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 Write-up

Title
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

This problem requires readers to fold a piece of paper 50 times.  After folding the paper fifty times, we must determine how thickness of that sheet of paper.

Plans to Solve/Investigate the Problem

My initial plan to solve the problem was to try to fold the paper a couple of times to see if a pattern existed between the number of folds and the thickness.

Investigation/Exploration of the Problem

Deciding that it was almost impossible to fold a piece of paper fifty times, I came up with an exponential formula for determining the number of sheets of paper based on the number of folds.  The following spreadsheet was created, including the formulas for each category.

 number of folds number of sheets reams inches feet yards miles 2 (folds) ^ number of folds number of sheets/500 sheets in a ream 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 1073741.824 89478.48533 29826.16178 16.94668283 29 536870912 1073741.824 2147483.648 178956.9707 59652.32356 33.89336566 30 1073741824 2147483.648 4294967.296 357913.9413 119304.6471 67.78673131 31 2147483648 4294967.296 8589934.592 715827.8827 238609.2942 135.5734626 32 4294967296 8589934.592 17179869.18 1431655.765 477218.5884 271.1469253 33 8589934592 17179869.18 34359738.37 2863311.531 954437.1769 542.2938505 34 17179869184 34359738.37 68719476.74 5726623.061 1908874.354 1084.587701 35 34359738368 68719476.74 137438953.5 11453246.12 3817748.708 2169.175402 36 68719476736 137438953.5 274877906.9 22906492.25 7635497.415 4338.350804 37 1.37439E+11 274877906.9 549755813.9 45812984.49 15270994.83 8676.701608 38 2.74878E+11 549755813.9 1099511628 91625968.98 30541989.66 17353.40322 39 5.49756E+11 1099511628 2199023256 183251938 61083979.32 34706.80643 40 1.09951E+12 2199023256 4398046511 366503875.9 122167958.6 69413.61286 41 2.19902E+12 4398046511 8796093022 733007751.9 244335917.3 138827.2257 42 4.39805E+12 8796093022 17592186044 1466015504 488671834.6 277654.4515 43 8.79609E+12 17592186044 35184372089 2932031007 977343669.1 555308.9029 44 1.75922E+13 35184372089 70368744178 5864062015 1954687338 1110617.806 45 3.51844E+13 70368744178 1.40737E+11 11728124030 3909374677 2221235.612 46 7.03687E+13 1.40737E+11 2.81475E+11 23456248059 7818749353 4442471.223 47 1.40737E+14 2.81475E+11 5.6295E+11 46912496118 15637498706 8884942.447 48 2.81475E+14 5.6295E+11 1.1259E+12 93824992237 31274997412 17769884.89 49 5.6295E+14 1.1259E+12 2.2518E+12 1.8765E+11 62549994825 35539769.79 50 1.1259E+15 2.2518E+12 4.5036E+12 3.753E+11 1.251E+11 71079539.57

The last couple of columns are metric conversions from inches to miles.  I kept converting in order to create a whole number that did not have scientific notation.  The number highlighted in yellow would indicate how many miles one sheet of paper would be if folded fifty times.

Extensions of the Problem

One of the extensions discussed was how far this sheet of paper would be relative to the sun.

Author & Contact
Mesha Rainey
M Rainey

Link(s) to resources, references, lesson plans, and/or other materials
None

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