Sequences

How many squares are in the border of a 10 × 10 grid? How many different ways can you count them?

Try it on paper, then step through the working.

\(4 \times 10 - 4 = 36\)

4 sides of 10. Each corner is counted twice, so take 4 away.

\(4 \times 9 = 36\)

4 groups of 9, each starting at a corner.

\(10 + 8 + 10 + 8 = 36\)

The top and bottom rows, then the two sides without their corners.

\(4 \times 8 + 4 = 36\)

4 sides of 8 and the 4 corners.

\(10 \times 10 - 8 \times 8 = 36\)

The whole grid take away the inside.

Write each way of counting the border as an expression for an \(n \times n\) grid, and show that the expressions are equal.

Try it on paper, then step through the working.

\(4n - 4\)

4 sides of \(n\). Each corner is counted twice, so take 4 away.

\(4(n-1) = 4n - 4\)

4 groups of \(n - 1\).

\(n + (n-2) + n + (n-2)\) \(= 4n - 4\)

Two rows of \(n\) and two sides of \(n - 2\).

\(4(n-2) + 4\) \(= 4n - 8 + 4\) \(= 4n - 4\)

4 sides of \(n - 2\) and the 4 corners.

\(n^2 - (n-2)^2\) \(= n^2 - (n^2 - 4n + 4)\) \(= 4n - 4\)

The whole grid take away the inside.

A square jigsaw is made of corner pieces, edge pieces and middle pieces. Jigsaw 2 is 2 pieces long and 2 pieces wide. How many of each type of piece are there in jigsaw 100?

Try it on paper, then step through the working.

\(4\)

Every jigsaw has 4 corner pieces.

\(4 \times 98 = 392\)

Each side of jigsaw \(n\) has \(n\) pieces and 2 of them are corners, so there are \(4(n-2)\) edge pieces.

\(98^2 = 9604\)

The middle pieces make a square with \(n - 2\) on each side: \((n-2)^2\).

\(4 + 392 + 9604 = 10000\) \(= 100^2\)

The three types add up to the total number of pieces.

Two patterns are built from squares. For each one, find \(u_{10}\) and then \(u_n\).

Try it on paper, then step through the working.

\(u_n = n^2\), so \(u_{10} = 100\)

Squares of side 1, 2, 3, 4.

\(u_n = (n+1)^2\), so \(u_{10} = 121\)

Squares of side 2, 3, 4, 5, one more than the pattern number.

What is the general rule for the entire square, for the blue squares and for the red squares? Show that the blue squares and the red squares add up to the entire square.

Try it on paper, then step through the working.

\((n+1)^2\)

It has \(n + 1\) squares on each side.

\(n^2\)

They make an \(n\) by \(n\) block.

\(2n + 1\)

One row and one column around the red block.

\(n^2 + 2n + 1 = (n+1)^2\)

Expanding \((n+1)^2\) gives the same expression, so they add up to the entire square.

Each pattern is a square with its four corners missing. How many small squares are in Pattern 10? Find two ways to count Pattern \(n\).

Try it on paper, then step through the working.

5, 12, 21

Count the small squares in each pattern.

\(5 \times 5 - 4 = 21\)

Fill in the corners. Pattern 3 becomes a 5 by 5 square. Then take the 4 corners away.

\(3 \times 3 + 4 \times 3 = 21\)

A 3 by 3 square in the middle and four strips of 3.

\(12 \times 12 - 4 = 140\) and \(10 \times 10 + 4 \times 10 = 140\)

A 12 by 12 square take away 4 corners, or a 10 by 10 square and four strips of 10.

\((n+2)^2 - 4\) and \(n^2 + 4n\)

The filled-in square is 2 wider than the pattern number. The middle square is \(n\) by \(n\).

Both ways count the same squares, so the two expressions must be equal.

Pattern 1 is 1 row of 2 squares. Pattern 2 is 2 rows of 3. Pattern 3 is 3 rows of 4. How many squares are in Pattern 10? What about Pattern \(n\)?

Try it on paper, then step through the working.

2, 6, 12

Count the squares in each pattern.

\(3 \times 4 = 12\)

Pattern 3 is 3 rows of 4. Each row has one more than the pattern number.

\(3 \times 3 + 3 = 12\)

Pattern 3 is a 3 by 3 square with one more column of 3.

\(10 \times 11 = 110\) and \(10 \times 10 + 10 = 110\)

10 rows of 11, or a 10 by 10 square with one more column of 10.

\(n(n+1)\) and \(n^2 + n\)

\(n\) rows of \(n + 1\), or an \(n\) by \(n\) square with one more column of \(n\).

Both ways count the same squares, so the two expressions must be equal.

Each pattern is a cube built from small cubes. How many small cubes are in Pattern 10? What about Pattern \(n\)?

Try it on paper, then step through the working.

1, 8, 27

Pattern 2 is 2 layers of 4. Pattern 3 is 3 layers of 9.

\(3 \times 3 \times 3 = 27\)

Pattern 3 has 3 layers, and each layer is a 3 by 3 square.

\(10 \times 10 \times 10 = 1000\)

10 layers, each a 10 by 10 square.

\(n \times n \times n = n^3\)

\(n\) layers, each an \(n\) by \(n\) square.

A flat square pattern grows with \(n^2\). A cube pattern grows with \(n^3\).

A 4 × 4 × 4 cube is painted blue on every side. How many of the small cubes have 0, 1, 2 or 3 blue faces? What about an \(n \times n \times n\) cube?

Try it on paper, then step through the working.

\(8\)

The 8 corner cubes.

\(12 \times 2 = 24\)

Along each of the 12 edges, without the corners.

\(6 \times 4 = 24\)

The middle of each of the 6 faces.

\(2 \times 2 \times 2 = 8\)

The cube hidden inside.

\(8 + 24 + 24 + 8 = 64\) \(= 4^3\)

Every small cube has been counted.

\(8\), \(12(n-2)\), \(6(n-2)^2\), \((n-2)^3\)

For an \(n \times n \times n\) cube, in the same order.

A 4 × 4 × 4 cube is built from 64 small cubes. How many of the small cubes are on the outside, with at least one face showing? Count them two ways. What about an \(n \times n \times n\) cube?

Try it on paper, then step through the working.

\(4^3 - 2^3 = 64 - 8 = 56\)

All the small cubes take away the ones hidden inside.

\(8 + 24 + 24 = 56\)

Add the corner cubes, the edge cubes and the face cubes.

\(n^3 - (n-2)^3\)

This is the border problem in 3D. The border of a grid was the whole grid take away the inside.

Both ways count the same cubes, so they must give the same number.

Dots are arranged in triangles. How does the number of dots grow from one pattern to the next? How many dots are in Pattern 10?

Try it on paper, then step through the working.

1, 3, 6, 10

Count the dots in each pattern.

add 2, add 3, add 4

Each pattern is the one before with one new row, and the new row is one dot longer each time.

1, 3, 6, 10, 15, 21, 28, 36, 45, 55

Pattern 5 adds a row of 5. Pattern 6 adds a row of 6. Carry on to Pattern 10.

These are the triangle numbers. The rule is term-to-term: to get the next one, add the next whole number.

Oranges are stacked in a pyramid. The top layer is 1 orange, the next layer is a triangle of 3, then a triangle of 6. How many oranges are in a pyramid 4 layers tall? How many in 6 layers?

Try it on paper, then step through the working.

1, 3, 6, 10

Each layer is a triangle number.

\(1 + 3 + 6 + 10 = 20\)

Add the four layers.

1, 4, 10, 20, 35, 56

Each taller pyramid adds the next triangle number: add 3, add 6, add 10, add 15, add 21.

\(20 + 15 + 21 = 56\)

Add the fifth and sixth layers to the 4-layer pyramid.

The rule is term-to-term again: to get the next pyramid, add the next triangle number.

A strip is filled with squares (length 1) and dominoes (length 2). A strip of length 3 can be filled in 3 ways. How many ways are there for a strip of length 5? For length 10?

Try it on paper, then step through the working.

1, 2, 3, 5

List every way for lengths 1, 2, 3 and 4.

\(1 + 2 = 3\) and \(2 + 3 = 5\)

Each number is the sum of the two before it.

\(5 + 3 = 8\)

Every strip of length 5 ends with a square or a domino. Take off a square and a strip of length 4 is left: 5 ways. Take off a domino and a strip of length 3 is left: 3 ways.

1, 2, 3, 5, 8, 13, 21, 34, 55, 89

Keep adding the two before.

This is the Fibonacci pattern. The rule is term-to-term: add the two terms before. You do not find a term from its position.

Squares of side 1, 1, 2, 3 and 5 are fitted together. Each new square sits along the whole of the longest side. What size is the next square? What is the area of the rectangle after 6 squares?

Try it on paper, then step through the working.

1, 1, 2, 3, 5, 8

Each new square sits along the last two squares, so its side is the sum of the two before.

\(5 \times 8 = 40\)

After 5 squares the rectangle is 5 by 8. Its sides are neighbours in the sequence.

\(1 + 1 + 4 + 9 + 25 = 40\)

The rectangle is made of the squares, so their areas add up to the same number.

\(8 \times 13 = 104\) and \(40 + 64 = 104\)

The next square has side 8, and the rectangle is 8 by 13.

The sides follow the Fibonacci rule, a term-to-term rule: add the two before. The areas of the squares add up to the last side times the next one.