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Paper Folding 

Problem Statement
If a piece of paper could be folded in half fifty times, what would be the thickness of the folded paper? A ream (500 sheets) of paper is 2 inches thick.

Problem setup

First of all, it would be impossible to fold a piece of paper 50 times.  We tried and only got to 5 folds before it got too bulky.  But the problem says “if” it could be folded, so we’ll assume it could.  I imagine an Excel spreadsheet will be  needed.

 

Plans to Solve/Investigate the Problem

I’ll do a spreadsheet to help me figure it out.

 

Investigation/Exploration of the Problem

We made a spreadsheet on Excel as follows:

 

# of folds

# of sheets

# of Reams

Inches

Feet

Yards

Miles

0

1

0.002

0.004

0.000333333

0.000111111

6.31313E-08

1

2

0.004

0.008

0.000666667

0.000222222

1.26263E-07

2

4

0.008

0.016

0.001333333

0.000444444

2.52525E-07

3

8

0.016

0.032

0.002666667

0.000888889

5.05051E-07

4

16

0.032

0.064

0.005333333

0.001777778

1.0101E-06

5

32

0.064

0.128

0.010666667

0.003555556

2.0202E-06

6

64

0.128

0.256

0.021333333

0.007111111

4.0404E-06

7

128

0.256

0.512

0.042666667

0.014222222

8.08081E-06

8

256

0.512

1.024

0.085333333

0.028444444

1.61616E-05

9

512

1.024

2.048

0.170666667

0.056888889

3.23232E-05

10

1024

2.048

4.096

0.341333333

0.113777778

6.46465E-05

11

2048

4.096

8.192

0.682666667

0.227555556

0.000129293

12

4096

8.192

16.384

1.365333333

0.455111111

0.000258586

13

8192

16.384

32.768

2.730666667

0.910222222

0.000517172

14

16384

32.768

65.536

5.461333333

1.820444444

0.001034343

15

32768

65.536

131.072

10.92266667

3.640888889

0.002068687

16

65536

131.072

262.144

21.84533333

7.281777778

0.004137374

17

131072

262.144

524.288

43.69066667

14.56355556

0.008274747

18

262144

524.288

1048.576

87.38133333

29.12711111

0.016549495

19

524288

1048.576

2097.152

174.7626667

58.25422222

0.03309899

20

1048576

2097.152

4194.304

349.5253333

116.5084444

0.06619798

21

2097152

4194.304

8388.608

699.0506667

233.0168889

0.13239596

22

4194304

8388.608

16777.216

1398.101333

466.0337778

0.264791919

23

8388608

16777.216

33554.432

2796.202667

932.0675556

0.529583838

24

16777216

33554.432

67108.864

5592.405333

1864.135111

1.059167677

25

33554432

67108.864

134217.728

11184.81067

3728.270222

2.118335354

26

67108864

134217.728

268435.456

22369.62133

7456.540444

4.236670707

27

134217728

268435.456

536870.912

44739.24267

14913.08089

8.473341414

28

268435456

536870.912

1073741.82

89478.48533

29826.16178

16.94668283

29

536870912

1073741.82

2147483.65

178956.9707

59652.32356

33.89336566

30

1.074E+09

2147483.65

4294967.3

357913.9413

119304.6471

67.78673131

31

2.147E+09

4294967.3

8589934.59

715827.8827

238609.2942

135.5734626

32

4.295E+09

8589934.59

17179869.2

1431655.765

477218.5884

271.1469253

33

8.59E+09

17179869.2

34359738.4

2863311.531

954437.1769

542.2938505

34

1.718E+10

34359738.4

68719476.7

5726623.061

1908874.354

1084.587701

35

3.436E+10

68719476.7

137438953

11453246.12

3817748.708

2169.175402

36

6.872E+10

137438953

274877907

22906492.25

7635497.415

4338.350804

37

1.374E+11

274877907

549755814

45812984.49

15270994.83

8676.701608

38

2.749E+11

549755814

1099511628

91625968.98

30541989.66

17353.40322

39

5.498E+11

1099511628

2199023256

183251938

61083979.32

34706.80643

40

1.1E+12

2199023256

4398046511

366503875.9

122167958.6

69413.61286

41

2.199E+12

4398046511

8796093022

733007751.9

244335917.3

138827.2257

42

4.398E+12

8796093022

1.7592E+10

1466015504

488671834.6

277654.4515

43

8.796E+12

1.7592E+10

3.5184E+10

2932031007

977343669.1

555308.9029

44

1.759E+13

3.5184E+10

7.0369E+10

5864062015

1954687338

1110617.806

45

3.518E+13

7.0369E+10

1.4074E+11

11728124030

3909374677

2221235.612

46

7.037E+13

1.4074E+11

2.8147E+11

23456248059

7818749353

4442471.223

47

1.407E+14

2.8147E+11

5.6295E+11

46912496118

15637498706

8884942.447

48

2.815E+14

5.6295E+11

1.1259E+12

93824992237

31274997412

17769884.89

49

5.629E+14

1.1259E+12

2.2518E+12

1.8765E+11

62549994825

35539769.79

50

1.126E+15

2.2518E+12

4.5036E+12

3.753E+11

1.251E+11

71079539.57

 

Therefore, the piece of paper would be 71,079,540 miles tall/big.  That’s one big piece of paper.

 

Extensions of the Problem

One could figure out how long it would take to drive to the top of the piece of paper.  Sounds like a lot of trouble, though.

Meg Ramsey
mramsey@rockdale.k12.ga.us

Link(s) to resources, references, lesson plans, and/or other materials
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