19 Apr 2013

Prize Maths Quiz: Sylver Coinage {9} Game (PMQ15)

 J.J. Sylvester

Alice had been teaching her friend Charlotte how to play the Sylver Coinage Game. Just then, Bill walks in, fiddling with what looked like a broken Rubik’s Cube.
“Don’t ask! I just wanted to see how it works.”
“And, no doubt, to solve it using your unique brand of brute force!” Alice smirked.
Bill couldn't think of a witty reply, so pretended to ignore the quip.
“Bill, Charlotte wishes to beat you at the Sylver game.” The girls giggled.
“Oh no! All those numbers make my head buzz.”
“So you accept defeat?”
“Did I say that? No way!” Bill looked at Charlotte, who seemed to be entranced in a Sylver daze. “Come on then, let’s play your little game.”
After everyone was sitting comfortably, Charlotte had first pick in the first game. Charlotte picks the number 9. Alice sinks her head in her hands.
“We hadn't played that opening, Alice, had we?” beamed Charlotte, as if she’d invented a new colour or something.
“There was a reason for that!” But then Alice noticed that Bill looked deep in thought. He’s... not... sure! Bill seemed to be stuck in thought.

The Question

Can you help Bill? What is his best move? Assuming both players play the best possible moves from then on, who should win?

As there are many possible games, I would like to see one complete game with good reasons for the choices made. There is, however, one good move that Bill must make on his first turn.You can express your moves as {9, a, b, c} and so on and use standard brackets for the numbers still in play, such as (1, 2, 3, 4....)

I have already linked to yesterday’s Sylver Coinage Game question and it is worth going through how the game works and following the link to the Wolfram Demonstration of the game. However, that application has some serious restrictions that limit its usefulness in this particular PMQ game. You will see what I mean when you try it! Remember that Charlotte’s move of {9} removes from the game only those multiples of 9; all the other numbers are still in the game.

As this PMQ does not have a unique solution, feel free to ask any questions to clarify how a game progresses. You may also be interested to learn that the mathematical analysis of this game contains many unanswered questions.

Enjoy the challenge!

You may discuss the question below, but no full answers, please, until the competition has closed!

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How to Enter

Send your complete solution by email to pmq15sylvester@giftedmaths.comThis email address shall be removed after the competition closes to avoid spam. This PMQ15 competition closes on MONDAY 22 April at 23:59 GMT - one extra weekday from now on.

The Prize

The prizes for this PMQ are 3 free places in our Online Maths Club for 3 months. The very first correct solution will receive a prize plus two others randomly selected from all the other correct answers. The email time stamp shall determine the order of entries received. All winners can have their name posted and a link to their own online profile at their favourite social network or their own blog.

Quick Rules

Look at the expanded rules on our PMQ page.
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