How long is a piece of string?

Logic Level 3

Bernice has a long piece of string. She cuts it into two pieces, then cuts each piece into two each again, then cuts each new piece into two each once again a third time, so that she's left with 8 pieces of string. The point at which she cuts a piece of string is never more than 2cm from the exact center of each piece being cut.

What is the greatest possible difference between the longest and shortest pieces of string after she has finished cutting?

7cm 8cm 6cm 12cm

2 solutions

Joel Toms
Aug 2, 2015

The actual length is irrelevant*, except of course that it must be sufficient to achieve the maximal error. Let the original length be 100cm.

Then, the shortest piece will be made as follows:

100cm --> 50cm; shortest possible = 48cm.
48cm --> 24cm; shortest possible = 22cm.
22cm --> 11cm; shortest possible = 9cm.

Similarly, the longest piece will be made as follows:

100cm --> 50cm; longest possible = 52cm.
52cm --> 26cm; longest possible = 28cm.
28cm --> 14cm; longest possible = 16cm.

Now, 16cm - 9cm = 7cm.

*If you don't believe this, try a different starting length or check an algebraic proof.

Maggie Miller
Aug 2, 2015

Let the original length be $2l$ ,

To maximize the difference between the shortest and longest lengths, always cut the leftmost and rightmost pieces 2cm left of center.

The length of the shortest string is then $\frac{\frac{l-2}{2}-2}{2}-2$ cm while the length of the longest string is $\frac{\frac{l+2}{2}+2}{2}+2$ cm. The difference in these lengths is $\frac{\frac{4}{2}+4}{2}+4=\boxed{7\text{cm}}$ .

Thanks for this. I spent quite a while trying to work out how to write another solution in the same place, but you saved me the trouble. Cheers!

Joel Toms - 5 years, 10 months ago

How long is a piece of string?

2 solutions

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