math grade 1 problem solving

50 math problems for 1 st graders

Learn how to master word problems for 1st Graders with these 50 first grade math problems! Answers included.

Author Michelle Griczika

Published September 20, 2023

• Key takeaways
• Addition & Subtraction: Adding and subtracting numbers is like solving puzzles and helps us with real-life situations.
• Word Problems: Solving word problems makes math fun and helps us become better readers and problem solvers.
• Place Value: Understanding place value helps us read and write numbers correctly and compare their values.
• Measurement & Time: Measuring objects and telling time help us understand the world around us and manage our daily routines.
• Geometry (Shapes): Recognizing shapes and their properties helps us see the world in a new way and understand how things are put together.

Table of contents

• Addition and Subtraction
• Word Problems
• Place Value
• Geometry (Shapes)
• Measurement

Did you know that ladybugs have spots on their backs?

If a ladybug has 2 spots on its back and it gets 1 more spot, how many spots will the ladybug have in total?

The lady bug will have 3 spots!

First grade word problems are like fun puzzles that help us use our math skills to solve real-life situations. They are essential to 1st grade math, and practicing them can help us become math superstars!

• We can use addition and subtraction to determine how many cookies are left in a jar, how many friends are on the playground, or even how many apples are in a basket.
• Understanding place value is like being a number detective. It helps us read big numbers and know the value of each digit, so we can count things like how many toys we have or how many days until our birthday.
• Measuring things and telling time is like being an explorer in the real world. We can measure how tall we are, how long a pencil is, or even how much water is in a cup. And knowing how to read a clock helps us know when it’s time for fun activities or when it’s time to go to bed.
• Shapes are like building blocks of the world around us. We can learn about squares, triangles, and circles by looking at objects we see every day. It’s like being a shape detective, finding out how many sides and corners they have, and how they fit together like puzzle pieces in 1st grade math problems.

All of these skills are necessary for 1st grade math practice! Use these 50 math problems to go over important math skills needed to strengthen your first graders understanding and knowledge.

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• Kindergarten

Measurement and data

Sample questions, 50 math problems for first graders.

Today, we are going to practice different 1st grade skills. There are word problems for 1st graders measurement, addition and subtraction, and more! Once you are done, check your work with our answer sheet!

Section 1: Addition & Subtraction

Section 2: word problems, section 3: place value, practice 1st grade math with doodlemath.

Want more practice with 1st grade math? DoodleMath is an award-winning math app that’s proven to double a child’s rate of progression with just 10 minutes of use a day!

Filled with fun, interactive questions aligned to state standards, Doodle creates a unique work program tailored to each child’s needs, boosting their confidence and skills in math. Try it free  today!

Section 4: Geometry (Shapes)

Section 5: measurement.

How many inches long is the caterpillar?

How many inches long is the pencil?

How many inches long is the spoon?

How many inches tall is the building?

How many inches tall is the superhero?

Section 6: Time

What time is the clock below telling?

Parents, sign up for a DoodleMath subscription and see your child become a math wizard!

Lesson credits

Michelle Griczika

Michelle Griczika is a seasoned educator and experienced freelance writer. Her years teaching first and fifth grades coupled with her double certification in elementary and early childhood education lend depth to her understanding of diverse learning stages. Michelle enjoys running in her free time and undertaking home projects.

Time answer sheet

Measurement answer sheet

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Word Problems (Grade 1)

Grade 1 Word Problems(examples, solutions, songs, videos, games, activities) Examples, solution, videos, and songs to help Grade 1 students learn how to use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Related Pages More Lessons for Grade 1 Common Core for Grade 1

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Common Core: 1.OA.1 and 1.OA.2

Suggested Learning Target

• I can model addition and subtraction word problems using objects, drawings, and equations with unknown numbers in different positions.
• I can solve addition and subtraction word problems using objects, drawings, and equations.
• I can solve word problems with unknown numbers in different positions (e.g., 6 + ? = 8, ? + 2 = 8, 6 + 2 = ?).
• I can represent a problem in multiple ways including drawings and or objects/manipulatives (e.g., counters, unifix cubes, Digi-Blocks, number lines)
• I can take apart and combine numbers in a wide variety of ways
• I can make sense of quantity and be able to compare numbers
• I can use flexible thinking strategies to develop the understanding of the traditional algorithms and their processes
• I can solve a variety of addition and subtraction word problems
• I can use _ or ? to represent an unknown in an equation
• I can add three whole numbers whose sum is less than or equal to 20.
• I can solve word problems with three whole numbers using objects, drawings, and equations.
• I can add numbers in any order and be able to identify the most efficient way to solve the problem

Addition Examples:

Result Unknown Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now? 2 + 3 = ?

Change Unknown Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over to the first? 2 + ? = 5

Start Unknown Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before? ? + 3 = 5

Subtraction Examples:

Result Unknown Five apples were on the table. I ate two apples. How many apples are on the table now? 5 – 2 = ?

Change Unknown Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat? 5 – ? = 3

Start Unknown Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before? ? – 2 = 3

Word Problems. Addition with pictures (up to sum 20) Example: Lily had 5 apples and her aunt gave her 9 apples later. How many apples did Lily have finally?

Subtraction by counting back - word problems with pictures Example: There are 5 books on the desk. Betty took 2 away. How many books are left on the desk?

Word problems - subtraction with pictures - cross out - 1 digit Example: There are 9 cars in the shop and 8 of them are sold. How many cars are left?

Word problems - Subtraction with pictures - cross out (numbers to 20) Example: Mary had 11 toys and she gave 5 of them to Emma. How many toys did Mary have finally?

Word problems - adding multiple one-digit numbers Example: A furniture store sold 6 tables, 2 bookcases and 1 bed. How many pieces of furniture did the store sell in all?

Addition word problems that add up multiple values to sums less than or equal to ten. Example:

Laurem gave Troy two yellow pencils, three blue pencils and one green pencil. How many pencils did Lauren give him in all?

Ms. Ellis gave Lauren and Troy some cookies. Lauren ate two sugar and two chocolate chip cookies. Troy ate three oatmeal raisin and two sugar cookies. How many cookies did they eat in all?

Subtraction Strategies This video gives suggestions for solving subtraction story problems using a math mountain, equation, and circle drawing. Example: I have 8 peanuts. Then I eat 5 of them. How many peanuts are left?

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Word Problems Activities for 1st Grade

Strengthen your child's word problems skills with interactive educational resources for word problems for 1st graders online. These learning resources include fun games and worksheets with eye-catching visuals and characters. Get started to help your 1st grader master this concept by engaging their critical thinking.

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• Number Sequence (13)
• Counting (80)
• Counting Objects Within 5 (30)
• Counting Objects Within 10 (23)
• Counting Objects Within 20 (11)
• Compare Numbers (69)
• Compare Numbers Using Place Value (23)
• Compare 2-Digit Numbers (6)
• Order Numbers (2)
• Skip Counting (38)
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• Addition Properties (29)
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• Add Using A Number Line (19)
• Count On To Add (22)
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• Doubles And Near Doubles Addition Strategy (35)
• Make 10 Strategy (18)
• Add Three Whole Numbers (52)
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• Subtraction (448)
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• Subtract Using Multiples Of 10 (1)
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Addition Word Problems

Word Problems on Adding 3 Numbers Game

Learn to solve word problems on adding 3 numbers by playing this game.

Solve Count On Scenarios Game

Begin the exciting journey of becoming a math wizard by learning how to solve count on scenarios.

Select the Correct Addition Expression Worksheet

Reinforce math concepts by selecting the correct addition expression.

Find the Sum by Part-Part-Whole Model Worksheet

Assess your math skills by finding the sum by 'Part-Part-Whole' model in this worksheet.

Subtraction Word Problems

Subtraction Scenario Game

Take a look at subtraction scenarios with this game.

Take Away Scenario Game

Use your subtraction skills to solve 'Take Away' scenarios.

Select the Correct Subtraction Expression Worksheet

Solidify your math skills by selecting the correct subtraction expression.

Subtract Using Part-Part-Whole Model Worksheet

Combine math learning with adventure by solving to subtract using 'Part-Part-Whole' model.

All Word Problems Resources

Solve Word Problems on Adding Three Numbers Game

Learn to solve word problems on adding three numbers through this game.

Word Problems to Subtract Multiples of 10 Game

Help your child take flight by learning how to solve word problems to subtract multiples of 10.

Create and Solve Your Own Story Problem Worksheet

Help your child revise subtraction by creating and solving your own story problem.

Solve Word Problems on Take From Scenarios Game

Enjoy the marvel of math-multiverse by learning to solve word problems on "Take From" scenarios.

Complete the Model Using Clues Worksheet

Assess your math skills by completing the model using clues in this worksheet.

Write the Equation using Clues Worksheet

Be on your way to become a mathematician by practicing to write the equation using clues.

Solve Using Part-Part-Whole Model Worksheet

Learn subtraction at the speed of lightning by practicing to solve using 'Part-Part-Whole' model.

Represent 'Add To' Scenarios Game

Learn to solve math problems through our 'Represent 'Add To' Scenarios' game.

Complete the Part-Part-Whole Model Worksheet

Learners must complete the 'Part-Part-Whole' model to enhance their math skills.

Represent 'Put Together' Scenarios Game

Have your own math-themed party by learning how to represent 'Put Together' scenarios.

Word Problems on Subtraction Game

Add more arrows to your child’s math quiver by solving word problems on subtraction.

Part-Part-Whole Model Worksheet

In this worksheet, learners will get to practice the 'Part-Part-Whole' model.

Solve 'Count On' Word Problems Game

Take the first step towards building your math castle by solving 'Count On' word problems.

Word Problems on Take Away Game

Kids must solve word problems on 'Take Away' to play.

Answer Using Part-Part-Whole Model Worksheet

Pack your math practice time with fun by answering using 'Part-Part-Whole' model.

Write Subtraction Equation using Clues Worksheet

Assess your math skills by writing subtraction equations using clues.

Represent 'Add To' Word Problems Game

Take a deep dive into the world of math with our 'Represent 'Add To' Word Problems' game.

Solve Word Problems to Find the Difference Game

Kids must solve word problems to find the difference to practice subtraction.

Find the Sum Using Part-Part-Whole Model Worksheet

Combine math learning with adventure by solving to find the sum using 'Part-Part-Whole' model.

Solve Subtraction Word Problems Worksheet

Assess your math skills by solving subtraction word problems in this worksheet.

Represent 'Put Together' Word Problems Game

Unearth the wisdom of mathematics by learning how to represent 'Put Together' word problems.

Solve 'Change Unknown' Scenarios Game

Begin the exciting journey of becoming a math wizard by solving 'Change Unknown' scenarios.

Write Equations Using Part-Part-Whole Model Worksheet

Learn number sense at the speed of lightning by writing equations using 'Part-Part-Whole' model.

Identify Clues to Solve the Word Problems Worksheet

Reveal the secrets of math wizardry by identifying clues to solve word problems.

Compare and Solve to Find the Unknown Game

Have your own math-themed party by learning how to compare and solve to find the unknown.

Subtract to Compare Game

Let your child see the world through math-colored shades by subtracting to compare.

Solve Addition Word Problems Worksheet

Assess your math skills by solving addition word problems in this worksheet.

Story Problems on Comparison Worksheet

Print this worksheet to practice story problems on comparison like a math legend!

Addition Word Problems on Finding the Total Game

Learn to solve addition word problems on finding the total.

Represent Scenarios with More Game

Enter the madness of math-multiverse by exploring how to represent scenarios with 'More'.

Part-Part-Whole Model Problems Worksheet

Print this worksheet to practice 'Part-Part-Whole' model problems like a math legend!

Real-Life Scenarios as Subtraction Sentence Worksheet

In this worksheet, learners will get to practice real-life scenarios as subtraction sentences.

Addition Word Problems on Put-Together Scenarios Game

Use your skills to solve addition word problems on put-together scenarios.

Represent Scenarios with Fewer Game

Enjoy the marvel of mathematics by exploring how to represent scenarios with 'Fewer'.

Creating Addition Story Problems Worksheet

Help your child revise addition by creating addition story problems.

Solve Subtraction Problems Using Bar Model Worksheet

Combine math learning with adventure by solving subtraction problems using bar models.

Represent 'Add To' Situations Game

Dive deep into the world of addition with our 'Represent 'Add To' Situations' game.

Solve to Find How Many More Game

Learn to solve real world problems through our 'How Many More' game.

Add within 10: Summer Word Problems Worksheet

Fun summer-themed worksheet offering engaging word problems for adding numbers within 10.

Solving Story Problems on Subtraction Worksheet

Learn subtraction at the speed of lightning by solving story problems on subtraction.

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Check Out These 50 First-Grade Math Word Problems of the Day

Desmond saw 5 bunnies.

Opening your daily math lesson with a Math Word Problem of the Day is an excellent way to set the stage for learning. We all know that word problems are difficult for young learners to grasp, even when the mathematical operation portion of the problem is basic.

Incorporate these first grade math word problems one day at a time at the start of your math block to build confidence, critical thinking skills, and a learning community. Students will get used to reading slowly for meaning, while also identifying key information. Encourage students to write out equations and draw pictures to explain their thinking, since this helps them see the light when they are stuck!

Topics covered include addition, subtraction, multiplication, and comparison. All you need to do is post one of these first grade math word problems on your whiteboard or projector screen. Then let kids take it from there!

Want this entire set of word problems in one easy document? Get your free PowerPoint bundle by submitting your email here .

50 First Grade Math Word Problems

1. I had 6 pencils, and my teacher gave me 4 more. How many pencils do I have now?

2. Gina’s dog got 3 treats on Sunday and 0 treats on Monday. How many treats did Gina’s puppy get in all?

3. Joel went to the zoo with his family. During the first hour he was there he saw a bear, 2 tigers, and 3 lions. How many animals did Joel see in his first hour at the zoo?

4. Jackson sorted his toy cars by color. He has 6 blue cars, 5 green cars, and 4 black cars. How many cars does Jackson have in all?

5. Ben has 2 green balloons and 4 yellow balloons. How many balloons does he have altogether?

6. There are 3 kids in the Clark family. Tina is 3, Joshua is 4, and Samantha is 7. If you add up all their ages, what is the sum of the Clark kids?

7. If you go for a swim and 6 of your friends come along, how many friends are swimming in total?

8. Rachel’s mom had some flowers in a vase. 3 of the flowers wilted and Rachel’s mom took them out of the vase. Now there are 5 flowers in the vase. How many flowers were in the vase to start with?

9. Hayden’s cat had a litter of kittens. 3 kittens were gray, 2 kittens were spotted, and 7 kittens were black. How many kittens did Hayden’s cat have?

10. Pedro brought in 3 red leaves and 6 yellow leaves from the playground. How many leaves does he have in all?

11. Gabriella read 3 books on Monday, 6 books on Tuesday, and 4 books on Wednesday. How many books did Gabriella read in all?

12. If you have 3 cats, 2 guinea pigs, and a bunny. How many cute little noses do they have altogether?

13. If there are 3 inches of snow on the ground in the morning and we get 3 more inches of snow by dinner time. How many inches of snow did we get that day?

14. My cat has 4 paws and my brother’s dog has 4 paws. How many paws are there in all?

15. I had 10 pennies, but I lost 2 of them. How many pennies do I have now?

16. Santiago read 7 books over the summer. Ryan read 5 books. How many more books did Santiago read than Ryan?

17. Andrew put 10 stickers on his notebook. When he got to school he noticed some of the stickers had fallen off. Now Andrew only has 6 stickers on his notebook. How many stickers fell off Andrew’s notebook?

18. Nicole likes to help her mom pick tomatoes from their garden. She counted 9 tomatoes in the garden. 6 tomatoes were red and the rest were green. Nicole and her mom picked all the red tomatoes. How many green tomatoes did Nicole and her mom leave in the garden?

19. My sister and I have 20 pennies. If my sister has 10 pennies, how many pennies do I have?

20. The zoo had 8 tigers. 3 of the tigers moved to another zoo. How many tigers were left?

21. Esther read 3 poems. Magenna read some more poems. Altogether they read 7 poems. How many poems did Magenna read?

22. Haley’s dad bought 8 cheeseburgers. Haley ate 1 of them. How many cheeseburgers does Haley’s dad have left?

23. If you boil 7 eggs in water, and the number of eggs that float is one more than the number that sink, how many eggs float?

24. Rasheed loves to eat jellybeans. His favorite jellybeans are the yellow ones. There were 12 jellybeans in his bag. Rasheed removed all the yellow jellybeans and ate them, leaving 6 jellybeans in his bag. How many yellow jellybeans did Rasheed eat?

25. The gym teacher had 5 basketballs. The next week the gym teacher got some new basketballs. Now the gym teacher has 9 basketballs. How many new basketballs did the gym teacher get?

26. Jamal has 6 toy airplanes and his brother has 4 toy airplanes. How many more toy airplanes does Jamal have than his brother?

27. Antonio has some marbles. His brother Alex gives him 5 more. Now Antonio has 8 marbles. How many marbles did Antonio have to begin with?

28. If you have an 8-pack of crayons and you give your friend 3 of them to use during drawing time. How many crayons do you have in your pack now?

29. Emily has 4 pink erasers and some white erasers. She has 7 erasers in all. How many white erasers does Emily have?

30. Angel serves pizza at her birthday party. The pizza has 12 slices. 8 slices of pizza are eaten by Angel and her guests. How many slices of pizza are left?

31. If you have 9 toys on the floor and your little brother has 6 toys on the floor. How many more toys on the floor do you have?

32. There are 8 windows in the classroom. Some of the windows have decorations on them, 2 of them don’t have any directions. How many windows have decorations?

33. On Saturday, you brought home some fish from the pet store. If 15 of your 18 fish have stripes. How many of your fish are without strips?

34. 8 birds flew to the top of a fence. Some birds flew away and 6 birds stayed. How many birds flew away?

35. There were 6 books on Noah’s shelf. Olivia took some of the books. Now there are 2 books on the shelf. How many books did Olivia take?

36. Ethan has some folders in his backpack and 4 folders in his desk. He has 8 folders altogether. How many folders are in his backpack?

37. Liam has 8 t-shirts. 5 of them have superheroes on them, and the rest are solid colors. How many of Liam’s t-shirts are solid colors?

38. Mary was putting together a 20 piece puzzle. After she finished, she discovered that only 18 pieces were there. How many pieces were missing?

39. Nicholas has 7 cousins. Some of his cousins are girls and 3 of his cousins are boys. How many girl cousins does Nicholas have?

40. It snowed for 6 hours on Monday and 4 hours on Tuesday. How many more hours did it snow on Monday?

41. Charlie’s mom baked 12 chocolate chip cookies for dessert. Charlie ate 2 cookies and his mom and 1 cookie. How many chocolate cookies were left?

42. Melanie has 16 purple pens. Dante has 10 blue pens. Melanie has ____ more pens than Dante has.

43. Sofia has 75 pennies in her bank. How many more pennies will she need to have 100 pennies in her bank?

44. There were 9 cups of soda on the table. Some of the cups were knocked over, and 6 were still standing. How many cups of soda were knocked over?

45. Griffin has 20 board games. Some were under his bed, and 15 were in his closet. How many board games were under Griffin’s bed?

46. Antonio spotted 3 deer sitting at the top of a hill, but all he could see were their eyes. How many eyes did Antonio see in all?

47. Desmond saw 5 bunnies. He counted all of their ears. How many bunny ears did Desmond count?

48. Katie counted all of her toes, and then she counted all her mom’s toes. How many toes did Katie count altogether?

49. Which weighs more? A 15-pound red fox or a 24-pound wild turkey?

50. Which weighs more: a 150-pound white-tailed deer or a 110-pound kangaroo?

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Grade 1 Word Problems

Grade 1 word problems to help your students apply their understanding in maths. Word problems are the perfect test for children. It’s one thing being able to solve equations, but it’s a whole new level when you can apply your understanding in real world situations. Use these addition and subtraction word problem worksheets to help your students practice. They even get to draw math pictures!  Simply click on the images below to download. To keep up to date with quality resources, follow my TpT store!

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1st Grade Addition Word Problems Up to 20

Welcome to our 1st Grade Addition Word Problems Worksheets. Here you will find a wide range of free printable addition Worksheets, which will help your child practice solving a range of addition problems using numbers with a sum of up to 20.

Want to test yourself to see how well you have understood this skill?.

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1st Grade Addition Word Problems

Addition problems with a sum of up to 20.

Each sheet consists of adding two or three numbers with a total of up to 10, 15 or 20.

There is a space on each sheet for working out, using whichever method you wish your child to use.

There are also UK versions of some of the worksheets which use pounds (£) instead of dollars($).

Using these sheets will help your child to:

• recognise addition word problems;
• add with numbers up to 10, 15 or 20.

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• Addition Word Problems up to 10
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1st Grade Addition Word Problems up to 10

• Addition Word Problems to 10 Sheet 1
• PDF version
• Addition Word Problems to 10 Sheet 2

This sheets has word problems with 3 addends.

• Addition Word Problems to 10 Sheet 3

1st Grade Addition Word Problems up to 15

• Addition Word Problems to 15 Sheet 1
• Addition Word Problems to 15 Sheet 2
• Addition Word Problems to 15 Sheet 3

First Grade Addition Word Problems up to 20

• Addition Word Problems to 20 Sheet 1
• UK Version Sheet 1
• UK PDF version
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• UK Version Sheet 2
• Addition Word Problems to 20 Sheet 3
• UK Version Sheet 3

Looking for some easier worksheets?

Take a look at our Addition sentences to 12.

On this page, your child will learn to work out basic addition sums up to 12 by counting objects.

• Addition Sentences to 12

Looking for some harder worksheets?

Take a look at our 1st Grade Addition facts Worksheets page with numbers up to 12+12.

On this page, your child will learn to work out basic addition sums to 12+12.

• Addition Facts to 20 Worksheets

These sheets involve solving a range of addition word problems within 100.

• Addition Word Problems 2nd grade

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

First Grade Place Value & Counting Worksheets

Here you will find a range of Free Printable First Grade Place Value games.

Using these sheets will help your child learn to:

• understand place value to 100.
• Math Place Value Worksheets Tens and Ones
• Basic Math Worksheets - Ordering 2-digit numbers
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More First Grade Addition Worksheets

Here you will find some more of our 1st Grade Addition Worksheets.

• Number Bonds to 10 and 12
• Addition Fact Practice to 12
• Adding tens
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1st Grade Math Word Problems

Here you will find a range of math word problems aimed at first grade level. Each problem sheet is based on an interesting theme such as parties or the seaside.

Using these first grade math worksheets will help your child to:

• Add and subtract with numbers to 12;
• order numbers to 100;
• solve a range of math problems.

All the math problem sheets in this section support Elementary math benchmarks.

• Math Problems for Children 1st Grade
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Addition Word Problems to 20 Online Quiz

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August 16, 2024

The Paradox of 1 – 1 + 1 – 1 + 1 – 1 + …

Why a mathematician thought this infinite series explained how God created the universe

By Jack Murtagh

How do we resolve a centuries-old paradox? The answer tells us as much about mathematicians as about mathematics.

Ralf Hiemisch/Getty Images

Here’s a math problem that everybody can solve: What is 1 − 1? 0. So far so good. If we then add a 1, the sum grows, but if we subtract yet another 1, we’re back at 0. Let’s say, we keep doing this forever:

1 – 1 + 1 – 1 + 1 – 1 + ...

What is the resulting sum? The question seems simple, silly even, but it puzzled some of the greatest mathematicians of the 18th century. Paradoxes surround the problem because multiple seemingly sound arguments about the sum reach radically different conclusions. The first person to deeply investigate it thought it explained how God created the universe. Its resolution in modern terms illustrates that mathematics is a more human enterprise than sometimes appreciated.

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Take a guess at what you think the infinite sum equals. I’ll give you multiple choices:

A. 0 B. 1 C. ½ D. It does not equal anything

The argument for 0 comes naturally if we include suggestive parentheses:

(1 – 1) + (1 – 1) + (1 – 1) + ...

Recall that in mathematics the order of operations dictates that we evaluate those inside parentheses before evaluating those outside. Each (1 − 1) cancels to 0, so the above works out to 0 + 0 + 0 +..., which clearly amounts to nothing.

Yet a slight shift of the brackets yields a different result. If we set aside the first 1, then the second and third terms also cancel, and the fourth and fifth cancel:

1 + (–1 + 1) + (–1 + 1) + (–1 + 1) + …

Again, all the parentheticals add up to 0, but we have this extra positive 1 at the beginning, which suggests that the whole expression sums to 1.

Italian monk and mathematician Luigi Guido Grandi first investigated the series (the sum of infinitely many numbers) in 1703. Grandi, whom this particular series is now named after, observed that by merely shifting around parentheses he could make the series sum to 0 or 1. According to math historian Giorgio Bagni, this arithmetic inconsistency held theological significance for Grandi, who believed it showed that creation out of nothing was “perfectly plausible.”

The series summing to both 0 and 1 seems paradoxical, but surely option C (½) is no less troubling. How could a sum of infinitely many integers ever yield a fraction? Yet ultimately Grandi and many prominent 18th-­century mathematicians after him thought the answer was ½. Grandi argued for this with a parable: Imagine that two brothers inherit a single gem from their father, and each keeps it in their own museum on alternating years. If this tradition of passing the gem back and forth carried on with their descendants, then the two families would each have ½ ownership over the gem.

As proofs go, I wouldn’t recommend putting the gem story on your next math test. German mathematician Gottfried Wilhelm Leibniz agreed with Grandi’s conclusion, but he tried to support it with probabilistic reasoning. Leibniz argued that if you stop summing the series at a random point, then your sum up to that point will be either 0 or 1 with equal probability, so it makes sense to average them to ½. He thought the result was correct but acknowledged that his argument was more “metaphysical than mathematical.”

Swiss mathematician Leonhard Euler employed more complicated methods to argue for ½ and addressed those who disagreed in a rather defensive paragraph in his 1760 paper De seriebus divergentibus (translation: “On Divergent Series”). Euler asserted that “no doubt can remain that in fact the series 1 − 1 + 1 − 1 + 1 − 1 + etc. and the fraction ½ are equivalent quantities and that it is always permitted to substitute one for the other without error.” So a lot of smart people were strongly in favor of option C.

Infinite series like this one have flummoxed thinkers dating back at least to the ancient Greeks with Zeno of Elea’s paradoxes of motion. In a well-known example, Zeno observed that to walk a path, one must first traverse half of it, then must traverse half of the remaining distance (¼ of the total path), and then half of the remaining distance (⅛), and so on. One can keep subdividing forever, which suggests that every time we walk a path we complete an infinite number of actions in a finite amount of time—a paradox.

While philosophers still debate the metaphysics of Zeno’s paradoxes some 2,400 years later, mathematicians did make a substantial leap toward resolving them and the mystery of Grandi’s series in the late 19th century. From the foundations of calculus emerged clarifying definitions about when infinite series sum to finite values. Finding the answer begins with looking at partial sums—add the first two terms, then the first three, then the first four, and so on. If these intermediate sums continue to get closer and closer to a fixed value, then we say the series “converges” to that value. Let’s apply this to the series in Zeno’s paradox, which sums half of a path plus a quarter of a path plus an eighth of a path, and so on.

1 / 2 + 1 / 4 + 1 / 8 + 1 / 16

The first two terms sum to 0.75, the first three terms sum to 0.875, and the first four sum to 0.9375. If we summed the first 10 terms, we’d get 0.9990234375. The partial sums get closer and closer to 1, so the series converges to 1. Although we can conceive of a path as an infinite number of distances, calculus confirms that it still ultimately amounts to one path.

The partial sums of Grandi’s series oscillate between 0 and 1 without ever homing in on a single value. So modern mathematicians would choose option D (Grandi’s series does not sum to anything).

The resolution of Grandi’s series raises a sociological question. Why does the mathematical community accept the partial-sum approach but not Leibniz’s probabilistic argument or some other prescription for summing an infinite series? Although they may look alike and smell alike, summing an infinite series is not the same as addition. Addition does not change when you shift ­parentheses around—for example, 1 + (2 + 3) = (1 + 2) + 3—but many series, including Grandi’s, do. For convenience, mathematicians borrow words like “summing” and “equals” from addition to discuss series, but under the hood what they really mean when they say Zeno’s series “sums to 1” or “equals 1” is that the partial sums converge to 1, no more and no less.

The partial-sum definition of convergence isn’t arbitrary. The math community prefers it to alternatives for good reasons. It alleviates a lot of the paradoxes that beset earlier mathematicians who studied infinite sums, and it preserves many of the nice properties that finite addition enjoys. But other definitions of convergence are useful as well. For instance, rather than asking what number the partial sums approach, the Cesàro summation method takes the average of the first two partial sums, then the first three partial sums, and then the first four partial sums, and so on ad infinitum, and asks what those averages approach. If you apply this tweaked method to a convergent series like Zeno’s, it will always give you the same answer. But it sometimes will give a different answer when applied to series that do not converge under the standard definition. In particular, Grandi’s series has a Cesàro sum of ½.

Many other summation methods appear in the mathematical literature. In reality, we can’t physically add an infinite number of things, so summation methods simply provide principled ways of assigning values to infinite series. The partial-sum definition holds deserved status as the default, but it occasionally helps to have other options.

Curiously, Grandi’s series sums to ½ under most alternative methods. So a colloquial answer to our opening question might be: Grandi’s series does not sum to anything, but if it did it would sum to ½.