28 Feb 2013

Prize Maths Quiz (PMQ8): Announcement and Feedback Request

The next Prize Maths Quiz (PMQ8) will be posted on Friday 1 March at 02:02 GMT.

The competition will close, as usual, on the Sunday at 23:59 GMT. Please use our official GMT clock on the right-hand column.

See you Friday!

Oh yes, feedback! I have had a small amount of feedback regarding the timings of these PMQs. There has been a suggestion that I should include more weekdays and not have it as a 'weekender' quiz. The reasoning here is that many people go away for the weekend and seem to be in a more analytical frame of mind during weedays. However, it strikes me that few people seem to turn off their electronic gadgets and that Gifted Mathematics does have a mobile version - I even have a QR Code in the right-hand column now.

So, please let me know if you have any strong feelings either way. Comments box is below this article. I could extend it a day, into either Monday or Thursday, or leave it as it is.

Thanks

Richard




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27 Feb 2013

Prize Maths Quiz: Winners of PMQ7!

The latest PMQ, Five Circles in a Triangle, probably had the longest solution so far, involving both geometry and a bit of number theory. Once again, like last week, we have only one Winner.

Congratulations go to:

G. McR.

An email has been sent and I look forward to welcoming a new member into our online mathematics club.

The next competition, PMQ8, will start on Friday.



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Midweek Maths: A Purse Full of Ducats

Three men are sitting round a table. They each have some 1-ducat coins in front of them and there is a purse full of similar coins in the middle of the table.


Alessandro turns to Bruno and says:”If I add the ducats in the purse to my money I will have twice as many ducats as you.” Then Bruno turns to Cesare and says:”If I add the coins in the purse to my money, I will have three times as many ducats as you.” Finally, Cesare says to Alessandro:”If I add the ducats in the purse to those I already have, I will have four times as much money as you.”

How many ducats does each man have and how many are in the purse? Assume the smallest integer solution.

As in the previous question, Fibonacci did not use any algebra to solve this question. For students who feel that algebra is some kind of mental torture, try following a verbal algorithm nearly a page long!

25 Feb 2013

Math-e-Monday: A Merchant from Pisa

This week I am reading the Liber Abaci (The Book of Calculations) by Leonardo Pisano, better known as Fibonacci. The book, published in 1202, contains the famous problem of rabbits breeding in such a way as to generate the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13 and so on. However, the Liber Abaci also contains a wealth of other problems, and this week I shall post a small selection of questions inspired by those found in the book.

The question below is fairly easy and can be done by a lower secondary, or even upper primary, student. Interestingly, Fibonacci did not have a well-developed algebra, so his solution is a numerical recipe or algorithm. As part of Fibonacci's promotion of the Hindu-Arabic numerals, he has numerous questions involving fractions. Many such questions are designed to show the ease with which fractions can be manipulated using his 'new' system of numbers compared with using Roman numerals. This is one such question.

22 Feb 2013

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!

21 Feb 2013

Prize Maths Quiz (PMQ7): Announcement for Friday 22 February

The next Prize Maths Quiz (PMQ7) will be posted on Friday 22 February at 03:14 GMT.

The competition will close, as usual, on the Sunday at 23:59 GMT. Please use our official GMT clock on the right-hand column.

See you Friday!




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20 Feb 2013

Gifted Mathematics: New Schedule and More Questions

This is a brief announcement about some changes to the way questions will be posted on Gifted Mathematics. This website has been up just under two months and I am pleased with the general growth in readers and users, and in my meeting many new people online.

I initially resisted putting too much emphasis on grading the questions; after all, a gifted 10 year-old may well be able to do questions aimed at 16 year-olds. I also quickly realised that there is a large community of adults who enjoy maths games and puzzles. Indeed, some of the extensions to questions require some sophisticated techniques to solve, and it has been personally rewarding to find online collaborators willing to explore these mazes. But I also know that a large cohort of students will discover these questions through their teacher, and such teachers need to know that the questions used are at an appropriate level.

Therefore, from the 1st March 2013, Gifted Mathematics shall publish questions using the following schedule:

Monday: Math-e-Monday (random amusing maths puzzles)
Tuesday: Upper Primary
Wednesday: Lower Secondary (about AMC 8 and UKMT Junior MC level)
Thursday: Middle Secondary (about AMC 10 and UKMT Intermediate MC level)
Friday: Prize Maths Quiz, for Upper Secondary (about AMC 12 and UKMT Senior MC level)

The Prize Maths Quiz will remain free for anyone to participate. I will most likely have the question in two parts so that as many students as possible can take part.

Answers to various questions will usually be found within the comments section, either on this website or on the social web. I will only publish formal solutions to the PMQs and any other questions that inspire some interesting extensions.

I am also writing some Practice Test Papers at the various levels. These will be published soon and the Mini Practice Papers will be available for free download.

To keep up to date with mathematics questions for your particular age group, it may be a good idea to sign up to our RSS feed or email subscription found in the right-hand column. If there is substantial growth in the number of subscribers I will consider splitting the RSS feed into age groups.

Any suggestions or feedback, feel free to leave a comment below.

Best Wishes

Richard
GiftedMathematics.com

Midweek Maths: Two Diamonds

We continue on the theme of geometry. Like the previous question, the one below can be done in different ways: with pure trigonometry; with geometric transformations; and even using coordinate geometry.

This is true of many questions in mathematics. An important skill in mathematics is not just the ability to solve problems, but the capacity to quickly go through different possible methods and to pick the most efficient. This is a skill that is difficult to teach but essential in being able to save time in a mathematics competition – time that will be precious for the more challenging questions. One way for you to measure the most efficient method is to actually do the question below in three different ways! Try it and see for yourself which method seems faster.

It is also important to develop some self-knowledge about which branches of mathematics you find come most naturally. Some students prefer number theory whereas others may like geometry or probabilities. In team competitions it is vital to distribute each given question to the person who is the ‘expert’ in that field. Sometimes, the best team is not always the sum of the best individuals.

19 Feb 2013

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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18 Feb 2013

Math-e-Monday: Nested Hexagons

As always, something to ease into the week; after all, PMQ6 seemed like a hard nut to crack!

The diagram shows a large regular hexagon with a smaller hexagon nested within it. The vertices of the smaller hexagon are the mid-points of the sides of the larger hexagon.

What is the ratio of the area of the large hexagon to the total area shaded in blue?

A slightly easier question is: what is the ratio of the area of the smaller hexagon to the total area shade in blue?

Why is the 2nd question quicker to do?

Can you now do the 1st question without any calculations?




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15 Feb 2013

Prize Maths Quiz: A Tricky Tower (PMQ6)

Alice was feeling sick and lying in bed in the middle of the afternoon. She hated feeling sick and she hated lying in bed. Well, that wasn’t entirely true; she loved feeling soft and warm in her bed, but today she was feeling hot and irritated.

Bill suddenly bounced through the door.

“Hello Alice! How are you? I just popped round to cheer you up a bit – Mother said you had a slight fever.”

Bill was being chirpy; it seemed like bad manners to be chirpy on a day when Alice wasn’t.

“Look, I brought you a present!” Bill handed Alice a big box covered in coloured triangles and shapes. Before Alice had a chance to ask, Bill volunteered to explain, “It’s a jigsaw puzzle. A big jigsaw puzzle! I hope you like it.”

Alice thought Bill wanted a pat on the head and a fresh bone. The pattern on the front looked interesting but, as Alice opened the box, she frowned.

14 Feb 2013

Prize Maths Quiz (PMQ6): Announcement for Friday 15 February

The next Prize Maths Quiz (PMQ6) will be posted on Friday 15 February at 01:21 GMT.

The competition will close, as usual, on the Sunday at 23:59 GMT. Please use our official GMT clock on the right-hand column.

See you Friday!




Don’t forget to follow Gifted Mathematics on Google+, Facebook or Twitter. Shorter questions posted on our Google+ Community and Facebook..

You can also subscribe to our Bookmarks on StumbleUpon and Pinterest. Many resources never make it onto the pages of Gifted Mathematics but are stored in these bookmarking websites to share with you.



13 Feb 2013

Prime Numbers, Prime TV and a Prime Valentine’s Day

It is somewhat surreal to see the discovery of the largest prime number paraded on prime time broadcast media. Mathematicians around the world are asked to explain the significance of this discovery in layman’s terms, which is up there with physicists trying to explain what the Large Hadron Collider actually does.

Below you’ll see a pretty sparky Sky News interview by Eugenia Cheng, a senior lecturer at the University of Sheffield. Just notice at the very end how jolly pleased with themselves the newscasters seem to be.

The new record was discovered by Dr Curtis Cooper at the University of Central Missouri. This is the third time that Dr Cooper has entered the record books, and one must hope that he is using his university’s spare computing power to do so. Indeed, this discovery was not done in isolation but as part of a global internet project known as GIMPS (Great Internet Mersenne Prime Search). You too can join in; just go to the GIMPS website, sign up, download the software and you’re up and running; or rather, your computer will be.

Midweek Maths: You are in a maze of twisty little passages, all alike


This week I’ve created my own adventure game. I didn’t mean to, it’s just turned out that way. In the name of finding some amusing (and frustrating) strategic games I headed towards some old favourites, only to discover that they had been dismantled and sold as scrap.

In 1984 Infocom and Douglas Adams released an adventure game version of the Hitchhiker’s Guide to the Galaxy. Employing the same wit and tangential narrative of the original books and radio plays, the game was a huge hit. Then graphics arrived and people forgot how to read and write and the age of text-based interactive fiction, better known as adventure games, became a chapter in geek history.

BBC Radio 4 (yes, radio!) did have a graphical version of this game, which worked about a year ago (or maybe two) but has been terminated. Ironic that pressing the “DON’T PANIC!” button now reveals the laconic phrase “zmachine xml invalid”; this is as terminal as it gets. So, if anybody can find a working version of this game hiding in some corner of the net, then let us know, otherwise it’s back to the Douglas Adams page for the original text version.

11 Feb 2013

Prize Maths Quiz: Winners of PMQ5

Thank you to everyone who submitted their answer to PMQ5 (Mercury and the Sum of a Series).

Before I announce the winners, I need to mention a slight change to how I do this from now on. I used to wait for the winners to respond to my congratulatory email so that I could list their names and countries. However, some don't bother replying, so that it looks as if nobody won! So, from now on, I shall announce the winners using just their initials or part of their email address. Then, once the winners have responded and the actual claimant student has accepted the prize, I will list the names as they join our online classroom.

Having got that out of the way, this week's winners are:

K.R.
M.K.
J.K.

An email has already been sent.

For the first time, we have had more submissions than prizes, so to those who didn't win this week, please try again next Friday.

Well done!

Richard
GiftedMathematics.com


Math-e-Monday: A Mouse Trap Strategy

Many of the problems found in mathematics competitions require some playing around before they can be confidently solved. Many such questions do not obey the school test paradigm of question-method-answer; many of them demand that you do a bit of thinking!

This week we’re going to look at some strategic games. Why, for example, is noughts-and-crosses (tic-tac-toe) a boring game? It is dull because once you’ve understood the strategy you know that you will only win against someone who is clueless! Two knowledgeable players will always draw... until they get bored and go do something else.

8 Feb 2013

Prize Maths Quiz: Mercury and the Sum of a Series (PMQ5)

“Hello. Who are you?”

“My name is Alice. What’s yours?”

“I’m Mercury; welcome to my wedding.”

“Wedding?! You seem so young to be getting married!”

“I know, I’m supposed to look young, but actually I’m very old.” Mercury looked around as if he was about to slip out a secret. “All my family have been telling me for centuries to get married, settle down, raise some kids, but I just want to fly away.”

“Me too! I love to fly!!” Alice felt as if her words had just echoed around the whole universe and that everybody had heard them. She blushed a little.

At that moment, a thunderous clap jolted both Alice and Mercury from their reverie and they both turned their head to stare at what appeared to be the Sun slowly coming down to land in the middle of the... party! Alice had been so charmed by her new friend that she hadn’t noticed that she was in the middle of some kind of carnival.

7 Feb 2013

Prize Maths Quiz PMQ5: Announcement for Friday 8 February

The next Prize Maths Quiz (PMQ5) will be posted on Friday 8 February at 16:25 GMT.

The competition will close, as usual, on the Sunday at 23:59 GMT. Please use our official GMT clock on the right-hand column.

See you Friday!




Don’t forget to follow Gifted Mathematics on Google+, Facebook or Twitter. Shorter questions posted on our Google+ Community and Facebook..

You can also subscribe to our Bookmarks on StumbleUpon and Pinterest. Many resources never make it onto the pages of Gifted Mathematics but are stored in these bookmarking websites to share with you.



4 Feb 2013

Math-e-Monday: Sequences A

Last week, we had questions on geometry; this week is going to be sequences and series. From the point of view of mathematics competitions, you really need to be comfortable with linear and quadratic sequences, geometric sequences, and recurrence relations such as the Fibonacci sequence.

In the higher papers you will also need to be confident in handling series, especially summing convergent series. All the above topics can be found in various school curricula but they are given far more prominence in maths competitions. You also need to be able to rewrite numbers in different forms so that the pattern becomes clear; a kind of numerical juggling.


The Question

Today, we have two sequences. Find the next two terms in the following:


Have fun!




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1 Feb 2013

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.

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