25 Jan 2013

Prize Maths Quiz: Tinkering With Triangles (PMQ3)

Alice was sitting on the floor toying with her number cards. She was using them to make pretty shapes and then see if she could make number patterns with the shapes.

Her attention was suddenly distracted by a clatter behind her. “What are you doing?” Bill looked like he was in the middle of destroying her torch; though more likely nearer the end.

“Sorry, I was just taking it apart to see how it works.”

“Huh?! There are no moving parts in a torch. Is that your big discovery?”

“Ah, but there is... there's the switch!”

“And what good is a torch without a switch?” Alice stood up and leered down at Bill, who seemed to have surrounded himself with an octopus’s garden made from bits of her torch; her ex-torch. Actually, it was also her lost torch; where had he found it?

“It wasn’t working, so I looked inside and, with a bit of force I bent this little switch here so that it now touches the battery. I’ll show you once I put it all together.”

If you put it back together!” snapped Alice. She looked at Bill with a puzzled stare; she was searching for something in her mind. Tinkering! That’s the word, Alice thought. She went back to her numbers, with the idea in her head that perhaps Bill wasn’t always such a pea-brain. Then again, he might just be pretending to be clever by breaking things.

She flopped herself back down on her big cushion and surveyed the numbers scattered around her. Perhaps this was her garden and perhaps she too was a tinker. Time to put some order into this chaos.

Now let me think...

The Question

Place the numbers 1 to 9 into the triangular grid (shown above) in such a way that the sum along each side equals 21. You must use every number.

a)    What is the sum of the three corners?

b)    Find a complete solution and show your method of solution.

You may give your final answer either as three rows of four digits, each starting at a vertex, or a string of numbers starting at any vertex and going around the triangle.

Most open rounds of national mathematics competitions are either multiple-choice papers or single answers written into a form. This doesn't mean the questions themselves are easy, it just means you never need to show your thinking process. However, if you do very well in such competitions you may be invited to follow-up events. The further up the mathematical ladder you climb, the more open-ended questions you will find. Your method of solution becomes as important as the final answer - or answers, in the plural. You are not expected to solve the puzzles in your head; even the people who set the papers can't do the questions in their head. But they know that there are different methods of solution and that some are quicker and more efficient than others. Perhaps above all other skills, finding the best method of laying out a problem is a key measure of your mathematical thinking. And that's why I want to see full solutions and a final answer.

How to Enter

Send your complete solution by email to [Email removed. Competition now closed. PMQ4 starts next Friday]. This email address shall be removed after the competition closes to avoid spam. This PMQ3 competition closes on Sunday 27 January at 23:59 GMT.

The Prize

The prizes for this PMQ3 are 4 free places in our Online Classroom for 2 months. The very first correct solution will receive a prize plus three others randomly selected from all the other correct answers. The email time stamp shall determine the order of entries received.
Quick Rules

Look at the expanded rules on our PMQ page.
Anybody can enter our Prize Maths Quiz; adults and students.
Use the official server clock in the right column to avoid late entries.
All emails and email addresses sent to us will be deleted after the winners have been processed.
DO NOT submit your entries to the comments section at the bottom of this post.
You CAN discuss it there but you must email us to enter the competition.
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Enjoy the challenge!
Send us your solution! You can't win if you don't participate.
Then tell your friends!

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