Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

"Tell me the next two numbers in each of these seven minor spells",
chanted the Mathemagician, "And the great spell will crumble away!"
Can you help Anna and David break the spell?

Exploring and predicting folding, cutting and punching holes and
making spirals.

White Box

Age 7 to 18 Challenge Level:

White Box

The White Box contains a number of filled triangles.

Your challenge is to find the locations of those filled triangles in the grid.

You can fire rays into The White Box and observe where the rays exit using the Experiment Window.

Some rays will pass straight through the Box but some will be deflected by the filled triangles.

You can use the Hypothesis Window to test your ideas. Clicking on a triangle once marks the triangle as empty, clicking again fills it.

NOTES AND BACKGROUND

Anyone attempting to understand anything about the nature of atoms and molecules has to try to come to terms with their almost inconceivably tiny nature. A single hair is about the same width as a million atoms; a glass of water contains about 10 000 000 000 000 000 000 000 000 molecules. The minute size means that they're beyond the range of even the most powerful microscope, so to gain
understanding about the ways atoms are arranged within crystals and molecules, scientists need to use indirect methods.

One method involves firing rays of sufficient energy to break bits off the molecules. These particles will of course be smaller still, but they may be easier to recognise. The experimenter can then use the pieces like a jigsaw to explore how they fitted together in the original molecule.

A less destructive process involves not breaking up the structure, but deducing its arrangement by observing the areas where rays fired into it emerge. Some rays pass straight through, while others are deflected by atoms within the structure. The White Box interactivity models this process and makes an intriguing challenge as well.

Why do this problem?

The White Box interactivity models the scientific process of deducing the structure of atoms and molecules. It is a motivating and intriguing context in which learners can make and test hypotheses.

Possible approach

Set up the White Box so that it is relatively small to start with and perhaps just one triangle is filled. Explain how it works, giving learners some background about the process they will be modelling. This first set-up could be a challenge for the whole group working together so that everyone gets a feel for the way the interactivity works.

Ideally, then, learners will work in pairs using the interactivity at a computer. You might give them some time to explore for themselves and then you could pose one or more of the following questions (most of which refer to multi-atom configurations):

if an incoming ray A(in) produces B(out), does B(in) produce A(out)?

if an input ray A emerges at B can you be certain of any atomic position(s)? If not, how many ways can you find of producing that effect?

if an input ray appears to go straight through what other explanations might be possible?

is it possible to have an arrangement which has two possible exit points for an input ray?

is it possible for a ray to be affected by the same atom more than once? If so, what is the largest number of deflections you can generate?

can a ray ever get trapped inside so that it never emerges at all?

are there any arrangements of atoms where it is impossible to be certain of the arrangement?