# Patterns and Sequences Short Problems

### Printing Error

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

Weekly Problem 17 - 2011
Every third page number in this book has been omitted. Can you work out what number will be on the last page?

### Many Matildas

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

Weekly Problem 23 - 2011
MatildaMatildaMatil... What is the 1000th letter?

### Street Lamps

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

Weekly Problem 9 - 2007
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?

### Starting Fibonacci

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

Weekly Problem 40 - 2012
What is the first term of a Fibonacci sequence whose second term is 4 and fifth term is 22?

### Mean Sequence

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

Weekly Problem 38 - 2009
This sequence is given by the mean of the previous two terms. What is the fifth term in the sequence?

### Hexagon Line

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

Weekly Problem 5 - 2016
How many hexagons are required for the perimeter of the whole shape to have length 1002cm?

### 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?

### Pattern Snake

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

Weekly Problem 27 - 2010
This pattern repeats every 12 dots. Can you work out what a later piece will be?

### Luis' Seven

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

Weekly Problem 6 - 2016
Luis writes down seven consecutive positive integers. The sum of the three smallest numbers is 33. What is the sum of the three largest numbers?

### Triangle of Squares

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

Weekly Problem 42 - 2012
What number appears immediately below 400 in the triangle?

### Expanding Pattern

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

Weekly Problem 36 - 2010
How many squares are needed to continue this pattern?

### Lumber9

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

Weekly Problem 30 - 2016
The base of a pyramid has n edges. What is the difference between the number of edges of the pyramid and the number of faces?

### Triangular Clock

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

Weekly Problem 31 - 2007
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?

### Night Watchmen

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

Weekly Problem 20 - 2008
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?

### 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?

### Producing an Integer

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

Weekly Problem 46 - 2011
Multiply a sequence of n terms together. Can you work out when this product is equal to an integer?

### Below 400

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

Weekly Problem 51 - 2006
Can you work out which number will appear directly below 400 in this pattern?

### Knights and Knaves

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

Weekly Problem 35 - 2010
Knights always tell the truth. Knaves always lie. Can you catch these knights and knaves out?

### Trolley Park

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

Weekly Problem 14 - 2016
In a supermarket, there are two lines of tightly packed trolleys. What is the length of one trolley?

### Fibonacci Deduction

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

Weekly Problem 52 - 2010
Leonard writes down a sequence of numbers. Can you find a formula to predict the seventh number in his sequence?

### Powerful Finale

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

Weekly Problem 24 - 2017
What is the last digit of $3^{2011}$?

### Alternating Sum

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

Weekly Problem 1 - 2008
Given that the number 2008 is the correct answer to a sum, can you find n?

### Fruit Line-up

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

Weekly Problem 8 - 2011
This grocer wants to arrange his fruit in a particular order, can you help him?

### Amazing

##### Stage: 3 and 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?

### Doubly Consecutive Sums

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

Weekly Problem 51 - 2017
How many numbers less than 2017 are both the sum of two consecutive integers and the sum of five consecutive integers?

### Chessboard Counters

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

Weekly Problem 3 - 2008
Counters are placed on an 8 by 8 chessboard... What fraction of the counters are on squares of the same colour as themselves?

### Draught Plans

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

Weekly Problem 9 - 2011
Barbara is putting draughts on a $4x4$ board in a particular way. Can you find the least number of draughts she needs to put down?

### 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?

### Big Fibonacci

##### Stage: 4 Short Challenge Level:

Weekly Problem 41 - 2012
The fifth term of a Fibbonaci sequence is 2004. If all the terms are positive integers, what is the largest possible first term?

### In One... and the Other

##### Stage: 4 Short Challenge Level:

Weekly Problem 44 - 2012
Consider two arithmetic sequences: 1998, 2005, 2012,... and 1996, 2005, 2014,... Which numbers will appear in both?

### Collatz 13

##### Stage: 4 Short Challenge Level:

Weekly Problem 46 - 2012
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?