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Lastly - Well

What are the last two digits of 2^(2^2003)?

Lesser Digits

How many positive integers less than or equal to 4000 can be written down without using the digits 7, 8 or 9?


Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers and so on?

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, 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.