Patterns and Sequences short problems

problem

Triangle numbers

Age

11 to 14

Challenge level

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

problem

Seven squares

Age

11 to 14

Challenge level

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

problem

Triangular clock

Age

11 to 14

Challenge level

Trinni rearanges numbers on a clock face so each adjacent pair add up to a triangle number... What number did she put where 6 would usually be?

problem

What numbers can we make now?

Age

11 to 14

Challenge level

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

article

Spaces for exploration

Alf Coles writes about how he tries to create 'spaces for exploration' for the students in his classrooms.

problem

Night watchmen

Age

11 to 14

Challenge level

Grannie's watch gains 30 minutes every hour, whilst Grandpa's watch loses 30 minutes every hour. What is the correct time when their watches next agree?

problem

What a coincidence!

Age

11 to 14

Challenge level

Consider two arithmetic sequences: 1998, 2005, 2012,... and 1996, 2005, 2014,... Which numbers will appear in both?

article

Train spotters' paradise

Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.

problem

Elevenses

Age

11 to 14

Challenge level

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

problem

Beach huts

Age

11 to 14

Challenge level

Can you figure out how sequences of beach huts are generated?

list

Teaching patterns and sequences

Age

11 to 16

These articles make the case for teaching in a way that emphasises understanding of underlying structures rather than simply noticing patterns in sequences.

problem

Tower of Hanoi

Age

11 to 14

Challenge level

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

problem

Growing surprises

Age

11 to 14

Challenge level

Can you find the connections between linear and quadratic patterns?

problem

Days and dates

Age

11 to 14

Challenge level

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

problem

Shifting times tables

Age

11 to 14

Challenge level

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

problem

Hexagon line

Age

11 to 14

Challenge level

How many hexagons are required for the perimeter of the whole shape to have length 1002cm?

problem

Summing consecutive numbers

Age

11 to 14

Challenge level

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

problem

Pattern snake

Age

11 to 14

Challenge level

This pattern repeats every 12 dots. Can you work out what a later piece will be?

problem

Sliding robot

Age

11 to 14

Challenge level

A robot moves along the number line. Where will it be after 2011 slides?

problem

1 step 2 step

Age

11 to 14

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?

problem

Knights and knaves

Age

11 to 14

Challenge level

Knights always tell the truth. Knaves always lie. Can you catch these knights and knaves out?

problem

How many rectangles?

Age

11 to 14

Challenge level

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?

problem

Frogs

Age

11 to 14

Challenge level

How many moves does it take to swap over some red and blue frogs? Do you have a method?

problem

Expanding pattern

Age

11 to 14

Challenge level

How many squares are needed to continue this pattern?

problem

Knockdown

Age

11 to 14

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?

problem

Even up

Age

11 to 14

Challenge level

Consider all of the five digit numbers which we can form using only the digits 2, 4, 6 and 8. If these numbers are arranged in ascending order, what is the 512th number?

problem

Charlie's delightful machine

Age

11 to 16

Challenge level

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

problem

Suit sequence

Age

11 to 14

Challenge level

A pattern repeats every six symbols. What are the 100th and 101st symbols?

problem

Picturing triangular numbers

Age

11 to 14

Challenge level

What do you notice about the sum of two identical triangular numbers?

problem

Fibonacci deduction

Age

11 to 14

Challenge level

Leonard writes down a sequence of numbers. Can you find a formula to predict the seventh number in his sequence?

problem

Doubly consecutive sums

Age

11 to 14

Challenge level

How many numbers less than 2017 are both the sum of two consecutive integers and the sum of five consecutive integers?

problem

Picturing square numbers

Age

11 to 14

Challenge level

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

problem

Fruit line-up

Age

11 to 14

Challenge level

This grocer wants to arrange his fruit in a particular order, can you help him?

problem

Square grid

Age

11 to 14

Challenge level

Can you work out what fraction of this grid is shaded?

problem

Seven squares - group-worthy task

Age

11 to 14

Challenge level

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

problem

Printing error

Age

11 to 14

Challenge level

Every third page number in this book has been omitted. Can you work out what number will be on the last page?

problem

Tiled floor

Age

11 to 14

Challenge level

This tiled floor has 109 purple tiles. How many tiles are there altogether?

problem

Coordinate patterns

Age

11 to 14

Challenge level

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

problem

Many Matildas

Age

11 to 14

Challenge level

MatildaMatildaMatil... What is the 1000th letter?

problem

Squares in rectangles

Age

11 to 14

Challenge level

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

problem

What numbers can we make?

Age

11 to 14

Challenge level

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

list

Patterns and sequences - short problems

Age

11 to 16

A collection of short problems on patterns and sequences.

problem

Street lamps

Age

11 to 14

Challenge level

Walking up a steep hill, I pass 10 equally spaced street lamps. How long do I take to walk from the first lamp to the last?

problem

Odds, evens and more evens

Age

11 to 14

Challenge level

Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions...

problem

Diagonals

Age

14 to 16

Challenge level

How many diagonals does a regular icosagon (20 sides) have?

problem

Series sums

Age

14 to 16

Challenge level

Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

problem

Difference sequence

Age

14 to 16

Challenge level

When will 2000 appear in this sequence?

problem

Attractive tablecloths

Age

14 to 16

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?

problem

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate a(n) and b(n) for n<8. What do you notice about these sequences? (ii) Find a relation between a(p) and b(q). (iii) Prove your conjectures.

problem

Painted cube

Age

14 to 16

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?

problem

12345

Age

14 to 16

Challenge level

Repeat a pattern of numbers to form a larger number. Can you find the sum of all the digits?

problem

Below 400

Age

14 to 16

Challenge level

Can you work out which number will appear directly below 400 in this pattern?

problem

Alternating sum

Age

14 to 16

Challenge level

Given that the number 2008 is the correct answer to a sum, can you find n?

problem

Mystic rose

Age

14 to 16

Challenge level

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

problem

Partly painted cube

Age

14 to 16

Challenge level

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?