### Six Is the Sum

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?

### (w)holy Numbers

A church hymn book contains 700 hymns. The numbers of the hymns are displayed by combining special small single-digit boards. What is the minimum number of small boards that is needed?

### ABC

In the multiplication calculation, some of the digits have been replaced by letters and others by asterisks. Can you reconstruct the original multiplication?

# Number Detective

### Why do this problem?

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.

### Possible approach

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.

### Key questions

Which question shall we ask first? Why?
What does 'multiple of five' mean?

### Possible extension

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.

### Possible support

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.