Math Detours · Grade 5 · Around the corner · 22

Fold a paper to the Moon

Each fold doubles the thickness. Seven folds is hard; forty-two would reach the Moon. Doubling is faster than it looks.

Name ____________________Date ____________

Tonight's idea

Fold a sheet of paper in half. It is now 2 layers thick. Fold it again: 4 layers. Again: 8. Every fold doubles the number of layers.

1 0 2 1 4 2 8 3 16 4 32 5 64 6 number of folds (layers shown above each stack)
The stack after 0 to 6 folds. Each stack is twice as tall as the one before.

Try it with a real sheet. Most people get stuck at about seven folds. By then there are 128 layers, and the little bundle is too thick to bend.

Doubling starts slowly and then runs away. Keep doubling: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. After ten folds there are 1024 layers. That is close to 1000, which is much easier to work with.

Ten folds make the stack about 1000 times thicker.

One sheet of paper is about 0.1 mm thick. That is thin: ten sheets make 1 mm. How many folds would it take to reach the Moon?

Watch one: all the way to the Moon

Start at 0.1 mm. Every ten folds, multiply by 1000.

10 folds: 100 mm, or 10 cm.
20 folds: 100 m.
30 folds: 100 km.
40 folds: 100,000 km.
41 folds: 200,000 km. 42 folds: 400,000 km.

The Moon is about 384,000 km away, so 42 folds would pass it. Using 1024 each time, not 1000, gives about 440,000 km. No real sheet can be folded that often, but the sum is right.

Your turn

  1. How many layers are there after (a) 8 folds, (b) 9 folds?
  2. A sheet is 0.1 mm thick. How thick is the stack after (a) 5 folds, (b) 8 folds?
  3. A piece of card is 1 mm thick. Imagine you could keep folding it. How many folds make a stack taller than a door 2 metres high? Show the doubling.
  4. A water lily doubles in size every day. On day 30 it covers the whole pond. (a) On which day did it cover half the pond? (b) On which day a quarter? (c) Why is the answer to (a) not day 15?

Stretch: rice on a chessboard

Put 1 grain of rice on the first square, 2 on the second, 4 on the third, doubling each time. Add up the first four squares. Then the first five. Spot the pattern. How many grains are on the first ten squares together?

Answers

  1. (a) 256. (b) 512. Keep doubling from 128.
  2. (a) 32 layers, so \(32 \times 0.1 = 3.2\) mm. (b) 256 layers, so \(256 \times 0.1 = 25.6\) mm, about as thick as a big book.
  3. 2 metres is 2000 mm. The stack goes 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048 mm. Ten folds give 1024 mm, which is not enough. Eleven folds give 2048 mm, so 11 folds.
  4. (a) Day 29. It doubles each day, so the day before it was half as big. (b) Day 28. (c) The lily does not grow by the same amount each day. Most of the growing happens in the last few days. On day 15 it covers only a tiny speck of the pond.

Stretch. The first four squares hold \(1+2+4+8=15\). The first five hold 31. Each total is one less than the next square: 15 is one less than 16, and 31 is one less than 32. The eleventh square holds 1024 grains, so the first ten hold 1023.

← PreviousNext: How to escape any maze →