# Exploring and Noticing Structure - Lower Secondary

Exploring and Noticing Structure is part of our Thinking Mathematically collection.

### Summing Consecutive Numbers

##### Stage: 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?

### What's Possible?

##### Stage: 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?

### Marbles in a Box

##### Stage: 3 and 4 Challenge Level:

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

### Attractive Tablecloths

##### Stage: 4 Challenge Level:

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

### 1 Step 2 Step

##### Stage: 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?

### Got It

##### Stage: 2 and 3 Challenge Level:

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

### Pick's Theorem

##### Stage: 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.

### Pair Products

##### Stage: 4 Challenge Level:

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

### Number Pyramids

##### Stage: 3 Challenge Level:

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

### Tilted Squares

##### Stage: 3 Challenge Level:

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

### Painted Cube

##### Stage: 3 Challenge Level:

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

### 12345

##### Stage: 3 Short Challenge Level:

Weekly Problem 50 - 2011
Repeat a pattern of numbers to form a larger number. Can you find the sum of all the digits?

### Too Close to Call

##### Stage: 3 Short Challenge Level:

Weekly Problem 24 - 2012
Can you put these very close fractions into order?

### Out of Sync

##### Stage: 2 and 3 Short Challenge Level:

Weekly Problem 15 - 2013
Albert Einstein could see two clocks. One changed 10 seconds early, the other changed 10 seconds late. For what fraction of the day did they show the same time?

### Stuck in the Past

##### Stage: 2 and 3 Short Challenge Level:

Weekly Problem 13 - 2013
2002 started on a Tuesday. In which years between now and 2015 will each date fall on the same day of the week as it fell that year?

### Arithmagons

##### Stage: 3 Challenge Level:

Can you find the values at the vertices when you know the values on the edges?

### Odds and Evens

##### Stage: 3 and 4 Challenge Level:

Is this a fair game? How many ways are there of creating a fair game by adding odd and even numbers?

### Royal Arm Wrestling

##### Stage: 3 Short Challenge Level:

Weekly Problem 15 - 2007
The Kings of Clubs, Diamonds, Hearts and Spades, and their respective Queens, are having an arm wrestling competition.

### Route to Infinity

##### Stage: 3 and 4 Challenge Level:

Can you describe this route to infinity? Where will the arrows take you next?

##### Stage: 4 and 5 Challenge Level:

Which of these roads will satisfy a Munchkin builder?

### Interactive Spinners

##### Stage: 3 and 4 Challenge Level:

This interactivity invites you to make conjectures and explore probabilities of outcomes related to two independent events.

### Keep it Simple

##### Stage: 3 Challenge Level:

Can all unit fractions be written as the sum of two unit fractions?

##### Stage: 3 Challenge Level:

What can you say about the angles on opposite vertices of any cyclic quadrilateral? Working on the building blocks will give you insights that may help you to explain what is special about them.

### Shifting Times Tables

##### Stage: 3 Challenge Level:

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

### Weekly Lies

##### Stage: 3 Short Challenge Level:

Weekly Problem 4 - 2010
Mr Ross tells truths or lies depending on the day of the week. Can you catch him out?

### Charlie's Delightful Machine

##### Stage: 3 and 4 Challenge Level:

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

### Almost Constant Digits

##### Stage: 3 Short Challenge Level:

Weekly Problem 46 - 2010
Consider a 10-digit number which contains only the numbers 1, 2 or 3. How many such numbers can you write so that every pair of adjacent digits differs by 1?

### Newspaper Sheets

##### Stage: 2 and 3 Short Challenge Level:

Weekly Problem 29 - 2011
From only the page numbers on one sheet of newspaper, can you work out how many sheets there are altogether?

### What Numbers Can We Make?

##### Stage: 3 Challenge Level:

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

### Vector Journeys

##### Stage: 4 Challenge Level:

Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?

### Magic Letters

##### Stage: 3 Challenge Level:

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

### Seven Squares

##### Stage: 3 and 4 Challenge Level:

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

### Alphabetical Angle

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 8 - 2014
If all the arrangements of the letters in the word ANGLE are written down in alphabetical order, what position does the word ANGLE occupy?

### Collatz-ish

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 35 - 2014
A sequence is generated using these rules. For which values of n is the nth term equal to n?

### Take a Leaf

##### Stage: 3 Short Challenge Level:

Weekly Problem 19 - 2015
An examination paper is made from five pieces of paper. What is the sum of the other page numbers that appear on the same sheet as page 5?

### Back in Time

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 29 - 2015
How many times a day does a 24 hour digital clock look the same when reflected in a horizontal line?

### Halloween Day

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 44 - 2015
One year there were exactly four Tuesdays and four Fridays in October. On what day of the week was Halloween.

### How Many Rectangles?

##### Stage: 3 Short Challenge Level:

Weekly Problem 37 - 2016
By drawing 5 horizontal and four vertical lines, one can form 12 rectangles. What is the greatest number of rectangles that can be formed by drawing 15 lines?

### Knockdown

##### Stage: 3 Short Challenge Level:

Weekly Problem 51 - 2016
Pegs numbered 1 to 50 are placed in a row. Alternate pegs are knocked down, and this process is repeated. What is the number of the last peg to be knocked down?

### Suit Sequence

##### Stage: 3 Short Challenge Level:

Weekly Problem 8 - 2017
A pattern repeats every six symbols. What are the 100th and 101st symbols?

### Not a Zero

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 45 - 2017
What is the last non-zero digit of $2^{57} \times 3^4 \times 5^{53}$?

### Exploring and Noticing Structure - Short Problems

##### Stage: 3 and 4

A collection of short Stage 3 and 4 problems on Exploring and Noticing Structure.