29 Mar 2013

Feynman Point Poems - Join In - Write Your Own!

When I was a kid, I would happily play around with both words and numbers – I still do. Both have their aesthetic appeal. Whether it is constructing and deconstructing mathematical puzzles or cryptic crosswords, they appear as small artefacts that reveal a grander architecture. Combine this with the serendipity of the internet, and Pi Day was just a hop, skip and jump away from Richard Feynman and pilish poetry.

A couple of weeks ago, John Baez posted a link to Cadaeic Cadenza by Mike Keith; an epic work, albeit a short story, due to it being written entirely in pilish. Pilish is a form of constrained writing such that the lengths of the words follow the decimal expansion of the number π; Cadaeic Cadenza itself being 3835 digits (and words) long. Mike Keith later outshone himself by publishing a pilish novel entitled “Not A Wake: a dream embodying π's digits fully for 10000 decimals”; but that’s another story.

Free Mathematics Mini Practice Paper: the Answers to LS2013A

I had previously published my first Mini Practice Paper (LS2013A) for Lower Secondary students. Now, I have uploaded the Answers (LS2013A-A)!

The Answers document includes an answer grid, solution methods plus some extension exercises to those questions that make interesting projects.

Mini Practice Paper Answers - Lower Secondary - LS2013A-A

More papers are coming for Middle and Upper Secondary.

Please feel free to leave any feedback below in the Comments box run by Disqus, and if these papers have been useful to you then please consider making a small donation. Thanks!

Prize Maths Quiz: An 18-Sided Polygon (PMQ12)

The diagram below shows a large regular hexagon with unit sides, within which has been constructed an 18-sided polygon shown as the dark area. This figure has been created by, firstly, constructing six squares within the hexagon, one from each side of the hexagon. These six squares overlap to create a number of different shapes. The dark highlighted area shows the region where four or more squares overlap.

Find the area of the highlighted 18-sided polygon. You may leave the answer as an exact expression.

28 Mar 2013

Pool Table Trick Shot: Upper Secondary Mathematics Competition Question

Below is a diagram of a simple pool table. It has a length of 2 metres, a width of 1 metre and pockets at the four corners A, B, C and D. When a ball strikes a side of the pool table, it bounces off at the same angle as it hit that side. A ball is initially at point P, 25 cm from pocket D along side AD. It is struck in such a way as to bounce off sides CD and BC and then land in pocket A.

At what distance from C does the ball strike the side BC?

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27 Mar 2013

Triangles in a Hexagon: Middle Secondary Mathematics Competition Question

The figure BCDEFG is a regular hexagon. The lines GC and BE are perpendicular and meet at point A.

a) Find the ratio of Area ADE : Area ABG.

b) Show that angle ADE is not a right angle.

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26 Mar 2013

Integral Triangles: Lower Secondary Mathematics Competition Question

The diagram below shows two right-angled triangles joined along one side. The lengths of all the sides – AB, BC, CD, AD and BD – are whole numbers and AB < BC. What is the smallest possible value of the length of side AC?

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25 Mar 2013

Shapes and Angles: Upper Primary Mathematics Competition Question

The diagram below shows four regular polygons: a pentagon sitting on top of a hexagon plus two equilateral triangles. What is the small angle between the two triangles?

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22 Mar 2013

Prize Maths Quiz: A Binary Message (PMQ11)

There are a number of skills needed within mathematics competitions that rarely occur in the standard school curriculum; enumeration is one of these. Enumeration is a fancy way of saying ‘counting’, but it’s counting with style!

In secondary school, students may come across combinations and permutations as part of a course in statistics. In mathematics, these are technical terms with precise meanings and formulas. However, there are many problems that are not immediately solvable using just these types of arrangements. What is more common is that you’ll get a question that requires enumerating all possible sets that fit some criteria and then figuring out how many arrangements there are of the elements of each set. Here is one example of such a question.

21 Mar 2013

Four Cubes Puzzle: Upper Secondary Mathematics Competition Question

This question is an extension of the Three Cubes of Three Colours that was posted earlier this week.

This time you have a number of cubes of two different colours. The diagram below has blue and yellow, but the colours themselves are obviously not important. You are required to use 4 cubes at a time and to glue them together so that two faces are perfectly joined together. You are also required to use at least one of each colour. How many distinct objects can you make with 4 cubes in two colours?

Remember, again, that these are 3-dimensional objects that you can rotate in space, so the question requires that you select just one arrangement from those that have rotational symmetry.

Free Mathematics Mini Practice Paper (LS2013A)

One of the stated aims of this Gifted Mathematics website is to publish practice papers modelled on mathematics competitions throughout the world. I am pleased to announce that the first such paper has now been published.

This new paper is called a Mini Practice Paper as it is about half the length of a full paper. One advantage of this is that it can be used within a whole classroom period without having to edit it down. However, it has the same proportion of easier and harder questions so should be challenging for most students.

This paper is aimed at what we call Lower Secondary, so is roughly equivalent to the American AMC8 papers and the UKMT Junior Mathematical Challenges. It is therefore suitable for students in about the age range of 11 to 14 years old, but younger gifted mathematicians should find plenty to enjoy too!

20 Mar 2013

Friends Round a Table: Middle Secondary Mathematics Competition Question

Ten friends are going out to dinner to celebrate their graduation. Alice has two close friends, Brenda and Clarice, and the three of them wish to sit next to each other. However, David also wants to sit next to Alice. If all ten friends sit around one large circular table, how many possible unique seating arrangements are there that make all four people happy?

Note that the questions says nothing about the chairs being numbered, or anything else that breaks the symmetry of the round table, so you may count all rotational symmetries as one arrangement.

Level: Middle Secondary (Yellow)

19 Mar 2013

Products of Integer Compositions: Lower Secondary Mathematics Competition Question

The partition of a natural number is a way of writing the number as the sum of positive integers. For example, (1+2) is a partition of 3. If the order of the numbers in the sum is important we call this a composition; if we only include unique sums then that is strictly a partition.

So, the number 3 can be composed into 4 sums: 3, 1+2, 2+1 and 1+1+1. However, the unique partitions of number 3 are just 3, 2+1 and 1+1+1. Note that the partition with parts 2 and 1 could be written in one of two ways; the convention is to order such sums with the highest numbers first. Although this is not vitally important, it is good practice so that it is easier to see that you have not missed out on any partitions - and that you haven't duplicated any.

Now, let us go back to the compositions of 3 and we shall replace the summation sign with a multiplication sign. We therefore get the following list:
3
2x1=2
1x2=2
1x1x1=1
The sum of these products is 3+2+2+1=8.

Now, repeat the above process for numbers 5 and 6. That is, list all the integer compositions of 5, replace the summation signs with multiplication signs, and finally add together all these products. Then do the same for number 6.

You now have three sums of the products of the compositions of 3, 5 and 6. What do you think this total will be for the numbers 4 and 7?

What do you think this total will be for the number 15?

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18 Mar 2013

Three Cubes of Three Colours: Upper Primary Mathematics Competition Question

You have three cubes, each one with a different colour. You are going to glue together the cubes so that one face fits perfectly against another face. No face can be attached to more than one other face, and no face can be turned so that part of it is not covered.

How many different 3-cube shapes can you make?

Now, remember that these are three-dimensional objects that you can pick up and rotate in space. So, make sure that your arrangements are truly different from each other even after rotating them. One possible arrangement is shown in the diagram below.

So the real question is, how many unique shapes can you make using these three cubes of three different colours?

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15 Mar 2013

Prize Maths Quiz: A Bag of Odd Numbers (PMQ10)

 "I want to show you something. I can't figure out this week's PMQ." Louis Mordell and Gabo Szego

Alice was getting ready to impress her smart cousin Zeta. She went downstairs and could see Zeta sitting in the garden immersed in a book.

“Cousin Zeta, I have a little puzzle for you.”

Zeta barely looked up,”A bag of tricks, no less!”

Alice started to fidget with her small bag and the counters rattled her nerves.

“In this bag are all the odd numbers...”

All of them?” interjected Zeta.

“Well, what I meant...” if you’d let me finish, thought Alice,”... was that there are all the odd numbers in order, 1, 3, 5 and so on. Now, I’m going to pick one of them at random and I’m just going to tell you what the average of the remaining counters is. Your mission is to figure out which number I took out.” Alice beamed; she’d been rehearsing this for the last half an hour.

“How many counters are there in total?” Zeta placed her book down, intrigued by her little cousin.

“Can’t tell you that!”

“So, you have a bag full of odd numbers. I don’t know how many, but they are consecutive numbers from 1 upwards. You take out one of the numbers, and again I have no idea what it is. The only thing you’re going to tell me is the mean of all the remaining numbers in your bag. Is that it?”

“Yes! Perfect!” beamed Alice. “Shall we play?”

“Go on then! But, let me pick the number.” Zeta smirked.

Alice thought about that for a few seconds. “OK, but you can’t look at it! Just pick it with your eyes closed and give it to me.”

“OK, let’s play!” Zeta shut her eyes so tightly that she made an ugly grimace. She picked a number from the bag and gave it to Alice. “Can I open my eyes now?”

“Yes, yes!” Alice had turned around and was busy scribbling a few calculations in a small pocket book. “OK, done! The average of all the numbers left in the bag is ... 51.2! Now you have to tell me which number you picked.”

Zeta seemed to blank out and stared at a picture in her mind.

“I have a pen and paper if you need them.” said Alice. She was just happy that Zeta didn’t just come out with the answer in seconds.

“I have a brain.” Was Zeta’s laconic reply.

“OK, I’m going to get a drink – it’s very hot out here. Would you like one?”

Can you find the number that Zeta pulled out from the bag of consecutive odd numbers?

How to Enter

Send your complete solution by email to pmq10descartes@giftedmaths.com. This email address shall be removed after the competition closes to avoid spam. This PMQ10 competition closes on MONDAY 18 March 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 2 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.

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.
We adhere to COPPA guidelines regarding children's online safety and security.
Enjoy the challenge!
Send us your solution! You can't win if you don't participate.

14 Mar 2013

A Pi Day Puzzle

Today is Pi Day. So here is a Pi Day Puzzle.

A square and a circle both have an area equal to pi.
Their centres are at the same point.
What is the area of the region of overlap?

Also try the Pi Day Challenge.

And I have also set up a Pi Day Challenge 2013 community at G+
(This will stay open for about a month)
Enjoy!

All the Grids of Sums: Upper Secondary Mathematics Competition Question

The diagram below is the same as that used in the Grid of Sums question. However, in this case, I would like you to find all the possible solutions.

To restate the rules, the number in each circle is the sum of the numbers in the two squares above it. You must use every number from 1 to 9 only once.

List all the possible solutions. How many unique solutions are there? You may ignore reflections.

You will come across many questions such as this one, especially in any follow-up papers to open maths competitions. At first glance, there seem to be a huge number of options. Putting the nine numbers in randomly, there are 9!/2 permutations – too many to enumerate. However, the grid has some restrictions that will help you reduce the number of options dramatically.

This question does require enumeration, but you need to find a systematic way of doing it so that the answers can be found within a reasonable time. It also means that you can be confident of having found all the solutions.

Level: Upper Secondary (Red)

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13 Mar 2013

Numbers in a Grid Puzzle: Middle Secondary Mathematics Competition Question

The grid below is to be filled with numbers, one in each cell and not necessarily integers.

The numbers are arranged in such a way that if you look at the number in one particular cell, the cell to the right is double that number, and the cell below it is half that number.

The four numbers at the corners of the grid add up to a positive integral cube number. What is the smallest possible value of the number in the middle of the grid?

Level: Middle Secondary (Yellow)

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12 Mar 2013

A Grid of Sums: Lower Secondary Mathematics Competition Question

The diagram shows a row of five squares above a row of four circles. The number in each circle is generated by adding the numbers in the two squares immediately above it.

Using only the single-digit numbers 1 to 9, complete the grid so that every number is used once and every circle is the sum of the two squares above it. The diagram has the number 6 already inserted.

This question requires a bit of logic so that you can go through all the possible permutations in a systematic and quick way.

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The Math-e-Monday Puzzle: Peculiar Periodic Numbers (at Science 2.0)

I had started a weekly question called Math-e-Monday. It was supposed to be a gentle problem to start off the week and to wake up those brain cells! However, as Mondays now have questions dedicated to upper primary/lower secondary level, it seemed like duplicating similar problems.

So The Math-e-Monday Puzzles are now at Science 2.0, a science blog and article website where I used to write regularly before I started teaching again. This is actually the second puzzle I have posted on there.

Peculiar Periodic Numbers - read it at Science 2.0.

11 Mar 2013

This is hard!: Upper Primary Mathematics Competition Question

In spite of the name, this is not really very hard; but it does need a bit of logic to do it quickly!

In this puzzle, the aim is to replace each of the letters in the diagram on the right with a single-digit number in such a way that the sum is arithmetically correct.

Each letter stands for one number; the same letter means it must be replaced with the same number; different letters mean different numbers. There are only 7 different letters so not all the numbers from 0 to 9 need to be used.

How many solutions are there? And what are the correct sums?

In this particular case, there is no need to show a method. However, there is a fairly quick way of doing it rather than just plugging numbers in randomly.

Level: Upper Primary (Green)

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8 Mar 2013

Prize Maths Quiz: Prime Generator (PMQ9)

A week of questions on prime numbers ends with... a question about prime numbers, of course!

Now, because the question has a number of mathematical symbols, I have created one large image rather than inserting lots of tiny images into the text, that never align properly anyway. If the text appears unclear in your browser then just click on any part of it and you can view a larger, clearer version. If the question itself is unclear, then that's an altogether different problem!

7 Mar 2013

Prize Maths Quiz: Winners of PMQ8!

We seem to be inching forward with just one winner per week. Never mind, the site and its social networks continue to grow so I shouldn't be too concerned.

Also, many of our members interact in our Mathematics Competitions Google+ community rather than on this website. Again, never mind, but it makes this site look unloved!

We don't have a strong presence on Facebook, so if you're on Facebook, help us out by sharing our posts and liking our page. Better still, join us on Google+!

Enough chit-chat, our Winner of PMQ8 is:

D. W. (Singapore)

Congratulations! An email has already been sent.

Next competition is PMQ9 starting Friday 8 March at 02:11 GMT.

See you Friday!

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!

 Click banner for details

A Squared Prime Problem: Upper Secondary Mathematics Competitions

There are prime numbers p, such that the number squared p2 is a factor of [(p-1)! + 1].

For which of the following primes is this true?

2, 3, 5, 7, 11, 13, 17.

In a competition, without electronic devices, this question should take approximately 10 minutes. Doing the question by computer is trivial - although interesting as p becomes larger - the aim here is to find the fastest technique to do it manually, as if it were a test question.

Level: Upper Secondary (Red)

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6 Mar 2013

Factorial Primes: Middle Secondary Mathematics Competitions

A factorial prime is a prime number of the form p = n! + 1 or p = n! – 1, where n is a positive integer.

Find the sum of all such distinct factorial primes under 50.

In a competition, without any electronic devices, this question should take about 2 minutes.

Level: Middle Secondary (Yellow)

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5 Mar 2013

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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4 Mar 2013

Primeless Road: Upper Primary Mathematics Competition Question

Primeless Road is a rather peculiar road. All the odd numbered houses are on one side of the road, and all the even ones are on the other side, both sides starting with the lowest number. Nothing very strange with that! However, none of the house numbers is a prime number.

Assuming each house has exactly one other house directly opposite, what is the number of the house opposite number 20?

Level: Upper Primary (Green)

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1 Mar 2013

Prize Maths Quiz: Fractions of Phi (PMQ8)

This week has been dedicated to the Liber Abaci by Fibonacci. In this book, Fibonacci introduces into Italy the ‘Indian figures’, 1 to 9, plus the 0, which the Arabs called ‘zephir’. In order to show the superiority of this numbering system compared to the Roman system, Fibonacci spends a considerable number of pages handling fractions. Trade within the Mediterranean involved a plethora of coinage, weights and measures and the wisdom of using a single number system to handle them all was not lost on the merchants of Pisa, and beyond.

A unit fraction is one where the numerator is 1 and the denominator is some other whole number. There are times when such fractions continue to be useful, especially with non-decimal units of measurement such as those based on 12, 24 or 60 units. Fibonacci shows how to handle both decimal fractions and sums of unit fractions.

For example, the circle to the left is subdivided in such a way to show that one-half is equal to a third plus a sixth. Alternatively, the other half is divided into 1/3 plus 1/9 and 1/18. Without any further calculations, the diagram itself also shows that we can express 1/3 as 1/6 + 1/9 + 1/18. And finally, we have 2/3 = 1/2 + 1/9 + 1/18.