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DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

Robotic Rotations

Age 11 to 16
Challenge Level

Ashlynn from ISF Academy in Hong Kong created all of the patterns:


For this picture, Ashylnn described which angles could have been used:
The smallest angle is 72$^\circ$
Multiples of 72$^\circ$ can also generate the same pattern

 Can the other patterns be made using more than one angle? Ashlynn said:
Yes. In this picture, a rotation angle of 20$^\circ$ generates 18 shapes.
Other angles that are multiples of 20 (but not multiples of 40, 60) can generate the same pattern. These are 100$^\circ$, 140$^\circ$, 220$^\circ$, 260$^\circ$

Notice that this works because 18$\times$20$^\circ$ = 360$^\circ$, which is a full circle. So after 18 rotations of 20$^\circ$ (so 18 shapes), we get back to where we started from.

To make fewer copies of the shape, we need an angle with a multiple of 360 earlier in its times table. For example, above, 5$\times$72$^\circ$=360$^\circ$, so 72$^\circ$ only generates 5 copies of the shape.

5$\times$144 = 720 = 2$\times$360, so again after 5 copies rotating through 144$^\circ$, you get back to where you started. That is why multiples of 72$^\circ$ work.

However, Ashlynn pointed out that some multiples of 20$^\circ$ generate 18 copies of the shape, but those which are multiples of 40$^\circ$ do not.
This is because, although 18$\times$40$^\circ$ = 720$^\circ$ which is 2 full circles, 9$\times$40$^\circ$ = 360$^\circ$. So after just 9 copies, the robot gets back to where it started from and begins repeating the same shapes on top of each other.