Patterns and Sequences Short Problems

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What sizes of rectangle contain exactly 100 squares? Can you find them all?

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

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?

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

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

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

Trinni rearranges 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?

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

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?

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

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

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

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

Can you find the connections between linear and quadratic patterns?

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

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

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

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?

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

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

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?

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

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?

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

How many squares are needed to continue this pattern?

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?

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

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?

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

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

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

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

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

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

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

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

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?

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

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

MatildaMatildaMatil... What is the 1000th letter?

A sequence is generated using these rules. For which values of n is the nth term equal to n?

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

In a supermarket, there are two lines of tightly packed trolleys. What is the length of one trolley?

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

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

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?

When will 2000 appear in this sequence?

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.

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?

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

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

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

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

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?

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

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

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

Simple additions can lead to intriguing results...

If a number is even, halve it; if odd, treble it and add 1. If a sequence starts at 13, what will be the value of the 2008th term?