How Thick Does It Get?

This puzzle was suggested by Malcolm Gladwell's book The Tipping Point.

Fold a piece of paper in half. It is now twice as thick as it was. Fold in half again. Now it is four times as thick as the original, unfolded sheet.

Continue folding the paper in half. Do you find that it is difficult to go much further? You could try a larger piece of paper, but you would still find it difficult to continue folding it in half.

Imagine that you didn’t have this limitation and that you could take a very large piece of paper and fold it in half 50 times. How thick would it be?

Before you try to calculate the result, take a guess as to about how thick you think it would be. How does your calculation compare with your guess?

Course:

Math [1]

Result/Solution(s)

Let’s look at the pattern we get by folding. After the first fold, the result is 2 sheets thick. After the second fold, it is 4 sheets thick, then 8, then 16 and so on:

2, 4, 8, 16 . . .

This sequence may be written as powers of 2:

2¹, 2², 2³, 2⁴. . .

If we fold 50 times, the thickness of the final result is 2⁵⁰ sheets. How thick is this?

We measured the thickness of a one ream package of paper we had here in the office. It was 5 cm (2 in) thick. Since there are 500 sheets in a ream, each sheet is 0.01 cm (.004 in) thick.

Now 2⁵⁰ is a very large number. It equals 1.1259 x 10¹⁵. This may also be written as

1,125,900,000,000,000

That’s a very thick pile of paper! Since each sheet is 0.01 cm thick, we can divide our result by 100 to find out how high it is in cm:

11,259,000,000,000 cm (4,432,677,165,400 in)

It’s still hard to imagine how big this is, so let’s divide by 100 again to convert to m:

112,590,000,000 m (369,389,763,780 ft)

Maybe we should try km. There are 1,000 m in a km, so dividing by 1,000 we have:

112,590,000 km (69,960,183 mi)

The distance from the Earth to the Sun is about 150,000,000 km (93,205,679 mi)

Does this result surprise you?

Math Puzzle [2]