# How many Times?

## 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 are there between 7:00 and midday?

How many are there between midday and midnight?

## Getting Started

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!