4 Oct 2013
A Borromean Number Puzzle: Professor Pailyn's Mathematical Quiz PMQ39
“Well, hello Alice. My name is Professor Pailyn.”
“Professor Pie-Lynne, pleased to meet you.” Alice felt like a little pixie shaking hands with a large bear, but he had a broad smile under that grizzly beard and looked friendly enough.
“So... what work do you do, Professor?” Alice was trying to be nosey to see what the inside of the house looked like, but she couldn’t see round the Professor - nor through him, of course.
“I’m a mathematician.”
“Really! That’s fantastic! I love maths. My cousin is a genius at it and I keep trying to trip her up with some fiendishly difficult puzzles.”
“Do you ever succeed?”
“Well... not yet!” Alice grinned a mischievous grin. She looked around the hall for any tell-tale signs of genius – and spotted a lovely shiny shape. “What’s that?” she enquired. The Professor had little choice but to usher the girl in.
“That’s a sculpture of three Borromean rings.”
“Bor-ro-me-an. It’s a fancy word for interlocking rings. The interesting thing is that if you cut one ring and remove it, the other two rings fall apart. So the three rings together are locked but any two of them are not.”
“That’s weird.” Said Alice, (and rhymes, thought Alice). She walked around the sculpture, looking at it from different angles, like experts did when trying to think of something clever to say.
“And if you’re wondering, they are not really circular rings; they are ellipses.”
“Ah, I know what those are; they are squashed circles!” That wasn’t quite the air of sophistication that Alice was aiming for, but it gave her a bright idea.”I know a puzzle with those Borrow-me-Anne rings. Do you have some paper so I can show you?” She hoped this tactic would make the Professor completely forget her silly squashed circles.
Professor Pailyn smiled and led Alice into the living room. There were still piles of boxes strewn around, the bookshelves were only partially filled but, where the books were missing, there were more exquisite little sculptures of magical shapes. Alice was enjoying herself. She sat down at what seemed to her a huge mahogany table and started to draw.
“Now, here are the three rings. At each junction there is a little circle and in each circle we have to put a number. As there are six little circles, we must use just the numbers from 1 to 6. Can you write those numbers in such a way that the sums of the four numbers on each large ring are the same?”
Professor Pailyn was thinking about how long to pretend to be thinking so as not to offend his bright young guest. He decided to show her a bit of algebra.
“Wow, that’s too easy doing it like that! I just solved it in my head.”
“I’m impressed that you can do that. Let me show you a slightly harder version of the same problem. Imagine that the six numbers have to be primes. What is the smallest possible total for the sum of the four numbers on each ring?”
“Mmm... I can’t do that in my head. I’ll need to write down the primes.”
“Let me go and make us some tea while you’re solving this. Can’t have cake without tea.” Professor Pailyn left Alice to her number juggling. She could hear him clinking around with tea things. She was trying to solve this before the Professor returned.
Did it! She grinned to herself and, although she felt like jumping up and down and rushing into the kitchen flapping her sheet of paper, she kept calm and quiet – or at least pretended to.
“That’s excellent! Well done.” Exclaimed the Professor as he laid down the tea tray.
“I’ve had another idea.” Said Alice.”Is there a maximum possible total for the sums on each ring?”
“Well, there are an infinite number of primes, so that question is going to be a bit tricky.” Alice looked a tad deflated that her smart idea wasn’t actually so very smart.”However, if we set an upper limit, then it’s still a good question. Let’s assume we can use any prime less than 100, and that the six primes are distinct.”
“All different, yes?” Alice sought to clarify.
“Yes. Now, what is the maximum possible sum of the four numbers on each ring such that all three sums are equal? That works!” They both grinned.”Let’s do it together.”
“I think I can do this.” Said Alice.
CHANGES TO PMQs
Note, I have renamed the Prize Maths Quiz as Pailyn's Mathematical Quiz. It's still a PMQ, just without the Prizes! When I can get together some tangible prizes I will announce it on that particular PMQ. There have been so few entries recently that it seemed best to abandon the idea for now. I also need to concentrate on my writing which, hopefully, will yield some prizes. Just treat it as a weekend puzzle and feel free to join the discussion in the comments below. Have fun!