### Number Detective

Follow the clues to find the mystery number.

### 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?

# All the Digits

## All the Digits

This represents the multiplication of a $4$-figure number by $3$.

The whole calculation uses each of the digits $0 - 9$ once and once only.

The $4$-figure number contains three consecutive numbers, which are not in order. The third digit is the sum of two of the consecutive numbers.

The first, third and fifth figures of the five-digit product are three consecutive numbers, again not in order. The second and fourth digits are also consecutive numbers.

Can you replace the stars in the calculation with figures?

A practical version of this activity is included in one of the Brain Buster Maths Boxes which contains hands-on challenges developed by members of NRICH and produced by BEAM. For more details and ordering information, please scroll down this page .

### Why do this problem?

This problem requires learners to think about place value and the way that standard column multiplication works. Although the problem can be done by trial and improvement, it is solved more efficiently if worked through systematically.

### Possible approach

You could start by showing the problem to the whole group and discussing what is required to do it. Do they understand what consecutive numbers are? Are they confident about the meaning of 'sum' and 'product'?

After this introduction the group could work in pairs on the problem so that they are able to talk through their ideas with a partner. This sheet is intended for rough working and the solution, and this sheet gives the blank calculation and digit cards to cut out. Give the children time to make a start and then after a suitable length of time, bring the group back together to talk about how they are getting on so far. This is a good opportunity to share some initial insights. For example, some pairs may have worked out which digits must be in the four-digit number, even if they don't know the order yet. Some may have started in a different way, for example by looking for the digit which could go in the units column of the four-digit number. Draw attention to those pairs that have adopted a system in their working which means they are trying numbers in an ordered way. This means that they are guaranteed not to leave out any possibilities. You could then leave learners to continue with the problem.

At the end, the whole class could discuss the steps in their reasoning and how they reached a full solution. Did they use all the information in the question right from the start? Which parts were most helpful and why?

### Key questions

What could the first figure of the product be if the multiplication is by $3$?
Which consecutive numbers could be in the four-digit number?
Which other digit could appear in the four-digit number?

### Possible extension

Challenge those pupils who finish quickly to prove to you that there is only one solution. How many solutions would there be if the clues about consecutive numbers did not hold?

### Possible support

Suggest working with digit cards and possibly a mini-whiteboard. This sheet gives the blank calculation and digit cards to cut out.

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. More information on many of our other activities can be found here.
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