The power of 2

In this activity, the students should initially use geoboards or square dot paper to help them work out the area of the striped squares.

The area of the striped square in figure 1 (which is the shape in iv) is the area of the surrounding 4 by 4 square, that is, 16 square units, minus the area of the four corner triangles. The area of each triangle is 2 square units (one-half of a 2 by 2 square). So the striped area is equal to 16 square units minus 4 x 2 square units. We usually write 8 square units as 8 units², so in this case, the area is 16 units² – 8 units² = 8 units².

The area of the striped square in figure 2 (which is the shape in iii) is the area of the striped square from figure 1 minus the area of the four corner triangles. The area of each of these triangles is 1 unit² (two halves of a 1 by 1 square). So the striped area in figure 2 is equal to 8 units² – 4 x 1 = 4 units².

The students need to repeat this process for the other striped squares and write a rule connecting successive striped square areas. They will see that the striped area in figure 1 is double the striped area in figure 2. This relationship can also be seen clearly by folding squares of paper, as illustrated in the diagram below.

Square A is twice the area of square B. Square A is the square that encloses square B.

So a simple rule is: the area of a square is twice the area of the square it encloses or, as given in the answers, the area of a square is double the area of the enclosed square.

In this activity, the students repeatedly fold and then cut pieces of paper. Each cut doubles the number of pieces of paper. These results can be shown in a table:

So, for 10 cuts, there will be 2¹⁰ pieces of paper. This is 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 1 024. We express this as “2 to the power of 10 equals 1 024”. Pressing the following sequence of keys on most calculators will give the value of 210 as 1 024.

Where this sequence of keys doesn’t work, experiment until you find a sequence that does. For example, the following sequence works with some calculators:

Scientific calculators can also be used to find the value of powers of numbers. We can express these algebraically as y^x, which we say as “any number, y, raised to the power of any number, x”. So the value of 5⁷ can be found by pressing the following keys:

Note that some scientific calculators use the key x^y instead of y^x.

This activity explores a simplified version of a mathematical puzzle that was invented in 1883 by the French mathematician Edouard Lucas (1842–1891). The puzzle is known as the Tower of Hanoi and sometimes as the Tower of Brahma. It was inspired by a Hindu legend that tells of the mental discipline demanded of young priests. The legend says that at the beginning of time, the priests in the temple were given 64 gold discs, each one a little smaller than the one beneath it. The priests were to transfer the 64 gold discs from one of three poles to another, via the second pole where necessary, in such a way that a disc could never be placed on top of a smaller disc. The legend goes on to say that when this task was finished, the temple would crumble into dust and the world would end.

The students are initially asked to see if they can work out the minimum number of moves to transfer only 3 lids (discs) rather than the 64 in the legend. To understand how this puzzle works, the students will find it helpful to try the puzzle with just 1 lid and then 2 lids. The following diagrams show the moves for 1, 2, and 3 lids.

If you look closely at the diagrams, you may notice a symmetrical pattern. Each diagram has a centre of rotational symmetry, marked with a dot. The pattern for the number of moves is shown in this table:

The pattern in the table shows that there are 2(number of lids), – 1 moves for any number of lids. When there are x lids, the number of moves can be expressed algebraically as 2^x – 1.

If we return to the Hindu legend associated with this puzzle, we see that to transfer 64 gold discs, 2⁶⁴ – 1 moves will be required. Altogether, this is 18 446 744 073 709 551 615 moves. If the priests worked continuously for 24 hours a day, 7 days a week, making one move every second, the complete transfer would take slightly more than 580 billion years. This is more than current estimates for the age of the universe, so the legend may well be correct in asserting that the world will end when the priests finish their task!

Activity 1

1.

a.

ii. 2 units2
iii. 4 units2
iv. 8 units2
v. 16 units2

b. 32 units2

2.

A possible rule is: double the area of the enclosed square.

Activity 2

1.

32. (2 x 2 x 2 x 2 x 2)

2.

At least 7. (26 = 64 and 27 = 128)

3.

Yes. 20 cuts gives 1 048 576 pieces, which is more than one million pieces. There are several ways that he may have used a calculator. One way is to press the following sequence of keys (pressing the “equals” key 19 times):

On some calculators, this is

Activity 3

1.

a. 7 moves

b. It takes 3 moves to shift a stack of 2 lids and 15 moves to shift a stack of 4 lids.

2.

It will take 31 moves to shift a stack of 5 lids. A rule for this is 2⁵ – 1. The table below shows how the rule works: