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A Problem of Time

Consider a watch face which has identical hands and identical marks for the hours. It is opposite to a mirror. When is the time as read direct and in the mirror exactly the same between 6 and 7?

Eight Dominoes

Using the 8 dominoes make a square where each of the columns and rows adds up to 8


The ten arcs forming the edges of the "holly leaf" are all arcs of circles of radius 1 cm. Find the length of the perimeter of the holly leaf and the area of its surface.

A Roll of Patterned Paper

Age 14 to 16
Challenge Level

Single Unit for Roll

In what follows I'm going to call the design above the "unit shape" and imagine it repeated endlessly along a line - rather like a stream of paper coming off a roll.

Here are two pieces from the roll :

Basic Pattern

Basic Pattern Rotated 180 degrees

The second piece has then been turned around (rotated 180 $^\circ$).

Your challenge:

Try to design a new unit shape (probably simpler than mine) to make a strip which looks the same after a 180 $^\circ$ rotation.

In other words, make it so that you could not say whether the torn off strip had or had not been rotated.

There are two possibilities: across and along

First the original strip could have a mirror across it.

This illustration shows the right side as the reflection of the left side

Basic Pattern reflected in a vertical mirror line

Can you create a unit shape so that the strip has reflection symmetry across a vertical mirror line ?

And where would the mirror line need to be to be placed ?

Now try a mirror 'along' the torn off strip

The mirror could be reflecting the top half,

Basic Pattern with a Horizontal Mirror Line image 1

or (below) the bottom half.

Basic Pattern with a Horizontal Mirror Line image 2

Can you make a strip that has reflection symmetry across a horizontal mirror line ?

If you'd like more of this sort of thing, try the problem called 'One Reflection Implies Another'.