# Working Systematically - Lower Secondary

### We're All Going on a Quadrilateral Hunt

##### KS 2 & 3 Short Challenge Level:

Weekly Problem 38 - 2013
A regular hexagon is divided into six equilateral triangles. How many quadrilaterals are there in the diagram?

### Island Hopping

##### KS 2 & 3 Short Challenge Level:

Weekly Problem 49 - 2009
What is the smallest number of ferry trips that Neda needs to take to visit all four islands and return to the mainland?

### Satnav Dilemma

##### KS 2 & 3 Short Challenge Level:

Weekly Problem 8 - 2012
How many routes are there in this diagram from S to T?

### Two and Two

##### KS 2 & 3 Challenge Level:

How many solutions can you find to this sum? Each of the different letters stands for a different number.

### Negative Dice

##### KS 2 & 3 Short Challenge Level:

Weekly Problem 16 - 2012
If the odd numbers on two dice are made negative, which of the totals cannot be achieved?

### Jack of Cards

##### KS 2 & 3 Short Challenge Level:

Weekly Problem 32 - 2013
What order did Jack have the cards in to make his card trick work?

### Flag-tastic

##### KS 2 & 3 Short Challenge Level:

Weekly Problem 2 - 2007
How many different flags can you make?

### Patchwork Quilt

##### KS 2 & 3 Short Challenge Level:

Weekly Problem 33 - 2013
Squares of the type shown are sewn together to make a quilt. How many different quilts can be made?

### Famous Five

##### KS 2 & 3 Short Challenge Level:

Weekly Problem 22 - 2007
The Famous Five have been given 20 sweets as a reward for solving a tricky crime.... how many different ways can they share the sweets?

### Alberta's Age

##### KS 2 & 3 Short Challenge Level:

Weekly Problem 26 - 2011
Alberta won't reveal her age. Can you work it out from these clues?

### Summing Consecutive Numbers

##### KS 3 Challenge Level:

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

### Shifting Times Tables

##### KS 3 Challenge Level:

Can you find a way to identify times tables after they have been shifted up?

### Double with 1-9

##### KS 3 Short Challenge Level:

Weekly Problem 31 - 2011
Can you find a number and its double using the digits $1$ to $9$ only once each?

### Isometric Rhombi

##### KS 3 Short Challenge Level:

Weekly Problem 31 - 2016
The diagram shows a grid of $16$ identical equilateral triangles. How many rhombuses are there made up of two adjacent small triangles?

### Central Sum

##### KS 3 Short Challenge Level:

Weekly Problem 53 - 2011
Find a number between 100 and 999 that has its middle digit equal to the sum of the other two digits. Can you find all possibilities?

### ACE, TWO, THREE...

##### KS 3 Challenge Level:

Can you picture how to order the cards to reproduce Charlie's card trick for yourself?

### Threes and Fours

##### KS 3 Short Challenge Level:

Weekly Problem 32 - 2016
What is the smallest integer which has every digit a 3 or a 4 and is divisible by both 3 and 4?

### Gridlock

##### KS 3 Short Challenge Level:

Weekly Problem 29 - 2006
No letter is repeated in any row, column or diagonal. Which letter is in the square marked with the star?

##### KS 3 Short Challenge Level:

Weekly Problem 35 - 2011
You are given lots of clues about a number. Can you work out what it is?

### Half and Half

##### KS 3 Short Challenge Level:

Weekly Problem 44 - 2016
Two of the four small triangles are to be painted black. In how many ways can this be done?

### Quick Ticket

##### KS 3 Short Challenge Level:

Weekly Problem 38 - 2006
What is the fewest coins that must change hands if I buy a ticket for 44p?

### Loose Change

##### KS 3 Short Challenge Level:

Weekly Problem 10 - 2009
In how many ways can you give change for a ten pence piece?

### Grid Without Lines

##### KS 3 Short Challenge Level:

Weekly Problem 40 - 2010
Can you remove the least number of points from this diagram, so no three of the remaining points are in a straight line?

### When Shall We Three Meet Again?

##### KS 3 Short Challenge Level:

Weekly Problem 4 - 2013
On my clock's display, the time has just changed to 02:31. How many minutes will it be until all the digits 0, 1, 2, 3 next appear together again?

### Latin Multiplication

##### KS 3 Short Challenge Level:

Weekly Problem 28 - 2015
Can you choose one number from each row and column in this grid to form the largest possibe product?

### American Billions

##### KS 3 Challenge Level:

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

### Different Digital Clock

##### KS 3 Short Challenge Level:

Weekly Problem 14 - 2013
At how many times between 10 and 11 o'clock are all six digits on a digital clock different?

### Glovely

##### KS 3 Short Challenge Level:

Weekly Problem 39 - 2006
What is the fewest gloves that Dilly needs to bring Granny to ensure that they have a pair?

### M, M and M

##### KS 3 Challenge Level:

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

### Kept Apart

##### KS 3 Short Challenge Level:

Weekly Problem 43 - 2010
The squares of this grid contain one of the letters P, Q, R and S. Can you complete this square so that touching squares do not contain the same letter? How many possibilities are there?

### 1 Step 2 Step

##### KS 3 Challenge Level:

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

### Isosceles Triangles

##### KS 3 Challenge Level:

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

### Making 11p

##### KS 3 Short Challenge Level:

Weekly Problem 20 - 2006
How many ways are there to make 11p using 1p, 2p and 5p coins?

### Musical Chairs

##### KS 3 Short Challenge Level:

Weekly Problem 42 - 2008
How many different ways could we have sat on the two remaining musical chairs at Gill's fourth birthday party?

### Pick's Theorem

##### KS 3 Challenge Level:

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

### Can They Be Equal?

##### KS 3 Challenge Level:

Can you find rectangles where the value of the area is the same as the value of the perimeter?

### Colourful Tiles

##### KS 3 Short Challenge Level:

Weekly Problem 21 - 2011
How many ways can you paint this wall with four different colours?

##### KS 3 Challenge Level:

How many different symmetrical shapes can you make by shading triangles or squares?

### Sticky Numbers

##### KS 3 Challenge Level:

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

### End of a Prime

##### KS 3 Short Challenge Level:

Weekly Problem 25 - 2016
A list is made of every digit that is the units digit of at least one prime number. How many digits appear in the list?

### Peaches Today, Peaches Tomorrow....

##### KS 3 & 4 Challenge Level:

Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?

### Medal Ceremony

##### KS 3 & 4 Short Challenge Level:

Weekly Problem 41 - 2013
The teacher has forgotten which pupil won which medal. In how many different ways could he give the medals out to the pupils?

### Even Squares

##### KS 3 & 4 Short Challenge Level:

Weekly Problem 27 - 2008
In the diagram in the question, how many squares, of any size, are there whose entries add up to an even total?

##### KS 3 & 4 Short Challenge Level:

Weekly Problem 24 - 2015
In how many ways can you move through the grid to give the digits 2009?

### Rolling Along the Trail

##### KS 3 & 4 Short Challenge Level:

Weekly Problem 32 - 2011
What could be the scores from five throws of this dice?

### Phone Call

##### KS 3 & 4 Short Challenge Level:

Weekly Problem 25 - 2015
How many different phone numbers are there starting with a 3 and with at most two different digits?

### Nine Colours

##### KS 3 & 4 Challenge Level:

You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.

### Charlie's Delightful Machine

##### KS 3 & 4 Challenge Level:

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

### Gridlines

##### KS 3 & 4 Short Challenge Level:

Weekly Problem 27 - 2015
How many triples of points are there in this 4x4 array that lie on a straight line?

### Domino Hexagon

##### KS 3 & 4 Short Challenge Level:

Weekly Problem 18 - 2017
Dominic wants to place the six dominoes above in a hexagonal ring. Which of the dominoes could be placed next to the one shown?

### Amazing

##### KS 3 & 4 Short Challenge Level:

Weekly Problem 6 - 2014
A maze has nine rooms, with gaps in the walls between them. How many ways are there to travel from X to Y?

### Switch On

##### KS 3 & 4 Short Challenge Level:

Weekly Problem 39 - 2015
In how many different ways can a row of five "on/off" switches be set so that no two adjacent switches are in the "off" position?

### No Square Sums

##### KS 3 & 4 Short Challenge Level:

Weekly Problem 12 - 2011
How many numbers do you need to remove to avoid making a perfect square?

### Fiction in Wonderland

##### KS 3 & 4 Short Challenge Level:

Weekly Problem 17 - 2014
Tweedledum, Tweedledee, Alice and the White Rabbit are having a conversation. How many of the statements they make are true?

### Wedding Morning

##### KS 3 & 4 Short Challenge Level:

Weekly Problem 45 - 2015
If Sam is getting married on the 9th of November 2015 aged 30, do you know which year Pat was born in?

### Circle of Lies

##### KS 3 & 4 Short Challenge Level:

Weekly Problem 34 - 2014
Can you work out how many of Pierre, Qadr, Ratna, Sven and Tanya are telling the truth?

### Middle Digit Mean

##### KS 3 & 4 Short Challenge Level:

Weekly Problem 16 - 2016
How many three digit numbers have the property that the middle digit is the mean of the other two digits?

### Mini-sodoku

##### KS 3 & 4 Short Challenge Level:

Weekly Problem 8 - 2015
How many ways are there of completing the mini-sudoku shown?

### What's Possible?

##### KS 4 Challenge Level:

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

### A Long Time at the Till

##### KS 4 & 5 Challenge Level:

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?