### Chess

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

### 2001 Spatial Oddity

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

### Screwed-up

A cylindrical helix is just a spiral on a cylinder, like an ordinary spring or the thread on a bolt. If I turn a left-handed helix over (top to bottom) does it become a right handed helix?

# Making Rectangles, Making Squares

### Why do this problem?

This problem offers the opportunity to assess learner's knowledge, to pose problems, to share ideas and follow different routes. At one level, it is possible to discuss rational and irrational numbers as well as utilise Pythagoras' theorem. At others you might explore symmetry or triangle animals. It links to a number of other problems on the site including Triangle Relations and Equal Equilateral Triangles .

### Possible approach

Just hand out the triangles without any indication of how they are related or formed. Working in small groups, ask the learners to "play" with the triangles for a few minutes and write on large sheets of paper (to share with the rest of the group) what they consider to be four key mathematical properties of the triangles.
Invite the class to walk around the room and look at what other groups have written, then invite them to add anything they feel is important to their own lists. After this encourage discussion of the key points and salient features. It is at this point the relationships between the two triangles can be established - including their equal areas.
After this you might choose to select a feature that has been mentioned such as:

"The two triangles can be put together to form a right-angled triangle."

to lead into the main problem questions.

Alternatively - why not give the groups more time to come up with problems of their own first? In my experience one of these is invariably about rectangles or squares - so you can end up with a problem that the group has posed themselves, yet meets your learning objectives (if they require this focus). I have often allowed groups to choose a problem from the class as a whole to work on. They can tackle the task as well as discuss how the problem as posed can be improved.

Different routes offer great opportunities for display and sharing.

### Key questions

• What properties do the triangles have on their own, or when joined together?
• Can you write down four things that you think are most mathematically significant about these shapes?
• Can you pose a question for someone else to answer that involves these shapes?

### Possible extension

By offering the group opportunities to pose their own problems it is possible to identify suitable challenges for the most able.

### Possible support

As with the extension opportunities, this is an ideal problem for taking those whose background knowledge is less well developed from a more suitable starting point, for example:
"What equilateral triangles can you make? or
"What rectangles can you make and what are the smallest/largest number of each triangle that is required in each case?"

Then of course there are all the problems based on symmetry. (For example, how many different symmetrical shapes can you make with just four triangles?)