Number Starters:A Very Strange Game: Four different actions depending on the number which appears. Abundant Buses: A game based around the concept of factors and abundant numbers. Add 'em: Add up a sequence of consecutive numbers. Can you find a quick way to do it? All The Nines: Add up all the multiples of nine in an elegant way. Ancient Mysteries: This activity requires students to memorise fifteen numbers in a three by five grid. Aunt Sophie's Post Office: Work out the number of stamps needed to post a parcel. Birthday Clues: Work out the date Will was born by answering some number questions. Can You Decide?: Recognise odd, even, square, prime and triangular numbers. Christmas Bells: If all the bells ring together at noon, at what time will they next all ring together? This problem requires the use of LCM. Christmas Eve: Is there a pattern in the number of palindromic numbers to be found less than powers of 10? ClockEquate: Can you use the digits on the left of this clock along with any mathematical operations to equal the digits on the right? Coins in Envelopes: Fifteen pennies are placed in four envelopes and the envelopes are sealed. It is possible to pay someone any amount from 1p to 15p by giving them one or more envelopes. Consecutive Squares: What do you notice about the difference between the squares of consecutive numbers? Dancing: Work out how many people were at the dance from the clues given. Dice Nets: Determine whether the given nets would fold to produce a dice. Dimidiate: Arrange the digits from 1 to 9 in alphabetical order. How many times can this number be halved? Divided Age: How old is a person if when her age is divided by certain numbers, the calculator display ending are as shown. Double Trouble: Begin with one, double it, double it again and so on. How many numbers in this sequence can you write down before the register has been called? Factuples: Spot the factors and the multiples amongst the numbers in the grid. Flabbergasted: If each number in a sequence must be a factor or multiple of the previous number what is the longest sequence that can be made from the given numbers? Flowchart: Use the flowchart to generate a sequence of numbers. Which number will reach 1 the fastest? Four Factors: Find four single digit numbers that multiply together to give 120. How many different ways are there of answering this question? Four Problems: For mathematical questions to get everyone thinking at the beginning of the lesson. Halve it: Start with 512. Halve it to get 256. Halve it to get 128. Continue as far as possible. Handshakes: If all the students in this room shook hands with each other, how many handshakes would there be altogether? Hot Numbers: Move the numbered cards to form five 2 digit numbers which are all multiples of three. Hotel Digital: A puzzle about the lifts in a hotel which serve floors based on the day of the week. House Numbers: The numbers on five houses next to each other add up to 70. What are those five numbers? Inbetween Table: Write down as many multiples of 3.5 as possible in 3.5 minutes. Last Day: The 31st of December is the last day of the year. What mathematical lasts do you know? Leap Year: A question about the birthdays of a child born on the 29th February. Letters in a Number: Questions about the number of letters in numbers. Meta Products: Which numbers when multiplied by the number of letters in the word(s) of the number give square numbers? Missing Terms: Find the missing terms from these linear sequences. Name Again: Work out what the nth letter will be in a recurring pattern of letters in a person's name Negative Numbers: Perform calculations involving negative numbers No Partner: Find which numbers in a given list do not combine with other numbers on the list to make a given sum. Number Riddles: Can you work out the numbers from the given clues. Numbers in words: Write out in words some numbers writen as digits (optional pirate theme) Odd One Out: From the numbers given, find the one that is the odd one out. Only One Number: Find other numbers that can be changed to 1 on a calculator using only the 4 key and any operation. Pears Make Squares: Find three numbers such that each pair of numbers adds up to a square number. Perfect Numbers: Six is a perfect number as it is the sum of its factors. Can you find any other perfect numbers? Plane Numbers: Arrange numbers on the plane shaped grid to produce the given totals Pyramid Puzzle: Arrange numbers at the bottom of the pyramid which will give the largest total at the top. Register: When the register is called answer with a multiple of 7. Ropey Snowballs: Arrange the numbers on the snowballs so that no two consecutive numbers are directly connected by rope. Satisfaction: Rearrange the numbers, row and column headings so that this table is mathematically correct. Scaramouche: Can you work out from the five clues given what the mystery number is? Seeing Squares: How many square numbers can be found in the grid of digits. Sign Sequences: Continue the sequences if you can work out the rule. Simple Nim: The classic game of Nim played with a group of pens and pencils. The game can be extended to the multi-pile version. Small Satisfaction: Arrange the digits one to nine in the grid so that they obey the row and column headings. Square and Even: Arrange the numbers on the cards so that each of the three digit numbers formed horizontally are square numbers and each of the three digit numbers formed vertically are even. Square Angles: Find a trapezium, a triangle and a quadrilateral where all of the angles are square numbers. Square Christmas Tree: Draw a picture of a Christmas tree using only square numbers. Square Pairs: Arrange the numbered trees so that adjacent sums are square numbers. Square Sequence: Write out as many square numbers as possible in 4 minutes. Squigits: A challenge to find numbers which have each of their digits as square numbers. The Power of Christmas: Find a power of 2 and a power of 3 that are consecutive numbers. The story of ...: Be creative and come up with as many facts about a number as you can think of. Tindice: How can you put the dice into the tins so that there is an odd number of dice in each tin? Twelve Days: A Maths puzzle based on the 12 Days of Christmas song. Two Numbers: Find the two numbers whose sum and product are given. Upside Number: Work out the phone number from the clues given. Venn Diagram: Arrange numbers on the Venn Diagram according to their properties. What are they?: A starter about sums, products, differences, ratios, square and prime numbers.
Small images of these Starters :: Index of Starters Number Advanced Starters:Back To The Factory: Find all the numbers below 1000 which have exactly 20 factors Barmy BIDMAS: A misleading way of stating the answer to a simple calculation. Calendar Riddle: Work out the date of my birthday from the clues in rhyme. Car Inequalities: Solve three simultaneous inequalities to find how many cars I own. Cube Ages: Calculate the mean age of the two fathers and two sons with the given clues. Difference Cipher: Find the mathematical word from the cipher Divisible by 11: Can you prove that a three digit number whose first and third digits add up to the value of the second digit must be divisible by eleven? HCF and LCM given: If given the HCF, LCM can you find the numbers? Is it a Number?: A mathematical object is a number if it ... Key Eleven: Prove that a four digit number constructed in a certain way will be a multiple of eleven. Nine Digit Numbers: How many different nine digit numbers are their that contain each of the digits from one to nine? Penny Bags: Can you place 63 pennies in bags in such a way that you can give away any amount of money (from 1p to 63p) by giving a selection of these prepacked bags? Unlucky Seven Eleven: Follow the instructions to multiply a chosen number then explain the result you get. Zero Even: Prove that zero is an even number. SuggestedCounting Quest Try this exercise to master counting up to 100 items with precision and confidence.
Curriculum for Number:Year 5Pupils should be taught to read, write, order and compare numbers to at least 1 000 000 and determine the value of each digit more... Pupils should be taught to know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers more... Pupils should be taught to establish whether a number up to 100 is prime and recall prime numbers up to 19 more... Pupils should be taught to read Roman numerals to 1000 (M) and recognise years written in Roman numerals more... Year 6Pupils should be taught to read, write, order and compare numbers up to 10 000 000 and determine the value of each digit more... Pupils should be taught to solve number and practical problems that involve other recently learnt mathematical skills more... Pupils should be taught to identify common factors, common multiples and prime numbers more... Years 7 to 9Pupils should be taught to use the concepts and vocabulary of prime numbers, factors (or divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation property more... Pupils should be taught to appreciate the infinite nature of the sets of integers, real and rational numbers. more... International BaccalaureateSee the Number and Algebra sub-topics, syllabus statements, exam-style questions and learning resources for the IB AA course here. Exam-Style Questions:There are almost a thousand exam-style questions unique to the Transum website. Feedback:Comment recorded on the 1 August 'Starter of the Day' page by Peter Wright, St Joseph's College: "Love using the Starter of the Day activities to get the students into Maths mode at the beginning of a lesson. Lots of interesting discussions and questions have arisen out of the activities. Comment recorded on the 28 September 'Starter of the Day' page by Malcolm P, Dorset: "A set of real life savers!! Comment recorded on the 6 May 'Starter of the Day' page by Natalie, London: "I am thankful for providing such wonderful starters. They are of immence help and the students enjoy them very much. These starters have saved my time and have made my lessons enjoyable." Comment recorded on the 16 March 'Starter of the Day' page by Mrs A Milton, Ysgol Ardudwy: "I have used your starters for 3 years now and would not have a lesson without one! Fantastic way to engage the pupils at the start of a lesson." Comment recorded on the 8 May 'Starter of the Day' page by Mr Smith, West Sussex, UK: "I am an NQT and have only just discovered this website. I nearly wet my pants with joy. Comment recorded on the 10 September 'Starter of the Day' page by Carol, Sheffield PArk Academy: "3 NQTs in the department, I'm new subject leader in this new academy - Starters R Great!! Lovely resource for stimulating learning and getting eveyone off to a good start. Thank you!!" Comment recorded on the 14 September 'Starter of the Day' page by Trish Bailey, Kingstone School: "This is a great memory aid which could be used for formulae or key facts etc - in any subject area. The PICTURE is such an aid to remembering where each number or group of numbers is - my pupils love it! Comment recorded on the 21 October 'Starter of the Day' page by Mr Trainor And His P7 Class(All Girls), Mercy Primary School, Belfast: "My Primary 7 class in Mercy Primary school, Belfast, look forward to your mental maths starters every morning. The variety of material is interesting and exciting and always engages the teacher and pupils. Keep them coming please." Comment recorded on the 25 June 'Starter of the Day' page by Inger.kisby@herts and essex.herts.sch.uk, : "We all love your starters. It is so good to have such a collection. We use them for all age groups and abilities. Have particularly enjoyed KIM's game, as we have not used that for Mathematics before. Keep up the good work and thank you very much Comment recorded on the 9 May 'Starter of the Day' page by Liz, Kuwait: "I would like to thank you for the excellent resources which I used every day. My students would often turn up early to tackle the starter of the day as there were stamps for the first 5 finishers. We also had a lot of fun with the fun maths. All in all your resources provoked discussion and the students had a lot of fun." Comment recorded on the 23 September 'Starter of the Day' page by Judy, Chatsmore CHS: "This triangle starter is excellent. I have used it with all of my ks3 and ks4 classes and they are all totally focused when counting the triangles." Comment recorded on the 14 October 'Starter of the Day' page by Inger Kisby, Herts and Essex High School: "Just a quick note to say that we use a lot of your starters. It is lovely to have so many different ideas to start a lesson with. Thank you very much and keep up the good work." Comment recorded on the 2 April 'Starter of the Day' page by Mrs Wilshaw, Dunsten Collage,Essex: "This website was brilliant. My class and I really enjoy doing the activites." Comment recorded on the 9 October 'Starter of the Day' page by Mr Jones, Wales: "I think that having a starter of the day helps improve maths in general. My pupils say they love them!!!" Comment recorded on the 28 May 'Starter of the Day' page by L Smith, Colwyn Bay: "An absolutely brilliant resource. Only recently been discovered but is used daily with all my classes. It is particularly useful when things can be saved for further use. Thank you!" Comment recorded on the 24 May 'Starter of the Day' page by Ruth Seward, Hagley Park Sports College: "Find the starters wonderful; students enjoy them and often want to use the idea generated by the starter in other parts of the lesson. Keep up the good work" Comment recorded on the 3 October 'Starter of the Day' page by Mrs Johnstone, 7Je: "I think this is a brilliant website as all the students enjoy doing the puzzles and it is a brilliant way to start a lesson." Comment recorded on the 5 April 'Starter of the Day' page by Mr Stoner, St George's College of Technology: "This resource has made a great deal of difference to the standard of starters for all of our lessons. Thank you for being so creative and imaginative." Comment recorded on the i asp?ID_Top 'Starter of the Day' page by Ros, Belize: "A really awesome website! Teachers and students are learning in such a fun way! Keep it up..." Comment recorded on the 26 March 'Starter of the Day' page by Julie Reakes, The English College, Dubai: "It's great to have a starter that's timed and focuses the attention of everyone fully. I told them in advance I would do 10 then record their percentages." Comment recorded on the 2 May 'Starter of the Day' page by Angela Lowry, : "I think these are great! So useful and handy, the children love them. Comment recorded on the s /Indice 'Starter of the Day' page by Busolla, Australia: "Thank you very much for providing these resources for free for teachers and students. It has been engaging for the students - all trying to reach their highest level and competing with their peers while also learning. Thank you very much!" Comment recorded on the 19 October 'Starter of the Day' page by E Pollard, Huddersfield: "I used this with my bottom set in year 9. To engage them I used their name and favorite football team (or pop group) instead of the school name. For homework, I asked each student to find a definition for the key words they had been given (once they had fun trying to guess the answer) and they presented their findings to the rest of the class the following day. They felt really special because the key words came from their own personal information." Comment recorded on the 19 November 'Starter of the Day' page by Lesley Sewell, Ysgol Aberconwy, Wales: "A Maths colleague introduced me to your web site and I love to use it. The questions are so varied I can use them with all of my classes, I even let year 13 have a go at some of them. I like being able to access Starters for the whole month so I can use favourites with classes I see at different times of the week. Thanks." Comment recorded on the 1 May 'Starter of the Day' page by Phil Anthony, Head of Maths, Stourport High School: "What a brilliant website. We have just started to use the 'starter-of-the-day' in our yr9 lessons to try them out before we change from a high school to a secondary school in September. This is one of the best resources on-line we have found. The kids and staff love it. Well done an thank you very much for making my maths lessons more interesting and fun." Comment recorded on the 17 June 'Starter of the Day' page by Mr Hall, Light Hall School, Solihull: "Dear Transum, Comment recorded on the 11 January 'Starter of the Day' page by S Johnson, The King John School: "We recently had an afternoon on accelerated learning.This linked really well and prompted a discussion about learning styles and short term memory." Comment recorded on the 17 November 'Starter of the Day' page by Amy Thay, Coventry: "Thank you so much for your wonderful site. I have so much material to use in class and inspire me to try something a little different more often. I am going to show my maths department your website and encourage them to use it too. How lovely that you have compiled such a great resource to help teachers and pupils. Comment recorded on the 3 October 'Starter of the Day' page by S Mirza, Park High School, Colne: "Very good starters, help pupils settle very well in maths classroom." Comment recorded on the 7 December 'Starter of the Day' page by Cathryn Aldridge, Pells Primary: "I use Starter of the Day as a registration and warm-up activity for my Year 6 class. The range of questioning provided is excellent as are some of the images. |
Notes:Spotting patterns is an important skill in many areas of life. The world of numbers contains many fascinating patterns and understanding them enables better problem solving strategies. From seeing patterns in the multiples of numbers shaded in a hundred square to spotting the recurring sequences of digits in decimal numbers there is a great deal for pupils to be introduced to. Number Teacher Resources:Number Grids: Investigate the properties of number with these interactive number grids. Number Line: This number line visual aid is designed to be projected onto a whiteboard for whole class exposition. Formal Written Methods: Examples of formal written methods for addition, subtraction, multiplication and division. Counter: A dynamic visual aid that counts! Choose the first term, common difference and the speed Maths Mind Reader: Investigate this amazing mind reading performance based on simple mathematical principles. Number Rhyme: A poem about the numbers one to nine. Just for fun can you make up a last line for each verse? HCF and LCM Calculator: A demonstration of how to find the highest common factor (HCF or GCD) and the lowest common multiple (LCM) of two numbers. Divisibility Tests 2-12: A visual aid designed to be projected in the classroom. Here you can find the quick ways of telling whether a number is exactly divisible by the numbers two to twelve. Playing Card Maths: Imagine you are on a desert island with nothing but a pack of playing cards. Do you have to stop learning mathematics? Pesto: Students classify numbers randomly appearing on the screen by holding up cards Prison Cell Problem: A number patterns investigation involving prisoners and prison guards. Hot Number Challenges: These hot numbers challenges are for students with no access to a computer and are presented in a form suitable to be projected onto a whiteboard in front of a class. Pascal's Patterns: See the number patterns on Pascal's Triangle with this colourful, interactive application. Satisfaction: This is quite a challenging number grouping puzzle requiring a knowledge of prime, square and triangular numbers. Binary Lights: Represent binary numbers with a row of ligts which can be turned on or off. Sieve of Eratosthenes: A self checking, interactive version of the Sieve of Eratosthenes method of finding prime numbers. Number Activities:Counting Quest: Try this exercise to master counting up to 100 items with precision and confidence. Number Grids: Investigate the properties of number with these interactive number grids. Knowing Numbers: Read, write, order and compare numbers from one to twenty. Number Jigsaws: Online, interactive jigsaw puzzles of grids of numbers. Times Tables: A collection of activities to help you learn your times tables in only 5 days. Prime Numbers Jigsaw: Interactive jigsaw puzzles of different types of grids containing prime numbers. Factor Trees: Create factor trees to find the prime factors of the given numbers. Consecutive Numbers: Find the consective numbers that are added or multiplied to give the given totals Roman Numerals Jigsaw: An online interactive jigsaw puzzle of a grid of Roman numerals. Square Pairs Game: A game for two players who take turns to select two numbers that add up to a square number. Fizz Buzzer: The digital version of the popular fizz buzz game. Press the buzzers if they are factors of the counter. Product of Primes: Match the primes that multiply to give the products. A drag and drop activity. Equatero: Find the expression from a series of guesses and clues. Prime Labyrinth: Find the path to the centre of the labyrinth by moving along the prime numbers. Mix and Math: Determine the nature of adding, subtracting and multiplying numbers with specific properties. Missing Terms: Can you work out which numbers are missing from these number sequences? Pick The Primes: Pick the prime fruit from the tree as quickly as possible. Practise to improve your personal best time. Pairs 240: Find the pairs of numbers that multiply together to give a product of 240 in this collection of matching games. Twelve Days: How many gifts did my true love send to me according to the traditional Christmas song 'Twelve Days of Christmas'. HCF and LCM: Practise finding the highest common factor (H.C.F), sometimes called the greatest common divisor, and the lowest common multiple (L.C.M) of two numbers. Don't Shoot The Square: You will need to be quick on the draw to shoot all of the numbers except the square numbers. Leapfrog: An investigation of the minimum number of moves required to make the blue and green frogs swap places. Venn Diagram: Place each of the numbers 1 to 16 on the correct regions on the Venn diagram. Numbersina: Show that you know how many whatsits are in a thingamabob and other 'Numbers In A ...' answers. No Partner: Find which numbers in a given list do not combine with other numbers on the list to make a given sum. Dump-A-Dice Race: An online board game for two players involving prime and square numbers and making choices. Times Square: Practise your times tables with this self-checking multiplication grid. Dice Net Challenge: Drag the numbers onto the net so that when it is folded to form a cube numbers on opposite faces add up to prime numbers. Watsadoo: Rotate the cogs to catch the flying numbers in the correct sections. Number Skills Inventory: A checklist of basic numeracy techniques that every pupil should know. Divisibility Test: Practise using the quick ways to spot whether a number is divisible by the digits two to nine. Lemon Law: Change the numbers on the apples so that the number on the lemon is the given total. Prime Pairs Game: A game for two players who take turns to select two numbers that add up to a prime number. Plane Numbers: Arrange numbers on the plane shaped grid to produce the given totals River Crossing: The traditional River Crossing challenge. Can you do it in the smallest number of moves? Goal Products: Arrange the numbered footballs on the goal posts to make three, 3-number products that are all the same. Powten: Practise multiplying and dividing by powers of ten without using a calculator. Roman Numerals Quiz: This online, self marking quiz tests your ability to convert Roman numerals. Scallywags and Scoundrels: Arrange the scallywags and scoundrels on the chairs so that the numbers of any two sitting next to each other add up to a prime number. Three Prime Sum: A self-marking challenge to write each of the given numbers as the sum of three prime numbers. Pandigital Numbers: An exercise exploring the properties of nine-digit numbers containing each of the digits 1 to 9. Thai Numbers Jigsaw: An online interactive jigsaw puzzle of a grid of Thai number symbols. Suko Sujiko: Interactive number-based logic puzzles similar to those featuring in daily newspapers. Prison Cell Problem: A number patterns investigation involving prisoners and prison guards. Chinese Numbers Jigsaw: An online interactive jigsaw puzzle of a grid of Chinese number symbols. Product Square: Arrange the given numbers in a three by three grid to obtain the diagonal, row and column products. What Are They?: An online exercise about sums, products, differences, ratios, square and prime numbers. Hot Numbers: Move the numbered cards to form five 2-digit numbers which obey the given rules. A ten-stage numeracy challenge. Satisfy: Place the nine numbers in the table so they obey the row and column headings about the properties of the numbers. Square and Even: Arrange the numbers on the cards so that each of the three digit numbers formed horizontally are square numbers and each of the three digit numbers formed vertically are even. Magic Square Puzzle: Find all of the possible ways of making the magic total from the numbers in this four by four magic square. Three Ways: Find three different ways of multiplying four different digits together to get the given target number. There are nine levels for this online challenge. Prime Square: Drag the numbers into the red cells so that the sum of the three numbers in each row and each column is a prime number. Pascal's Triangle: Get to know this famous number pattern with some revealing learning activities Satisfaction: This is quite a challenging number grouping puzzle requiring a knowledge of prime, square and triangular numbers. Binary Numbers: Learn how to convert numbers from decimal to binary and from binary to decimal. Nim: Nim is a mathematical game of strategy in which two players take turns removing objects from groups of objects. Delightfully Divisible: Arrange the digits one to nine to make a number which is divisible in the way described. Sieve of Eratosthenes: A self checking, interactive version of the Sieve of Eratosthenes method of finding prime numbers. Number Systems: Place the numbers in the correct sets in this concentric circles Venn diagram. Some Sums and Products: Number puzzles involving adding, multiplying and problem solving Finally there is Topic Test, a set of 10 randomly chosen, multiple choice questions suggested by people from around the world. Number Investigations:Steps: Investigate the numbers associated with this growing sequence of steps made from Multilink cubes. Lamp Posts: What is the greatest number of lamp posts that would be needed for a strange village with only straight roads? Maths Mind Reader: Investigate this amazing mind reading performance based on simple mathematical principles. lnvestigating stamps: How many different postage amounts can you make with the given stamps? Leapfrog: An investigation of the minimum number of moves required to make the blue and green frogs swap places. Calendar Maths Investigation: Investigate the connection between the numbers in a T shape drawn on this month's calendar. Palindromic numbers: How many steps does each number take to become palandromic? Number Stairs: Find the relationships between the numbers in the stairs on different size grids. Trapezia: Which numbers can be represented by groups of circles arranged in the shape of a trapezium? Digit Sums and Multiples: Investigate numbers which are multiples of the sum of their digits. Snooker Investigation: Investigate a special snooker table with only four pockets. Which pocket will the ball fall into for various sized snooker tables? Handshakes: If everyone in this room shook hands with each other, how many handshakes would there be? Decimal Products: Find two decimal numbers that add up to exactly one. What is the product of these two decimals? Number Videos:Prime Numbers Video: A reminder of what prime numbers and composite numbers are. Finding Prime Factors: A straight forward explanation from SLEP Square Tricks: Math Tricks for Fast Calculation - How to square numbers between 10 and 20 without memorisation. BIDMAS Misconception: This Maths problem went viral on Twitter and there has been debate about the answer. Presh explains why one interpretation is considered correct. Binary and The Tower of Hanoi: A video showing how binary counting can solve the Tower of Hanoi puzzle. Problems with Zero: Dividing by zero, zero divided by zero and zero to the power of zero - all pose problems! Log Tables: Logarithms and log tables - what Transum used before calculators! A Numberphile video. Number Worksheets/Printables:Divided Age Worksheet: Additional questions for the 6th December Starter of the Day about the decimal part of quotients. Lemon Law Worksheet: Can you find the solution to these Lemon Law challenges using a spreadsheet? Divisibility Tests Worksheet: This worksheet contains a list of the divisibility tests along with a fill-in-the-table exercise. Siam Symbols Worksheet: Work out which digit each of the ten symbols represents from the six given calculations. Pesto Cards: Cards to print for pupils to use in the 'Can You Decide?' activity. Online Psychic Cards: With these cards you can read the minds of your pupils. It's Mathemagic at its best! Pascal's Triangle Worksheet: Various forms of Pascal's Triangle ready for printing. Number External Links:Links to other websites containing resources for Number are provided for those logged into 'Transum Mathematics'. Subscribing also opens up the opportunity for you to add your own links to this panel. You can sign up using one of the buttons below: SearchThe activity you are looking for may have been classified in a different way from the way you were expecting. You can search the whole of Transum Maths by using the box below.
OtherIs there anything you would have a regular use for that we don't feature here? Please let us know. SuggestedSome Sums and Products Number puzzles involving adding, multiplying and problem solving
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