International Contest Game Mathematics Kangaroo 2013 Answer Sheet

Question

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When Matthew and Marten found their old model railway, Matthew quickly made a perfect circle from 8 identical track parts.

Marten starts to make another track with two of these pieces as shown in the picture. He wants to use as few pieces as possible to make a closed track.

How many pieces does his track consist of?

The way I thought about this problem is, each segment has an arc of $45$ degrees ($\frac{360}8$). So Marten starts with two segments attached in a way to nullify the net gain in angle. So he starts with $0$ degree after having used two tracks. So he needs $8$ more to move $360$ degrees, so he will need total $10$ tracks; But $10$ is not among the answer choices

asked Feb 24 '20 at 13:38

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3

$\begingroup$ The way I thought about this problem is, each segment has an arc of 45 degrees (360/8). So Marten starts with two segments attached in a way to nullify the net gain in angle. So he starts with 0 degree after having used two tracks. So he needs 8 more to move 360 degree, so he will need total 10 tracks; But 10 is not among answer choices $\endgroup$

Feb 24 '20 at 13:44
$\begingroup$ The angles may work out with 10, but do the lengths? $\endgroup$

Feb 24 '20 at 13:48
$\begingroup$ Can the new track pieces be flipped over? $\endgroup$

Feb 24 '20 at 14:42

$\begingroup$ The way I thought about this problem is, each segment has an arc of 45 degrees (360/8). So Marten starts with two segments attached in a way to nullify the net gain in angle. So he starts with 0 degree after having used two tracks. So he needs 8 more to move 360 degree, so he will need total 10 tracks; But 10 is not among answer choices $\endgroup$

Feb 24 '20 at 13:44
$\begingroup$ The angles may work out with 10, but do the lengths? $\endgroup$

Feb 24 '20 at 13:48
$\begingroup$ Can the new track pieces be flipped over? $\endgroup$

Feb 24 '20 at 14:42

Accepted Answer

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The way you would try to make this work with $10$ pieces would be to split the circle along a diameter, say vertical. On the ends of the left half you would put a piece turning down as it goes right. The two ends are now not in line, so you cannot close the track with the other semicircle, so $10$ pieces will not work. If you start with the left semicircle and put two pieces on each side, one turning down and then one turning up, you can close the end with the top semicircle, so $12$ pieces will work. enter image description here

answered Feb 24 '20 at 14:30

Ross MillikanRoss Millikan

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2

$\begingroup$ Diagram helped me wrap my head around this problem, so looks like your segments c and h consist each of 4 pieces, so turning around 180 degrees. $\endgroup$

Feb 24 '20 at 14:59
$\begingroup$ Yes, that is correct. To try to do the $10$ piece solution you have to delete $f$ and $g$, but then the semicircle $h$ will not line up properly. $\endgroup$

Feb 24 '20 at 15:11