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A Change in Code

Age 14 to 16
Challenge Level Yellow star
  • Problem
  • Getting Started
  • Student Solutions
  • Teachers' Resources

Why do this problem?

Code-breaking is often about partial conclusions gradually adding up to possibilities. This problem is unlikely to be done instantly by most students, so discussion should bring up lots of helpful thoughts to share around a group, energising explanation and stimulating individuals into new reasoning and strategy.

Possible approach

Ask students to look at the code and the column of original values and to share their first thoughts. Hopefully including the insight that whole numbers have stayed as whole numbers after the increase.
The three codings make progressively more demanding challenges.
Maintain an emphasis on the deductive process that establishes the solution rather than merely confirming that a particular multiplier works, though verification should of course be part of the process.
 
A teacher comments:
After some initial thought and discussion all (Year 9 set 1) made good progress and found a number of different ways into the problem. The second part of the problem raised points which led neatly into reverse percentages.

Key questions

What could all the original numbers be divided by without producing a decimal anywhere in the results column ?

Possible extension

Able students may like to design similar coded columns for each other to crack.
Discussion may include an exploration of how many values need to be seen coded before the solution multiplier is known for sure.

Possible support

Students who are not ready for this problem without some preliminary activity should generate simple two-digit 'codings' for themselves and swap these around the group for others to crack.

Encourage exploration to discover the multipliers that tend not to produce many decimal options, and then pick the original numbers so that not even these decimal residuals appear.

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The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.

NRICH is part of the family of activities in the Millennium Mathematics Project.

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