## 30 Apr 2013

### Another Prime Number Cascade: Lower Secondary Mathematics Competition Question

This is a similar question to that posted yesterday, so the best place to start is to read that puzzle first.

However, in this case, you must fill the grid so that all ten prime numbers are different. What is the smallest possible prime that can go in the circle at the bottom of the triangle?

Just to recap the rules, you start by picking four different primes, one for each of the circles in the top row. For each circle below, take the two numbers immediately above it; let’s call them p and q. Calculate the three sums (p+q), (p+q+1) and (p+q-1). If one, or more, of these sums is also a prime number then pick one of them and insert it into the previously empty circle. Note that it is possible that none of the sums are prime, in which case your two numbers p and q are in the wrong places and you will need to change at least one of them.

The puzzle is to create a triangle filled with ten different prime numbers. Using the rules above, what is the smallest possible prime that can go in the last circle at the bottom of the triangle?

## 29 Apr 2013

### Prime Number Cascade: Upper Primary Mathematics Competition Question

The diagram below shows an inverted triangle of circles. Inside each circle, you need to place a number according to the following rules.

Start at the top row of four circles and insert a different prime number inside each one. The number inside each of the other circles is chosen using the two circles immediately above it. Firstly, find the sum of those two numbers, then add 1 and subtract 1 from this total so that you have three numbers to choose from. For example, if the numbers are 13 and 17, calculate the sum (30) and then add and subtract 1 from this to get 31 and 29. You thus have the three numbers: 29, 30 and 31. In this case, two of the numbers are prime number – 29 and 31 -  so you have a choice as to which one you place inside the empty circle. Continue in this manner until you reach the bottom.

The puzzle is how to fill the diagram with ten prime numbers, using the rules above, in such a way as to get the smallest possible prime in the last circle at the bottom of the triangular grid. What is the value of this smallest prime number in the bottom circle?

Just to clarify, your four starting primes along the top row must all be different. However, the final solution may have numbers that appear more than once.

You may also like to start with this slightly easier version. If you could pick any prime numbers along the top row, and if you are allowed to repeat the same number multiple times, using the same rules to create each new circle what is the smallest possible prime in the bottom circle?

## 26 Apr 2013

### Prize Maths Quiz: Tetrahedral Constructs (PMQ16)

A regular tetrahedron is constructed within a unit cube in such a way that its four vertices coincide with those of the cube – see the diagram.

Question Alpha

What is the volume of the tetrahedron?

Question Omega

If you have been following this week’s questions, then you’ll find the above pretty simple. So, let’s increase the difficulty dial! Imagine constructing another regular tetrahedron using the other four vertices of the unit cube. The space where the two tetrahedra overlap is known as the core. What is the name of the polyhedral core? What is the volume of this core?

## 25 Apr 2013

### Slicing a Triangular Prism: Upper Secondary Mathematics Competition Question

Below is a diagram of a regular triangular prism. Triangles ABC and DEF are equilateral and parallel to each other. A cut is made that slices the prism into two pieces in such a way that the plane passes through points A, B and F.

What is the ratio of (volume ABEFD) : (volume ABCF) ?

## 24 Apr 2013

### Slicing a Hexagonal Prism: Middle Secondary Mathematics Competition Question

The diagram below shows a regular hexagonal prism (not to scale). Each side of the hexagonal faces has a length of 1 unit. The height of the prism BP is equal to the length BQ.

A slice is taken across the prism in such a way that the plane includes the line segments AB and DE. This plane also cuts the prism at points C and F. The intersection of the plane and the prism forms another hexagon ABCDEF.

Find the area of the hexagon ABCDEF.

## 23 Apr 2013

### Paths on a Cube: Lower Secondary Mathematics Competition Question

Below is a diagram of a large cube constructed from eight smaller cubes. Each small cube has a side length of 2 cm. An ant wishes to travel from point Q to point S. The figure shows two routes: QRS and QTS.

Which of the two routes is the shortest and by how much?

Also, if the ant was able to burrow its way through the cubes, what would be the shortest direct straight-line distance from Q to S?

### Paths Round a Cube: Upper Primary Mathematics Competition Question

This week we’re going to be investigating some questions in three-dimensional geometry.

Below is a diagram of a large cube constructed from eight smaller cubes. An ant wishes to travel from point Q to point S along the route of shortest distance. The figure also shows the route QRS. This route is, indeed, the shortest possible path length if travelling along the surface of the cube. However, it is not the only possible such route.

How many routes are there in total, from Q to S, that have the same path length as the route QRS?

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## 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.

## 18 Apr 2013

### 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.

## 17 Apr 2013

### The Currency of Tripolina: Middle Secondary Mathematics Competition Question

The island of Tripolina has a strange currency. The tryad is denominated in powers of 3; they have coins worth 1, 3, 9, 27 and 81 tryads. Alice is walking around the local market and feels that her purse is really far too heavy. Back at the hotel she had counted out all her coins and found that she had five each of 1, 3, 9 and 27 tryads. At the market, she spotted an ornate vase that would make a lovely present for her mother. The vase cost 59 tryads. What is the maximum number of coins that Alice can use to pay for the vase without receiving any change?

Pleased with her purchase, Alice heads for a coffee shop. A coffee costs 5 tryads. Alice frowns when the change arrives and asks for the minimum number of coins. Which coin did Alice pay with? And what is the minimum number of coins she can receive in change?

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## 16 Apr 2013

### Combinations of Small Tiles: Upper Primary Mathematics Competition Question

You are given two tiles, one has dimensions 1x2 units and the other is 1x5 units.

You have as many of each tile as you need. You are asked to tile different areas using only these two tiles.

For example, you can make an area of 10 using just two 5s. You can make an area of 14 using two 5s and two 2s or with seven 2s.

Two questions

a) Which whole numbers cannot be made using just these two tiles?

b) In how many different ways can you make the number 30?

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### Linear Combinations of 5 and 7: Lower Secondary Mathematics Competition Question

You are given two tiles, one has dimensions 1x5 units and the other is 1x7 units.

You have as many of each tile as you need. You are asked to tile different areas using only these two tiles.

For example, you can make an area of 10 using just two 5s. You can make an area of 12 using one 5 and one 7. You cannot make an area of 11 using just these two tiles.

The question is: what is the largest possible area that cannot be made using just the tiles 1x5 and 1x7?

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## 12 Apr 2013

### Prize Maths Quiz: Minimal Tilings (PMQ14)

Today’s PMQ comes in two parts. Firstly, take a square with sides 7 units in length. You are required to tile this with smaller squares, each with whole number sides, in such a way as to use the minimum number of smaller squares. One extra condition is that the distinct tiles used must have no common factors, or put another way, the gcd of any two different tiles must be equal to 1.

The diagram shows one possible tiling for a square with sides of 6 units. At one extreme, we could just use unit squares, in which case we would need 36 tiles. At the other extreme, we could use one tile with sides 5 units and 11 unit tiles, making a total of 12. Neither of these will give us the minimum number of tiles; they are just presented as examples of how to approach this problem. However, notice that the tiling in the diagram satisfies the second condition. There are three distinct tiles used - of sides 1, 2 and 3 - and their pairwise gcd equals 1. There are solutions that use tiles of sides 2 and 4 but this would give a pairwise gcd of 2.

The second step in the question is to find all the different arrangements of the minimum number of tiles. These must be unique layouts, so that tilings with rotational symmetry count as just one arrangement. If we were actually tiling a floor, each arrangement could be rotated four times to give the appearance of different layouts, but it is still just a single unique design.

To recap the questions, you are given a square with sides of 7 units. You must find the minimum number of smaller squares with integer sides that can completely cover this larger square with no overlaps and no gaps. The sides of the distinct squares used can have no pairwise common factor other than 1. Then, using these minimum number of squares, how many unique arrangements are there?

[edited for clarity on 13 April]

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

How to Enter

Send your complete solution by email to pmq14nash@giftedmaths.comThis email address shall be removed after the competition closes to avoid spam. This PMQ14 competition closes on MONDAY 15 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

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## 11 Apr 2013

### Dissecting a Cube: Upper Secondary Mathematics Competition Question

This week we have been looking at the sums of squares; here is a problem regarding the sums of cubes.

You have a cube with sides of length 5 units. You need to dissect it into smaller cubes, all with integer sides, in such a way that you can reconstruct the original cube. What is the minimum possible number of smaller cubes? List how many cubes of each size there are.

Remember that it is not enough for the sums of cubes to be arithmetically correct, the cubes need to all fit together to make the original cube with sides of length 5 units.

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## 10 Apr 2013

### Sums of Squares in Rectangles: Middle Secondary Mathematics Competition Question

In this question, we’re going to combine the sums of squares with a geometric interpretation.

Firstly, let’s look at how to write a number as sums of squares. This is called a number partition into squares. Let’s look at the number 12 and start with the most obvious partition into sums of unit squares.

12 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1

It is important to do this in some kind of order, so let’s see how many partitions we can write using the next highest square, 22.

12 = 22 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
12 = 22 + 22 + 1 + 1 + 1 + 1
12 = 22 + 22 + 22

We have one more square that we can use and that’s 32.

12 = 32 + 1 + 1 + 1

That’s it! There are just 5 different ways to partition the number 12 into sums of squares. The next step is to look at the geometry of these partitions. Below are just three of these partitions. We can see that all the squares fit together to make rectangles with areas also equal to 12. However, the layout with the least number of squares has just 3 shapes. This is what we shall be looking for: the sum of squares with the least number of squares and that they fit within a rectangle with the same area as our original number.

Let’s see what happens to the number 11. It has similar partitions to 12 but with one unit removed.

11 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
11 = 22 + 1 + 1 + 1 + 1 + 1 + 1 + 1
11 = 22 + 22 + 1 + 1 + 1
11 = 32 + 1 + 1

Now, if we try to create rectangles with the sums of squares, we find that there are gaps. Imagine those rectangles for the number 12 with one unit square removed. However, this means that the smallest rectangles are all 12 units in area, apart from one. The only rectangle that has the same area as the number itself is the sum of unit squares.

So, let me recap. We are going to partition some numbers into their sums of square numbers. Then we are going to fit together the actual squares to create rectangles that have the same area as the original number. If there is more than one such rectangle we shall select the one with the smallest number of squares.

Try this with the numbers 30 to 40 inclusive. What do you notice about the prime numbers? What do you notice about the square numbers?

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## 9 Apr 2013

### Unique Sums of Squares: Lower Secondary Mathematics Competition Question

This is an extension to yesterday’s question. We are still looking at how positive whole numbers can be expressed as the sums of square numbers.

This time we are looking at how some numbers can be expressed as different sums of squares. Again, we are only allowed a maximum of four squares and a minimum of, obviously, just one.

For example, 9 can be simply expressed as 32, but it can also be made from 22 + 22 + 12. The number 11 can only equal 32 + 12 + 12; there are no other sums of squares that can equal 11 unless we use more than four of them.

So, our question this time is to find those numbers between 20 and 30 inclusive that can be expressed as the sums of squares in only one unique way and using a maximum of four squares.

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## 8 Apr 2013

### Sums of Squares: Upper Primary Mathematics Competition Question

Alice is playing a number game. She is trying to make every whole number as the sum of square numbers. For example, 14 = 32 + 22 + 12 and 15 = 32 + 22 + 12 + 12.

But she notices that numbers like 14 can be made using all different squares, whereas others, like 15, need to use the same square number twice. Indeed, some numbers may need to use the same square three times.

Now, can you find those numbers between 20 and 30 inclusive that can be made as the sum of different squares?

It doesn't matter how many squares you use - it could be two, three or four, or even just one!

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## 5 Apr 2013

### Prize Maths Quiz: A Triangular Tiling Puzzle (PMQ13)

The diagram below illustrates one large equilateral triangle composed of smaller equilateral triangles. The number shown within each triangle is the length of one side relative to the other triangles. Every triangle must have a non-zero integer value. If we doubled the sides of every triangle we would get the same pattern but with double the numerical values; the triangle shown is thus also the smallest possible with that particular tiling.

The order of such a triangle is the number of smaller triangles – in this case, the order is 8.

The size of the triangle is the length of the large triangle – in this case, the size is 5.

The sum of the triangle is the sum of the numbers within the smaller triangles. However, the rule to calculate the sum is that triangles pointing upwards are positive, while those pointing downwards are negative. So in this case, the sum is equal to 3 + 2 + 2 + 1 + 1 – 2 – 1 – 1 = 5.

## 4 Apr 2013

### A Convoluted Reciprocal: Upper Secondary Mathematics Competition Question

Using only the trigonometric functions on a calculator, including their inverses (if needed), find a sequence of buttons pressed such that the final result is the reciprocal of the initial input value. Also, for what range of input values does this work?

That is, we are looking for a function f such that f(x) = 1/x and f is a compound function of trigonometric functions and their inverses. We also need to find the domain of f.

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## 3 Apr 2013

### Overlapping Circles Puzzle: Middle Secondary Mathematics Competition Question

Below is a diagram showing five overlapping circles. Alice is trying to put one of the numbers 1 to 9 inclusive into each of the nine regions in such a way so that each circle has the same total sum and every region has just one number within it.

a) Is it possible? If so, how many solutions are there? List the order of the numbers, ignoring reflections in the vertical axis.

b) Also, prove that if we use the first nine odd numbers there are no solutions possible.

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## 2 Apr 2013

### Prime Shopper: Lower Secondary Mathematics Competition Question

Marcus is trying to carry three bags of shopping. Each bag weighs a prime number of kg and the total weight is 25 kg. However, one of the bags is far too heavy for him. He would like to redistribute the items in the bags in such a way that his left and right arms are carrying nearly the same weight and he still has three prime number weights.

How much does each bag weigh? And how much did each bag weigh before Marcus redistributed the items?

[Edited for clarity: 2 April 2013]

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## 1 Apr 2013

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

Here is a small number puzzle.

1 Across is a cube

3 Across is a square

1 Down is a square

2 Down is a cube

All four 3-digit numbers must be different.

Level: Upper Primary (Green)
and Hard (Black).

Difficulty is relative to the student level.