Building Bricks

An Investigation

Building Bricks A child is playing with building bricks like this one.

He decides to build the simple tower shown.

How many bricks would he need to build this tower?

If the last column was 14 bricks high

how many bricks would he need?

If the last column was 50 bricks high

how many bricks would he need now?

The German mathematician Carl Friedrich Gauss gave us an idea which might help to find the answers to such questions.

Suppose you are required to add together the numbers

1, 2, 3, -------- up to 100.

Then Gauss suggested the following

Let S = 1 + 2 + 3 + 4 + 5 + - - - - + 99 + 100

Reversing these he wrote

S = 100 + 99 + 98 + 2 + 1

Adding these gives

2 S = 101 + 101 + 101 + 101 + 101 + 101

There are 100 terms all equal to 101 so

2 S = 100 101

which on dividing by 2 gives S = 5050.

Apply Gauss’s method to the previous type of tower when the highest column is 140

bricks high, to determine how many bricks will be needed to build the tower.

How many bricks would be needed if the highest column was 184 bricks high?

Up and down towers:- Consider the ‘up and down’ tower shown.

The centre column is seven bricks high.

How many bricks are needed

to make this tower?

If the centre column was 10 bricks high,

how many small bricks would be needed?

If the centre column was 100 bricks, how

many bricks would be needed?

Skeleton Tower:- Now consider the tower shown. It has four arms leading away from the centre column which is 7 bricks high. How many small bricks will be needed to make this tower?

If the centre column of this kind of tower was 20 bricks high, how many small bricks would be needed to build it?

What number of small bricks would be needed if the centre column was 80 bricks high?

Some

Solutions

Building Bricks:- Solution

S = 1 + 2 + 3 + - - - + 139 + 140

S = 140 + 139 + - - - + 2 + 1

2 S = 140 141

S =

Up and down tower:-

S = 1 + 2 + 3 + 4 + 5 + 6

S = 6 + 5 + 4 + 3 + 2 + 1

762 S

98702

141140

2

76S

Therefore you will need 21 + 21 + 7=49 bricks

Up and down tower:- 10 bricks high

S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9

S = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1

2 S = 9 10

S=45

Therefore you will need 45 + 45 + 10 = 100 bricks

Up and down tower:- 1000 bricks high

S = 1 + 2 + 3 + 4 + - - + 99 + 100

S = 100 + 99 + - - + 2 + 1

2 S = 100 101

S=4950

Therefore you will need 4950 + 4950 + 100 = 10000 bricks

Skeleton Tower:- 7 bricks high

S = 1 + 2 + 3 + 4 + 5 + 6

S = 6 + 5 + 4 + 3 + 2 + 1

2 S = 6 7

S=

S = Therefore you will need 4 21 + 7 = 91 bricks

212

76

Skeleton Tower:- 20 bricks high

S = 1 + 2 + 3 + - - + 19

S = 19 + 18 + - - + 2 + 1

2 S = 19 20

S = 190

Therefore you will need 4 190 + 20 = 780 bricks

Skeleton Tower:- 80 bricks high

S = 1 + 2 + 3 + - - 78 + 79

S = 79 + 78 - - + 2 + 1

2 S = 79 80

S = 3160

Therefore you will need 4 3160 + 80 = 12720 bricks