The diagram is an alchemical symbol, reminiscent of a cosmogram or mandala. The interplay of the triad and tetrad within two circles leaves this open to numerous interpretations. However, this is the only construction that allows for an equilateral triangle; I have seen versions with an isosceles triangle.

One decoding of this symbol is to start at the smaller inner circle, designating our normal state of consciousness. The square then represents the distillation of the four elements, with the triangle signifying the three realms of existence; such as body, mind and spirit. The larger circle is the end of the great work; an expanded consciousness. Note that the centre of the larger circle is also above the centre of the smaller one.

However, let us leave behind such neoplatonic meditations on moral geometry and concentrate on the purely metric. Given that the smaller circle has a radius of unity, find the radius of the larger circle. Also, calculate the distance between the centres of the two circles.

## 31 May 2013

## 30 May 2013

### Sum of Digits Function: Upper Secondary Mathematics Competition Questions

The function Q(N) gives the sum of the digits of the positive integer N.

Calculate the value of Q(Q(Q(2012

A slightly easier version of the same question is to calculate Q(Q(Q(2013

Calculate the value of Q(Q(Q(2012

^{2012})))A slightly easier version of the same question is to calculate Q(Q(Q(2013

^{2013}))). Why is this easier, and what is the answer?## 29 May 2013

### Three Positive Integers: Middle Secondary Mathematics Competition Question

The three positive integers (37+K), (193+K) and (421+K) are all square numbers such that their square-roots are in an arithmetic sequence.

Find the values of K.

## 28 May 2013

### Free Mathematics Mini Practice Paper MS2013B: Answers and Solutions

The Answers and Solutions to the Free Mini Practice Paper MS2013B for Middle Secondary students has now been published.

You can download it directly from Dropbox:

Mini Practice Paper MS2013B-A Answers and Solutions.

Included are some extension questions to Q12.

Any comments, questions or queries about the wording, please let me know so can edit anything as quickly as possible. Many thanks.

Further Practice Papers coming soon; the next one shall be for Upper Secondary students.

You can download it directly from Dropbox:

Mini Practice Paper MS2013B-A Answers and Solutions.

Included are some extension questions to Q12.

Any comments, questions or queries about the wording, please let me know so can edit anything as quickly as possible. Many thanks.

Further Practice Papers coming soon; the next one shall be for Upper Secondary students.

### A Lonely Ant: Lower Secondary Mathematics Competition Question

An ant has been cast adrift from its colony and is obsessively pacing up and down a centimetre ruler. It walks from the 0 cm to the 10 cm mark on the ruler, then back again, at a constant speed of 1 cm/s. It also takes 1 second to turn around at each end. It repeats this pacing motion without rest.

At what number on the centimetre ruler will the ant be after precisely 30 minutes?

## 27 May 2013

### A Number Puzzle: Upper Primary Mathematics Competition Question

This addition requires that the letters A, B and C be replaced by single-digit numbers in such a way as to make the sum arithmetically correct. The same letter stands for the same number, and the three numbers are all different.

i) Which number should replace the letter B?

ii) Given that the 3-digit number CAC is a perfect square, find the numbers that correspond to A, B and C.

[edited. Sorry, had a mental coupling of two different questions! Sorted now. This here has just one solution.]

## 24 May 2013

### Prize Maths Quiz: You Set The Question! (PMQ20)

This week’s Prize Maths Quiz is really easy to write down.

**Submit your own mathematics competition question!**

During the past week, I have mentioned a couple of times that I would be trying this as an experiment. I would like

*you*to submit a question that could possibly be found within a mathematics competition. The question must involve dominoes as part of the problem. The actual branch of mathematics could be anything, such as combinatorics, sequences, fractions, geometry or anything else you can think of. However, the construction of the problem must involve some manipulation of dominoes. Examples can be found here, here and here.

You must submit your original question in full plus the answer and solution. As it usually takes longer to devise a question than to solve it, just for this week I have extended the deadline to Thursday night. It will also take me longer to read through the submissions rather than just checking that the answers are correct. The best questions will be published on this website, with due acknowledgement and a link to your personal homepage. I reserve the right to edit the questions for language, so if English is not your first language, don’t worry, I shall correct any grammatical errors; the idea is more important than the precise wording. If your question involves a diagram, then submit it as an attachment or include a link to the URL of the image. However, I expect each question to be your original work. Each individual may submit a maximum of two questions; if I see too many from the same email address I shall delete unread any further submissions beyond the two - so pick your best.

Your question should also be solvable with just pen, paper and brain, without any electronic calculating device. It should also be solvable within a reasonable time, assuming it is in a proper test paper, so a maximum of 10 minutes for a really hard question. I know it is difficult to estimate solution time, but if you try the same question yourself and time it, then multiply this by a factor of 3 for extra thinking time.

If I have missed out anything, or you wish to clarify any points, please leave a comment below.

Have fun!

## 23 May 2013

### A Domino Wheel: Upper Secondary Mathematics Competition Question

This is the last of this week’s domino puzzles. As before, take one full set of dominoes and remove the tiles with blanks, leaving you with 21 tiles.

The diagram below shows 8 rectangles placed at the vertices of an octagonal wheel. Each rectangle is to have one domino tile placed inside it. Every tile consists of two squares, each with some pips inside it. The number in the centre of the diagram is the product of the 4 squares taken from the 2 dominoes that are diametrically opposite each other. All four such products must equal the same total and be formed from different tiles from your set of dominoes. Next, orient each tile so that the square with the smallest number of pips is closest to the centre of the diagram, thereby forming a kind of “inner circle”. Calculate the sum of these eight squares, the “inner sum”.

The aim is to find the largest possible “inner sum” of the lowest value squares of the 8 dominoes placed so that the product of the squares of diametrically opposite dominoes is constant.

## 22 May 2013

### More Domino Puzzles: Middle Secondary Mathematics Competition Questions

Like yesterday’s Domino Puzzle, you are given a full domino set of 28 tiles. Firstly, remove all the tiles that have a blank end, leaving you with 21 tiles. Now, by looking at each tile as a vertical arrangement of numbers, we are going to use them as fractions. For example, the tile [5,2] can be thought of as the fraction 5/2 or 2/5.

__Question A__This time we are going to add together two domino fractions, giving the sum of 6/5. The tile [6/5] is already used as our answer so, using the 20 remaining tiles, how many different sums can you find? The order of the fractions is not important.

Algebraically,

[A/B] + [C/D] = [6/5] = 6/5

where the square brackets [A/B] merely serve as a reminder that these are domino tiles being used as domino fractions.

## 21 May 2013

### Domino Puzzle: Lower Secondary Mathematics Competition Question

You are given a full domino set of 28 tiles. Firstly, remove all the tiles that have a blank end, leaving you with 21 tiles. Now, by looking at each tile as a vertical arrangement of numbers, we are going to use them as fractions. For example, the tile [5,2] can be thought of as the fraction 5/2 or 2/5.

Your task is to find all the combinations of 5 distinct tiles such that the product of 4 of the fractions is equal to the 5th fraction, which is itself equal to the number 3.

Algebraically, this equates to:

where the square brackets [A/B] merely serve as a reminder that these are domino tiles. The order of the 4 tiles on the left-hand side is not important.

__Short Question__Find all the solutions where the above product equals 3 and that include the tile [5/2].

__Project Question__Find all the solutions where the above product equals 3.

## 20 May 2013

### Slicing Through Chocolate: Upper Primary Mathematics Competition Question

Imagine you have a bar of chocolate and wish to cut it into individual pieces. Like the chocolate in the image, the bar is one solid piece but moulded to have distinct chunks.

If your bar is 3 chunks wide and 5 chunks long, how many cuts do you need to make so that you have all individual chunks?

A slice means one straight-line cut with a sharp knife. You may not stack pieces on top of, or next to, each other in such a way as to cut through multiple pieces with one cut.

In general, if a bar of chocolate is

*n*chunks wide and

*m*chunks long, how many cuts (as described above) would you need to make in order to separate it into individual pieces?

## 17 May 2013

### Prize Maths Quiz: The Middle Secondary Paper (PMQ19)

I haven't done this before. Admittedly, there has only been one other Practice Paper published so far, so here's a first. Instead of having one question for this PMQ, the aim is to get the highest marks on this week's Mini Practice Paper for Middle Secondary.

You can download the paper here. As it is a Mini Paper, it is just 12 questions designed to be done in 30 minutes. You have almost 4 days in which to submit your answers! Think of it as this weekend's puzzler.

When you submit your answers, just write a list of the questions and the letter of your chosen answer. I don't need to see your full calculations - just the answers. You can discuss the correct methods in the comments section below. I shall mark it as if it was a test paper.

The answers and solutions shall be published next week after this PMQ has closed.

Good Luck!

You can download the paper here. As it is a Mini Paper, it is just 12 questions designed to be done in 30 minutes. You have almost 4 days in which to submit your answers! Think of it as this weekend's puzzler.

When you submit your answers, just write a list of the questions and the letter of your chosen answer. I don't need to see your full calculations - just the answers. You can discuss the correct methods in the comments section below. I shall mark it as if it was a test paper.

The answers and solutions shall be published next week after this PMQ has closed.

Good Luck!

## 16 May 2013

### A Better Game With Four Lockers: Upper Secondary Mathematics Competition Question

Let’s return to Mrs Pixey’s class and the game she has been playing with her hapless students. Today is Thursday, and they have been playing this game for the past three days. So far, the students have won just 4 points, with Mrs Pixey having accumulated 8 points. The situation seems hopeless and the students feel that this is a very unfair game. During lunch, the four students sit together to discuss whether they should give up and forget about the prizes on offer. However, Debbie has a bright idea! Before we look at her plan, let’s remind ourselves of the game.

Each of the four students – Alex, Brenda, Chris and Debbie – has a school locker with their name on it. At the end of the day, Mrs Pixey places their homework books, one in each locker, in such a way that every book is inside the wrong student’s locker. The task of the four students is to each pick a locker, open it and see if they have found their own book. They all do this at the same time, having decided in advance who should open which locker. Each student who finds their own homework book gets 1 point, otherwise Mrs Pixey gets 1 point. At the end of the week, if the sum of the points gained by the students is greater than that won by their teacher, each student will receive a small prize.

## 15 May 2013

### Free Mathematics Mini Practice Paper for Middle Secondary (MS2013B)

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 second 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 Middle Secondary, so is roughly equivalent to the American AMC10 papers and the UKMT Intermediate Mathematical Challenges. It is therefore suitable for students in about the age range of 14 to 16 years old, but younger gifted mathematicians should find plenty to enjoy too!

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 Middle Secondary, so is roughly equivalent to the American AMC10 papers and the UKMT Intermediate Mathematical Challenges. It is therefore suitable for students in about the age range of 14 to 16 years old, but younger gifted mathematicians should find plenty to enjoy too!

### A Game With Four Lockers: Middle Secondary Mathematics Competition Question

We have already met Mrs Pixey and how she likes to annoy her children. However, Mrs Pixey is also a teacher and likes to play similar tricks with her students. She believes it keeps them alert to the less playful dangers that they will likely face when they grow up. It also stops them from falling asleep.

Today, she decides to play a game with four of her students: Alex, Brenda, Chris and Debbie. Each student has a locker with their name on it. Mrs Pixey places their homework books in the lockers in such a way so that every book is in the wrong locker.

She then asks the four students to discuss which locker each of them should open. Once they have all agreed on their choices they each open their selected locker and see whose homework book they find.

If a student finds his or her own homework book they get 1 point; if they do not find their own book, Mrs Pixey gets 1 point. At the end of the week, if the sum of the points won by the students is greater than the points won by Mrs Pixey, they each get a small prize.

## 14 May 2013

### No Lunch For You!: Lower Secondary Mathematics Competition Question

Yesterday, Mrs Pixey was being horrid to her long-suffering children by mixing up their lunch boxes. This morning, she thought of another fiendish prank. Again, she packs their lunch boxes in such a way so that all of them are in the wrong school bag, but this time she allows for the possibility that one bag may contain more than one lunch box. She rolls a die to help make her selection random.

What is the probability that at least one of her three children goes without lunch?

In reality, her kids have gotten used to their mother’s sense of humour and have learnt to double-check their bags as they leave the house to go to school. None of them went hungry.

## 13 May 2013

### A Lunch Box Trick: Upper Primary Mathematics Competition Question

Mrs Pixey likes to play tricks on her three kids. This morning she packs their personal lunch boxes, one in each of their school bags. However, she places every lunch box in the wrong bag!

In how many ways can she do this?

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## 11 May 2013

### Feynman Point Pilish Poetry: 11 May 2013 Anniversary

Richard Feynman was born on 11 May 1918. Today would have been his 95th birthday. This isn’t a paean to a physicist – it could be, but I’m sure millions of words have been published on this – but a game for wordsmiths and numberphiles. Within the highly restricted format of the Pilish Poem, the aim is to be both enlightening and eloquent; plus to have some fun playing around.

You can follow my train of thoughts at my first article on this so, without repeating myself, I’ll jump right in with the basic rules of what constitutes a Feyman Point Pilish Poem.

The Feynman Point is a sequence of six 9s that occurs within the expansion of Ï€ starting at the 762nd decimal place. Pilish writing in general is a technique whereby the lengths of consecutive words follow the decimal digits of Ï€. Numerous examples can be found at Mike Keith’s Cadaeic.net website.

Putting these two ideas together, I came up with the Feynman Point Pilish Poetry challenge. Each poem consists of just six 9-letter words. That’s it! A kind of haiku for maths geeks.

This challenge was thrown out to the socialnets a few short weeks ago. I didn’t restrict the subject matter of the poems; some are related to science, some are not. I think it would be interesting to see if I can compile an impressive collection by 2018, Richard Feyman’s centenary year. Below are the best, in my opinion, from the first iteration of Feynman Point Pilish Poems. Those poems not credited to an author are my own work, and are often included to inspire improvements rather than as examples of literary genius.

## 10 May 2013

### Prize Maths Quiz: Number Puzzle on a Truncated Cube (PMQ18)

Imagine a cube with a positive integer written on each of its six faces. Now imagine cutting the vertices of this cube; the resulting solid is known as a truncated cube, or truncated hexahedron, and is shown in the image. The six original squares are now octagons, but the integers remain unchanged, and the eight vertices have become triangles. Let the number on each triangular face be the product of the three numbers written on the octagons that share an edge with it.

The sum of the numbers on all the triangles is equal to 6006. Find the smallest possible sum of the original six numbers written on the faces of the cube.

## 9 May 2013

### The Math-e-Monday Puzzle: Squares from a Tetrahedral Die (on Science 2.0)

After a bit of a lull, I have published a new mathematical puzzle in my inaccurately-named and approximately-weekly Math-e-Monday column at Science 2.0.

The Math-e-Monday Puzzle: Squares from a Tetrahedral Die

I think this is a nice puzzle. I shall post a variant of it here on Gifted Mathematics once comments have closed at Science 2.0.

Have fun!

The Math-e-Monday Puzzle: Squares from a Tetrahedral Die

I think this is a nice puzzle. I shall post a variant of it here on Gifted Mathematics once comments have closed at Science 2.0.

Have fun!

### Divisors of Large Numbers: Upper Secondary Mathematics Competition Question

Find those single-digit integers greater than 1 that divide evenly into 79

^{79}+1.

Also, find the highest value of n such that 10

^{n}divides evenly into 79

^{79}+1.

As always, questions such as these should be done without a calculator. I think this question should take approximately 5 minutes.

## 8 May 2013

### A Sequence of Composite Numbers: Middle Secondary Mathematics Competition Question

Find a sequence of 123 consecutive composite numbers.

That's it!

Just a reminder that a composite number is any positive integer greater than 1 that is not a prime.

## 7 May 2013

### Number Grid Puzzle: Upper Primary Mathematics Competition Question

The diagram below shows a grid with the whole numbers written in order, one in each cell, and laid out in a spiral formation.

Which number is in the cell that is immediately to the right of the cell with the number 400 in it?

### FU.BU Number Puzzle: Lower Secondary Mathematics Competition Question

In the following multiplication, each of the letters F, B, U and W represents a unique digit in base ten.

If FU x BU = WWW

Find the sum F + B + U + W.

## 3 May 2013

### Prize Maths Quiz: Constructing a Set of Primes (PMQ17)

__The Question__Using each of the non-zero digits only once, it is possible to construct a set of only prime numbers. The set {3, 41, 659, 827} is one possibility, with the sum of its members being equal to 1530.

What is the smallest possible sum that such a set of primes can have? Find one such set.

Just remember that the digits 1 to 9 inclusive must all be used but only once. Also note that you should not need to consult any tables of primes for the above question. However, such a list may well be useful for the extension exercises below.

__Extension Exercises__

If we restrict our required sets to those with just three 3-digit primes, find the two sets with the minimum and maximum possible sums. Again, use just the non-zero digits once each.

## 2 May 2013

### Quadratic With Prime Coefficients: Middle Secondary Mathematics Competition Question

If

*p*and

*q*are prime numbers and the equation x

^{2}- px + q = 0 has distinct positive integral roots, find the values of

*p*and

*q*.

Is it possible for the above equation to have distinct

*rational*roots? Either find values for

*p*and

*q*or prove why this is not possible.

### Factors of 12345654321: Upper Secondary Mathematics Competition Question

This is probably the shortest question I've ever posted.

How many distinct factors divide 12345654321 exactly?

Include 1 and the number itself as factors and, of course, we're only looking at positive integers.

This is supposed to be done by hand within a timed competition – calculators should not be used, and should not be necessary!

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