Patterns and Sequences - Short Problems
What a Coincidence!
Consider two arithmetic sequences: 1998, 2005, 2012,... and 1996, 2005, 2014,... Which numbers will appear in both?
Expanding Pattern
How many squares are needed to continue this pattern?
Street Lamps
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?
Fibonacci Deduction
Leonard writes down a sequence of numbers. Can you find a formula to predict the seventh number in his sequence?
Suit Sequence
A pattern repeats every six symbols. What are the 100th and 101st symbols?
Triangular Clock
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?
Fruit Line-up
This grocer wants to arrange his fruit in a particular order, can you help him?
Printing Error
Every third page number in this book has been omitted. Can you work out what number will be on the last page?
Night Watchmen
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?
Pattern Snake
This pattern repeats every 12 dots. Can you work out what a later piece will be?
Sliding Robot
A robot moves along the number line. Where will it be after 2011 slides?
Square Grid
Can you work out what fraction of this grid is shaded?
How Many Rectangles?
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?
Tiled Floor
This tiled floor has 109 purple tiles. How many tiles are there altogether?
Knockdown
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?
Hexagon Line
How many hexagons are required for the perimeter of the whole shape to have length 1002cm?
Even Up
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?
Knights and Knaves
Knights always tell the truth. Knaves always lie. Can you catch these knights and knaves out?
Doubly Consecutive Sums
How many numbers less than 2017 are both the sum of two consecutive integers and the sum of five consecutive integers?
Diagonals
How many diagonals does a regular icosagon (20 sides) have?
Trolley Park
In a supermarket, there are two lines of tightly packed trolleys. What is the length of one trolley?
12345
Repeat a pattern of numbers to form a larger number. Can you find the sum of all the digits?
Newspaper Sheets
From only the page numbers on one sheet of newspaper, can you work out how many sheets there are altogether?
Below 400
Can you work out which number will appear directly below 400 in this pattern?
Collatz 13
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?
Collatz-ish
A sequence is generated using these rules. For which values of n is the nth term equal to n?
Alternating Sum
Given that the number 2008 is the correct answer to a sum, can you find n?
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Seven Squares - Group-worthy Task
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?