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Alternatively, 1 metre (3 ft 3 in) of string is spliced into the original string, and the extended string rearranged so that it is at a uniform height above the equator. The question that is then posed is whether the gap between string and Earth will allow the passage of a car, a cat or a
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In the second phrasing, considering that 1 metre (3 ft 3 in) is almost negligible compared with the 40,000 km (25,000 mi) circumference, the first response may be that the new position of the string will be no different from the original surface-hugging position. The answer is
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solution. In a version of this puzzle, string is tightly wrapped around the equator of a perfectly spherical Earth. If the string should be raised 1 metre (3 ft 3 in) off the ground, all the way along the equator, how much longer would the string be?
369:{\displaystyle {\begin{aligned}c+\varDelta c&=2\pi (r+\varDelta r)\\2\pi r+\varDelta c&=2\pi r+2\pi \varDelta r\\\varDelta c&=2\pi \varDelta r\\\therefore \;\varDelta r&={\frac {\varDelta c}{2\pi }}\end{aligned}}}
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This diagram gives a visual analogue using a square: regardless of the size of the square, the added perimeter is the sum of the four blue arcs, a circle with the same radius as the offset.
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As the string must be raised all along the entire 40,000 km (25,000 mi) circumference, one might expect several kilometres of additional string. Surprisingly, the answer is 2
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Even more surprising is that the size of the sphere or circle around which the string is spanned is irrelevant, and may be anything from the size of an atom to the
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Visualisation showing that the length added to the circumference (blue) is dependent only on the additional radius (red) and not the original circumference (grey)
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times the width of the lane, whether the circumference of the inside lane is the standard 400 m (1,300 ft) or the size of a galaxy.
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This observation also means that an athletics track has the same offset between starting lines on each lane, equal to 2
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Photograph showing the offset between starting lines of an athletics track
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that a cat will easily pass through the gap, the size of which will be
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or closed curve which does not intersect itself. If the shape is
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109:times the offset, times the absolute value of its
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440:. Courier Dover Publications. p. 2436.
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82:metres or about 16 cm (6.3 in).
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437:The world of mathematics, Volume 4
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30:is a mathematical puzzle with a
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51:Visual analogue using a square
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379:regardless of the value of
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434:Newman, James Roy (2000).
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477:Mathematical paradoxes
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414:Napkin ring problem
119:More formally, let
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164:{{{annotations}}}
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466:Categories
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383: .
357:π
346:Δ
330:Δ
326:∴
316:Δ
313:π
297:Δ
287:Δ
284:π
272:π
256:Δ
247:π
231:Δ
219:π
203:Δ
150: ,
87:Milky Way
394:See also
43:Solution
454:p. 2436
404:annulus
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133:Δr
129:Δc
103:complex
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472:Length
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410:length
408:chord
442:ISBN
105:, 2
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388:π
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148:R
145:π
139:R
125:r
121:c
107:π
95:π
76:π
74:2
71:/
68:1
57:π
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