The Sylver Coinage Game: Upper Secondary Mathematics

This is more a project than a single question. There is, however, a question at the end; and it will help you with tomorrow’s PMQ.

John Conway invented a number of mathematical games including one that has come to be called the Sylver Coinage game. I usually try to obfuscate the standard names of various theorems so that users actually try to solve the problems rather than searching for the answer. In this case, the game remains interesting even after reading the literature.

The game is played by two players; let’s call them Alice and Bill. The game starts with the set of all positive whole numbers. Alice picks the first number – this removes from the game all multiples of this number. Bill then picks a number – this removes from the game all linear combinations of the two numbers in play. The game continues in this fashion until one player is left having to pick the 1 – that player is the loser.

Prize Maths Quiz PMQ9: Announcement for Friday 8 March

Friday is almost upon us again and so is our Prize Maths Quiz; we're up to PMQ9 now.

The question will be posted on Friday 8 March at 02:11 GMT; this means it will still be Thursday evening or afternoon in the Americas. This week I am extending the competition to MONDAY at 23:59 GMT to see if some people prefer to do this on a Monday rather than the weekend.

I hope everyone has noticed that weekday questions Monday to Thursday are now colour-coded and graded from Upper Primary to Upper Secondary. In reality, age is little guide to how good a student is at such maths competitions, but I hope it helps teachers who give added assistance to those students keen on such challenges. The grading uses a mixture of AMC and UKMT competitions as guides.

See you Friday!

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Primes in a Sequence: Lower Secondary Mathematics Competition

We have a sequence of numbers defined as follows:

T(1) = 3, T(2) = 4, and T(n) = T(n-1) + T(n-2), where n is a positive whole number greater than 2.

T(1) is obviously a prime number. Find the next five prime numbers that appear in this sequence. Write down their sum.


Answer will eventually be posted below in the Comments section, but best to try it yourself first!


Level: Lower Secondary (Blue)

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Prize Maths Quiz: Five Circles in a Triangle (PMQ7)

Five Circles in a Triangle

To end a week of geometry problems, it is no surprise that PMQ7 is also a geometric puzzle.

The diagram shows five circles, each with integer radius, all touching the base of the large triangle. The four smaller circles all touch their two neighbouring circles, with the large circle touching all four. The two sides of the triangle each touch two of the circles.

Let the radii of the circles be a, b and c, such that a > b > c.

Given that c has a length of 4 units, find the area of the large triangle. You may leave it as an exact solution.

As always, I would like to see a method. As an interim solution, find the radii of all three circles.

Good luck!

Prize Maths Quiz: Winners of PMQ6!

PMQ6, the Tricky Tower, has been our most popular quiz question so far, a least in terms of traffic to the page. However, there was only one correct answer submitted.

Congratulations go to:

J.Y. (from the UK)

I will post the answer to the problem later this week, but there is a discussion on-going at our Mathematics Competitions Community on Google Plus. Unlike the standard Tower of Hanoi, colouring the discs creates some limitations in the possible moves. It also destroys the simple formulas that are associated with the standard puzzle.

This could form the basis for an interesting project. Start with a 2-colour Tower of Hanoi, with alternating colours, and see whether you can find a pattern such that you can calculate the number of moves needed to solve a 100-disc tower!

More on this coming soon...



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Prize Maths Quiz: Trapezium Artist (PMQ4)

Alice was playing out in the garden. She really was getting far too big to be playing on the swing; this made her sad as she liked swinging up as high as she could. However, the creaking noises troubled her and her mind turned to whether she could unhinge the whole structure and fall head-first onto the lawn. It struck her as safer to do this as a theoretical calculation than as an experiment. She knew it could be done, but didn't yet know how.

“Alice! Come on, we’re going shopping!” bellowed her mother. Oh no... why don’t my parents shop on the internet and have it all delivered? Now that’s what supermarkets are for; otherwise they are just dull-markets. And why is my big brother never around when he could be useful? Alice jumped off the swing and rolled around on the grass, making herself look as unpresentable as possible.

“Come on, darling! You know I hate all that traffic later in the day.” Her mother fussed trying to brush off clumps of dried grass from Alice’s dress. That didn’t work!

Shopping malls are ghastly plastic places with very little of interest for Alice. She usually tries to head for the computer shops but they are always full of tall smelly teenagers staring at games. “Alice, I know you love to wander off and I’m not wasting my time looking for you. If we get separated I’ll wait for you in this coffee shop. OK?” Alice was not really old enough to be let out on her own, but she was good at wandering off on her own.

Math-e-Monday: Half a Triangle

Most of the questions posted so far have been about numbers, so this week we’re going to concentrate on geometry.

The people who set geometry questions in mathematics competition papers like to mess with your eyes! They are not really optical illusions but, rather like a good magician, you can’t see the whole picture because you’re concentrating only on what you’re being shown. Geometric diagrams do usually show you everything you need but not always everything you know. My advice is to redraw the diagram and fill in everything you know that is related to the question; it could be trigonometry or circle theorems or construction techniques. Then make sure you read the question carefully to check for details that are written down but not illustrated in the given diagram.

Lastly, brush up on some of the more obscure geometric theorems. You can always prove things from first principles, but in a maths test that wastes precious time. It is better to be prepared with a basket full of theorems and then figure out which ones you need.

The Question

The diagram above shows a circle centre C (not drawn to scale). The line CQ is perpendicular to the diameter PR. The triangle PQR has double the area of triangle PSR. Find the angle PRS.

This is not a PMQ so feel free to discuss this puzzle in the comments section below.

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Prize Maths Quiz (PMQ3) Announcement for Friday 25 January

The next Gifted Mathematics Prize Maths Quiz shall be PMQ3 and will be posted on Friday 25 January 2013 at 07:03 GMT. The competition will close, as always at 23:59 GMT on the Sunday.

Please use our offical GMT clock in the right-hand column.

Please also read the rules, which have been recently updated and clarified, on our PMQ Prize page.

See You Then!




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