How Many Times?
1 out of 3
On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?
Problem
On a digital 24 hour clock, at certain times, all the digits are consecutive (in counting order). You can count forwards or backwards.
For example, 1:23 or 5:43.
How many times like this are there between midnight and 7:00?
How many are there between 7:00 and midday?
How many are there between midday and midnight?
Getting Started
Which digits will be possible? Why?
How will you know you've got all the different times?
Student Solutions
Jay and Ben (Mile Cross Middle School) sent in a correct solution:
Between midnight and 7:00 we found ten
0:12 1:23 5:43 6:54 2:34 3:45 3:21 4:32 4:56 2:10
There are no times between 7:00 and midday .
We found two times between midday and midnight. These are 12:34 and 23:45.
Syed (Foxford School and Community College) agrees with this answer and makes a statement about why you don't get times containing a 7, 8, or 9 in the solution:
The largest the tens digit of the minute number can be is 5, so the largest unit of the hour number is 6 in order for the time to have consecutive digits.
George (Rosebank Primary School, Leeds) also solved this one and explained his thinking well.
Jason (Priory Middle School, Dunstable) took a different view of this problem. Instead of only looking at single digit numbers, he also looked for consecutive two-digit numbers. This is what he found:
| For midnight to 7am | For 7am to midday | For midday to midnight | |
| 1:23 | 10:11 | 12:13 | 23:22 |
| 2:34 | 11:12 | 13:14 | 22:21 |
| 3:45 | 11:10 | 14:15 | 21:20 |
| 4:56 | 15:16 | 20:19 | |
| 2:10 | 16:17 | 19:18 | |
| 3:21 | 17:18 | 18:17 | |
| 4:32 | 18:19 | 17:16 | |
| 5:43 | 19:20 | 16:15 | |
| 6:54 | 20:21 | 15:14 | |
| 21:22 | 14:13 | ||
| 22:23 | 13:12 | ||
| 23:24 | 12:11 | ||
| 0:23 | |||
Some interesting patterns here Jason!
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
This problem will help consolidate children's understanding of the 24 hour clock notation. It could also be used to focus on ways of working systematically.
Possible approach
It would be good to have an interactive digital clock on the whiteboard for the duration of this lesson so that you and the class can refer to it whenever necessary. You may want to begin by asking a few oral questions based on the clock before moving on to the problem as it stands.
Explain the challenge to the class and ask children to suggest a few examples so that it is clear what is meant by consecutive. You may need to clarify that all the digits in the time need to be consecutive so, for example, 13:45 wouldn't count, as it only has three consecutive digits. Invite pairs of children to begin working on the first part of the problem. They could use mini-whiteboards to keep a record of the times they find.
After a short time, draw the group together to share ways of working. Some children may be recording answers as they occur to them, others may have some sort of system - for example starting with the earliest time and working 'upwards'. Discuss the benefit of a systematic approach - it means that we know when we have found all the solutions. Having talked about this, children will be able to apply a system to the other parts of the question.
In the plenary, as well as sharing solutions, encourage children to articulate reasons for their findings.
Key questions
Which digits will be possible? Why?
How will you know you've got all the different times?
Possible extension
Children could also investigate the times which have just three consecutive digits. 5 on the Clock is a problem that requires a similar systematic approach and also involves digital time.
Possible support
It might be useful for some children to have access to an interactive version of a digital clock themselves, perhaps at a shared computer.