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Where Can We Visit?

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?


Can you find the values at the vertices when you know the values on the edges?

Babylon Numbers

Can you make a hypothesis to explain these ancient numbers?

In the Bag

Age 11 to 14
Challenge Level


Why do this problem?

This problem introduces students to the limitations of making predictions based on small samples, and how predictions can become more reliable as the number of experiments increases. The problem challenges students to make predictions based on the information they gather. It offers students a chance to test various strategies for reaching the target in the most efficient way.

The can be used as a starting point for developing the skills of making and testing hypotheses. Students can suggest various possible strategies, then make decisions about the information that is required to compare their efficiency and how to analyse the data that is collected.

Possible approach

Place 10 marbles (or counters, multilink cubes...) in a bag/envelope/hat and ask students if they can predict the colours of the marbles.
To help them make accurate predictions allow them to pick one marble, record its colour and return it to the bag before repeating the process.
After this has been done 10 times:
"Do you now know the colours of the 10 marbles?"
"If not, why not?"
"Is there anything we can be sure of?"
Record the results of another 10 viewings.
"Are the results the same/different?" "Why?"
"Can you now predict the colours of the 10 marbles?"
Repeat a few more times.
"What can you do with the different sets of results?"
When the class become fairly confident that they can predict the contents of the bag, show them what the bag contains.

Demonstrate the interactivity and clarify the scoring system.

"Can you develop an effective strategy for reaching 1000 points?"

"Can you develop an effective strategy for reaching 1000 points in the least number of rounds?"
Allow students some time to test their strategies using the interactivity.

Bring students back together (possibly in a follow-up lesson).

Discuss, refine and list possible strategies. Ask students (possibly working in small groups) to select different strategies to test.


  • how they could test the effectiveness of their strategy
  • what data they will need to collect
  • the amount of data needed to ensure meaningful results
Ask groups of students to write down a plan for what they will do to test their strategy before they carry out the investigation. Give them time to collect, analyse and interpret the data before presenting their findings to other groups.
What methods of collection, analysis and representation were most appropriate and effective in communicating their findings? Discuss the merits and pitfalls of different approaches highlighting good choices and appropriate use of the data.

Key questions

Is it better to have just a few viewings before making a guess or is it better to have more viewings and improve your chances of guessing correctly?
How do you decide on the most effective strategy?

Possible extension

Ask students to articulate two clear strategies, test them and then produce a clear justification of why one strategy is better than the other.
Encourage students to read the article Understanding Hypotheses

Possible support

Do the introductory activity several times before moving on to the interactivity.
Discuss the value of averaging the results.