Why do this problem?
Code-breaking is often about partial conclusions gradually adding
up to possibilities. This
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.
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
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
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.
- What could all the original numbers be divided by without
producing a decimal anywhere in the results column ?
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.
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.