Question 1

“My goodness,” said Dad, “just look at the time. The hour hand is exactly on a minute, and the minute hand is just one minute behind it.”

“But it’s too early in the morning for that sort of a puzzle .” said Mum. “Anyway, if you think that’s strange, I was cooking the breakfast when I noticed that the hour hand was exactly on a minute and the minute hand was 2 minutes behind it.”

Just then, Jodie came into the room, rubbing her eyes and saying “I had a terrible sleep. I had nightmare and woke up to see that the hour hand on my clock was exactly on a minute and that the minute hand was exactly 3 minutes behind. I couldn’t get back to sleep for wondering if that happens at any other time.”

Her brother Craig was quick to say “Well, I woke up and it was still dark, so I put the light on to see the time, and the minute hand was exactly on a minute, and the minute hand was 4 minutes behind.”

“Come to think of it, “ said Dad, “I seem to remember that the hour hand was exactly on a minute and the minute hand was on the same minute when your mother and I turned in.”

What were the times that this clock-crazy family were talking about?

Suggestion: You might find it helpful to draw up a clock face like this one, which has all the minute positions marked. Just a thought!

Question 2

Ann is very much into beads and has made some really nice ones in bright primary colours.
“Wouldn’t it be nice if you strung them together to make a naturally mathematical design.” I said.
“And what is a naturally mathematical design.” she asked.
“Easy”, I replied, “use as many beads as you can, but don’t ever let a colour combination of two beads repeat itself. Here, look at this one.”

Question 3

The other day, for some reason I can’t remember, I happened to write down a list of four numbers:

Then I noticed that no two of the numbers add to the same total. Go on, check it out if you don’t believe me!

Okay, all the numbers are positive whole numbers, and no two of them add to the same total, but …

Part 1
7 is the largest number in my list. Is it possible to find four positive whole numbers no two of which add to the same total where the largest is less than 7? In fact, what is the least value that the largest number could have?

Part 2
What happens when we extend the size of the list?
What is the least value that the largest of a list of five numbers can have where no two of the numbers add to the same total?

Part 3
So, it goes on, but let’s put a limit on it. What is the least value that the largest of a set of ten numbers can have where no two of the numbers add to the same total?

Note: You may well notice a familiar pattern as you build the list of numbers. In your answer, it would be good if you were to explain the pattern ... and why it breaks down eventually.

If you click on the 1, 3, 6, 7 diagram, you can open up an applet that will help you with this investigation. When you have a list of  numbers that you think is  is correct, you should:

1. Hold down <Alt> and press the <Print Screen> key. This lets you copy the screen and then you can paste it into a Word document.
2. Use the crop tool in Word which allows you to remove the bits of the picture that you don't want.
3. Then cut the diagram out of your document (<Ctrl> + <X>) and go to the Paste Special menu where you can paste it back again in .jpeg format. Otherwise your files will get very large.