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### Schools Mathematics Grand Challenge

PROBLEM 28:

A new train timetable is being designed for the route between Maynooth and Dublin, and it is hoped to greatly increase the frequency of trains on the line. However, the timetable for departures from Maynooth must satisfy certain conditions.

1. The first train to leave Maynooth in the morning must depart at 7.00am exactly;
2. The gap between successive departures from Maynooth must be either exactly 5 minutes or exactly 17 minutes.

Working with these rules, what is the latest time before 12 noon at which no train can possibly leave Maynooth for Dublin? For example, no train can possibly leave at 7.11 or at 7.06.

Give your answer in the form HH.MM (H for hour, M for minute).

In this question, we want to know the latest time before 12 noon at which NO train departure from Maynooth could be scheduled.

For example, following rules 1 and 2, a train could be scheduled to depart at 7.05, 7.10, 7.15, 7,17, 7.22, ...

On the other hand, if we follow rules 1 and 2, no departure could be scheduled at the times 7.01, 7.02, 7.03, 7.04, 7.06, 7.07, 7.11, 7.19, ... We want the latest such time before noon.

PROBLEM 29:

How many positive whole numbers have the following properties:

1. No digit of the number is 0;
2. The sum of the digits of the number is equal to 9;
3. The number has an even number of even digits?

HINT: Since 0 (zero) is an even number you can have no even digits in the number. This means 9 and 111111111 are OK, but 27 isn't.

PROBLEM 30:

John is holding a party at his house and has bought one very large pizza. He wishes to share it out by making a number of straight line cuts across the pizza such as that shown below (Each cut must go from one point on the circumference of the pizza to a different point on its circumference).

If John makes 12 cuts through the pizza, and nobody takes more than one piece of pizza, what is the maximum number of people that can get a piece?

For example, if he made 3 cuts through the pizza, the answer would be 7, as shown below. Note, the pieces do not need to be the same size.

EXTRA "PROBLEM" 31:

There are no marks for this problem, but please let us know the number of your favourite problem by entering it as the answer to problem 31.