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Some points never touch, other points touch only once, other points touch twice.
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Age 14 to 16
Challenge Level 
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When the disc is in the corner there is one quarter of the
circumference (that is a length $2\pi/4 = \pi/2$) which does not
come in contact with the edge of the tray. As the disc rolls round
one circuit:
an arc of length $2$ units comes into contact,
then an arc of length $\pi/2$ gets missed,
then an arc of length $1$ unit comes into contact,
then an arc of length $\pi/2$ gets missed,
then an arc of length $2$ units comes into contact,
then an arc of length $\pi/2$ gets missed,
then an arc of length $1$ unit comes into contact.
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Some points never touch, other points touch only once, other points touch twice.
As an extension to this problem, can you now work out how much of the circumference of the disc never touches the edge of the tray?
The article 'A Rolling Disc' discusses variations of this problem.
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