Skip over navigation
Six Times Five
Age 11 to 14
Challenge Level 
Why do this problem?
This problem is short and encourages students to think about the meaning of place value and engages their logical thinking. It could be used as a starter to engage pupils as they come in to the classroom, though there are good extensions available for a full lesson.Possible approach
Put the problem on the board to allow pupils to familiarise
themselves with the problem.
Discuss as a group the possible forms of proper 6 digits
numbers. Students might like to decide whether numbers starting
with zero count as a proper six-digit number (no!).
Allow students to experiment to try to determine the number of
possible answers.
A good problem solving strategy is to make the problem
smaller, e.g.how many three- (or four-) digit numbers do not
contain a $5$, then to work out how to extend the solution
method.
Key questions
- Is $000001$ a six-digit number?
- How many six-digit numbers are there?
- How many choices do we have for the first digit?
- How many choices do we have for the second digit?
Possible extension
- How many six-digit numbers do not contain a $5$ or a $7$?
- How many six-digit numbers are there for which the digits increase from left to right (such as $134689$ or $356789$)?
- How many numbers less than $10$ million do not contain a $5$?
- Will your methods extend to similar problems? if so, can you express them algebraically?
- What other [interesting] questions could you ask starting "How many six-digit numbers..."?
Possible support
- You could ask the almost equivalent question "How many six-figure telephone numbers do not contain a $5$?". This encourages student to imagine dialling a number in sequence, which will may help them to see the different choices which can be made at each step of the process.
- Encourage students to adapt the problem to make it accessible: fewer digits, how many six-digit numbers are a multiple of $10$ (probably seen as a number with $0$ as last digit) or even, or a mult of $5$, or square, etc.
You may also like
Double Digit
Choose two digits and arrange them to make two double-digit numbers. Now add your double-digit numbers. Now add your single digit numbers. Divide your double-digit answer by your single-digit answer. Try lots of examples. What happens? Can you explain it?
Repeaters
Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.
Big Powers
Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.