. We can find the regularity here.

		the kind of unit square	the number of unit square	.
1-layer square		1 unit suqare	1 unit: 1=1*1	=1
2-layer square		1 unit square 2 unit square	1 unit: 4=2*2 2 unit: 1=1*1	=5
3-layer square		1 unit square 2 unit square 3 unit square	1 unit : 9= 3*3 2 unit : 4=2*2 3 unit : 1=1*1	=14
4-layer square		1 unit square 2 unit square 3 unit square 4 unit square	1 unit : 16=4*4 2 unit : 9=3*3 3 unit : 4=2*2 4 unit : 1=1*1	=30
100-layer square	...	...	1 unit :  100*100 2 unit : 99*99 3 unit : 98*98 ... 99 unit : 2*2 100 unit :1*1	=338350

Problem 4

In a large square 100 layers tall, form rectangles with horizontal bases. How many different sizes can you form? How many diferent rectangles of all sizes?

Answer...

We can compute the number of the rectangle using the computation. The formula is (mC2*mC2)-the number of square, when m is the line of square.

		number of rectangle	.
1-layer square		0	.
2-layer square		4	(3C2*3C2) - 5
3-layer square		130	(4C2*4C2) - 14
100-layer square	...	.	(101C2*101C2)

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