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This
problem offers opportunities to reinforce the language and
characteristics of numbers such as odd, even, multiple. However it
is also a problem which can be tackled in a systematic way, or,
more elegantly, through insightful reading of the clues. Comparing
the two methods can be a useful exercise in considering what
mathematical thinking looks like.

Put three two-digit numbers, such as $93$, $56$, $75$ on the
board and ask the children which is the odd one out and why. Listen
for explanations which include descriptions of place value, odds
and evens, and perhaps multiples.

Say that you are going to choose one of them secretly and they
can ask just one or two questions to see if they can find which one
it is. For each correct guess ask the children to justify their
answer.

Put $23$, $45$, $62$, $101$, $94$ on the board and tell then
you have chosen one of these. What questions would be good ones to
ask and why? Make a record of the questions on the board so that
the children can refer to them. Again, ask the children to justify
any conclusions they come to.

Then offer the problem to pairs of children. Say that you are
interested in the mystery number but also how they know and you
will be asking for a description of what they did and in what
order, so they may like to keep some notes.

Bring the children back together and ask a few pairs to
describe their method. Listen for those who go through the clues in
order eliminating the impossible numbers, and explain that working
systematically is a very important skill for a mathematician.

But also look out for the children who have scanned through
the clues to find one that is more useful and saves some work - for
example

'The sum of the two digits is a multiple of five.'

which means the magic number must be 46 or 64. After that it
only needs one more carefully chosen clue to distinguish between
them:

'The digit in the tens place is greater that the digit in the
unit (or ones) place.'

Bring out the importance of scanning through to see which
clues might be the most useful.

Which question shall we ask first? Why?

What does 'multiple of five' mean?

An obvious extension is for the children to make up their own examples for each other. They could repeat the format of the given problem, or play 'what's my number?' with a partner, where they try to find out what the mystery number is in the minimum number of guesses. Give an opportunity for them to describe why some questions are more useful than others.

Sometimes it can be confusing to be given too much information
at one time. Write the clues and the numbers out on separate cards.
Spread the number cards out and offer the clues one at a time,
encouraging discussion of the characteristics of each number in
response to the clue.