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Lower Bound

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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Sum Equals Product

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 1 [1/3]. What other numbers have the sum equal to the product and can this be so for any whole numbers?

Turning Tangles

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Look carefully at this video of a tangled light blue ribbon. The tangle was originally made by the Twisting and Turning method, but in the video it is manipulated in a number of ways. Can you explain what is happening? Can you see some properties of this tangle that the video is trying to demonstrate? Do you think these properties hold for any tangle?
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Can you work out the fraction associated with the tangle in the video? You'll need to decide on an initial configuration and orientation, so let's say that:
  • the zero tangle would have the two ropes parallel to the green axis;
  • twisting always occurs on the green axis in the direction of the green arrow;
  • turns are a quarter turn about the blue axis in the direction of its arrow.