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 Coin through a hole (Posted on 2006-05-29)
Take a piece of paper and cut out a perfect circle with diameter 1 inch.

What is the diameter of the largest unaltered coin which may be passed through the hole without tearing it?

(Consider the coin to be very thin, and the paper to be very flexible and tear resistant, but not at all stretchy.)

 See The Solution Submitted by Jer Rating: 3.0000 (5 votes)

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 summary | Comment 12 of 24 |

Assuming you start with a fold that would extend through the center of the hole, forming a semicircular arc, and then subsequently fold the paper going though points along the circumference so as to halve each previously formed arc (thus at each stage forming 2^n smaller arcs), you get a serrated edge that at the limit will approach a straight line with the same length as the semicircumference of the original circle, then at each stage:

As each new set of folds takes place, along the midpoints of the previous arcs, the number of arcs doubles, and the size of each one halves. The chord length will be sin A/2, where A is 180 degrees at stage zero and halves at each stage.  There will be 2^n chords, so at each stage the total of all the chords will be 2^n * sin 180/(2^(n+1)).

The total chord length (distance from cusp to cusp to cusp ...) at different stages will be:

` 1             1.414214 2             1.530734 3             1.560723 4             1.568274 5             1.570166 6             1.570639 7             1.570757 8             1.570786 9             1.570794 10            1.570796 11            1.570796 12            1.570796 13            1.570796 14            1.570796 15            1.570796 16            1.570796 17            1.570796 18            1.570796 19            1.570796 20            1.570796 21            1.570796 22            1.570796 23            1.570796 24            1.570796 25            1.570796 `

As you can see, depending on how many folds you can make (2^n) the closer you can get to a coin with diameter pi/2.  My initial guess had stopped after the first two folds (n=1) after the initial fold (n=0).  I doubt that you can really get much past n=2 with real paper.

This is similar to starting out with a metal semicircle with a hinge in the middle. Swing out the half arcs until the endpoints and the hinge are in a straight line.  Then install new hinges at the midpoints of all the formed arcs and continue from there.

 Posted by Charlie on 2006-05-30 11:21:16

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