Exploring the Mathematical Mysteries of 2026
On Weekend Edition yesterday, Puzzlemaster Will Shortz offered the following challenge:
This week’s challenge is a numerical one from Ed Pegg Jr., who runs the website mathpuzzle.com. Take the nine digits — 1, 2, 3, 4, 5, 6, 7, 8, 9. You can group some of them and add arithmetic operations to get 2011 like this: 1 + 23 ÷ 4 × 5 × 67 – 8 + 9. If you do these operations in order from left to right, you get 2011. Well, 2011 was 15 years ago. Can you group some of the digits and add arithmetic symbols in a different way to make 2026? The digits from 1 to 9 need to stay in that order.
He went on to say that he was aware of two different solutions. If you think you know one, you can submit your answer. (Editor’s note: I think there are more than two.)
It seems auspicious that the first NPR Sunday Puzzle challenge of the year is numerical. But there are other reasons that 2026 promises to be a great year…
In mathematics, 2026 is known as a happy number, because its digital sum is 2 + 0 + 2 + 6 = 10 → 1 + 0 = 1. In numerology, 2026 is known as Universal Year 1 — for the exact same reason! This implies that 2026 will offer a fresh start with new beginnings, leadership, and individuality.
The year 2026 is said to be semiprime, because it’s the product of exactly two prime numbers: 2 × 1013; less exciting, it can be expressed as the sum of two prime numbers in many different ways, of which my favorite is 23 + 2003 (the second addend is the first addend two 0s in the middle).
Below, you’ll find 15 more reasons to enjoy 2026.
- For what positive integers a and b is a2 + b2 = 2026?
- A “multiplicative date” is one in which the product of the month and date is equal to the (two-digit) year. How many multiplicative dates will there be in 2026?
- Write 2026 as a sum of consecutive positive integers. How many ways can you do it?
- A rectangular garden has integer side lengths and an area of 2026 square meters. What is the difference between the least and greatest possible perimeters?
- What is the sum of 1 + 2 + 3 + 4 + ··· + 2026?
- How long would it take you to count to 2026?
- Using only common mathematical symbols and operations and the digits 2, 0, 2, and 6, make an expression that is exactly equal to 100. (Bonus: make an expression using the four digits in order.)
- All possible four-digit numbers that can be made with the digits 2, 0, 2, and 6 are formed and arranged in ascending order. What is the median of those numbers?
- What fraction is equivalent to 0.2026?
- Create a 4 × 4 magic square in which the sum of each row, column, and diagonal is 2026. (Bonus points if you can do it with consecutive integers.)
- What is the units digit of 20262026?
- Try this: cube each digit of 2026, then add them; then cube each digit of the result, and add them; continue, ad infinitum. What eventually happens?
- For what positive integers a and b is ab – ab = 2026?
- Each dimension of a rectangular box is an integer number of inches. The volume of the box is 2026 in3. What is the minimum possible surface area of the box?
- What is the maximum possible product for a set of positive integers that have a sum of 2026?
Happy New Year! Good luck!
Problems Involving 6-7
Just in case you’re unaware of the 6-7 meme — how’s life under that rock, anyway? — here’s some background for you…
Some of you may be aware that I wrote a book for NCTM called One Hundred Problems Involving the Number 100. While those problems are great for the 100th day of school, it seems like a gap that there isn’t a set of problems celebrating middle schoolers’ current favorite meme. As I was contemplating remedies for this situation, my friend Bill Zahner, a professor at San Diego State University, sent me an email that said:
A couple of teachers [at my wife’s school] are doing 67-related activities in their class — like the 100 day activities, but a little more surreal. That inspired me to… generate an AMC-8 style math test where each question involves 67. I’m not sure who would enjoy this, but since you are the author of a book full of math jokes, I figure you are more likely than others to find this funny.
Bill’s correct; I do enjoy these kinds of things. And I know my audience, so I suspect you will, too.
In full disclosure, Bill didn’t write all the questions; he got a little help from AI. Still, the idea is pretty cool, and with Bill’s permission, I share with you…
My meager contribution to this effort are three 67 problems, two of which I had written prior to receiving Bill’s email, the third written after receiving inspiration from one of Bill’s problems.
- Using only the numbers 6 and 7, and combining them with arithmetic operations, create an expression equal to 67. One trivial example would be 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 7. Can you find a more elegant solution? What is the fewest 6s and 7s you could use to make such an equation?
- Between what two integers is the square root of 42?
- A sequence is recursively defined by f(x) = f(x – 1) + 2x – 69. If f(1) = 67, what is f(67)?
Of course, if you just want to blow kids’ minds, you could simply ask the following:
- What are the next two numbers in the sequence 1, 2, 3, 4, 5, __, __?
If you want to understand how cool 67 really is, check out this tutorial from the Wrath of Math.
Math Kennections
Since 2012, current Jeopardy! host and 74‑time Jeopardy! champion Ken Jennings has been writing Kennections, puzzles where the answers to five trivia questions share a common theme. Your job as puzzle solver? Answer the questions, and determine the theme! Sound simple? It is… to understand! But not necessarily to solve. You can try some of Ken’s puzzles at Mental Floss.
Last weekend, Ken Jennings appeared on the NPR Sunday Puzzle with Will Shortz to promote his new book, The Complete Kennections: 5,000 Questions in 1,000 Puzzles. (And I think we all know that I’m a sucker for books with numbers in the title.) Will created two original Kennections puzzles, and Ken played them on air; you can hear how he fared at Sunday Puzzle: Kennections.
But I get it. You don’t want to click away from this amazing post before you finish reading to the end. And who could blame you? So I created three Kennections puzzles that you can try below. They roughly proceed in order of difficulty. Your only hint is that they all have some type of association with mathematics. And I’ll provide you with the same instructions that Ken gives:
All five answers to the questions below have something in common. Can you figure it out?
Math Kennection 1
- Types of fears resulting from a lack of logic, reason, or sound basis in reality.
- In psychology, it can be Oedipus, God, or Napoleon.
- This is equal to the sum of its parts.
- Middle Earth is one example of this type of world.
- Word that follows Optimus, Coach, or Amazon.
Can you figure out what the five answers from Math Kennection 1 have in common?
Math Kennection 2
- Everyone knows that Superman’s cape is red. What color is Thor’s cape?
- Officially, it’s known as the MBTA, but what letter do Bostonians use to refer to the train system?
- What informal term is used to describe a table that defines a multiplication operation for an algebraic system?
- In biblical times, what “officer of arms” served as a messenger who delivered vital information to the subjects of a kingdom?
- What battle between the British Navy and the combined French and Spanish fleets, which was named after the cape near where it was fought, happened in October 1805?
Can you figure out what the five answers from Math Kennection 2 have in common?
Math Kennection 3
- According to the theme song for Cheers, “you wanna go where everybody knows your” what?
- What does a magician say after completing a trick?
- What adjective is used to describe a ship’s first voyage?
- What word could follow King, Astro, Silver, or Super?
- In the movie Inside Out, what fiery emotion was voiced by comedian Lewis Black?
Can you figure out what the five answers from Math Kennection 3 have in common?
Good luck! And if you’re feeling creative, I’d love to see an original Kennections puzzle that you create show up in the comments!
Math in Songs
An idea for a blog post about the math in song lyrics has been germinating at MJ4MF for a while, but with the passing of the mathematically musical — or should that be the musically mathematical? — Tom Lehrer earlier this week, now seems like an excellent time to bring this post to life. Given his background, Lehrer always correctly represented the math in his songs — even his description of subtraction in New Math is spot on, albeit a little hard to follow auditorily.
But for every correct mathematical statement in a song, there are dozens of errors that you can find elsewhere.
The Beatles sang about “Eight Days a Week.”
George Strait claimed that “4 – 3 = 0,” and Radiohead said that “2 + 2 = 5.”
And at the 16:55 mark of “Alice’s Restaurant Masscree,” Arlo Guthrie claims that he’s “been singing this song now for 25 minutes.”
Perhaps the most egregious math error in a song, though, occurs in Pi by Kate Bush, wherein she sings the digits of the world’s most famous transcendental number — yet, amazingly, she skips over 23 digits after getting the first 80 correct. Shown below are the first 115 digits of π; those in bold were omitted by Bush.
3.1415926535 8979323846 2643383279 502884197 1693993751 0582097494
4592307816 406286208 998628034 8253421170 6798214808 65132
Though Bush committed a mathematical error by omitting digits, she committed a far more egregious social error by writing and singing this song in the first place — ‘twould be generous to describe it as unlistenable.
One of the songs on our family’s road trip playlist is “The Other Side” from the movie The Greatest Showman. The song features P. T. Barnum (Hugh Jackman) offering Phillip Carlyle (Zac Efron) an opportunity to join the circus. Halfway through the song, they negotiate the details…
Phillip Carlyle:
Well, it’s intriguing, but to go would cost me greatly.
So, what percentage of the show would I be taking?
P. T. Barnum:
Fair enough. You’d want a piece of all the action.
I’ll give you 7 [percent]; we can shake and make it happen.
Phillip Carlyle:
I wasn’t born this morning; 18 will be just fine.
P. T. Barnum:
Why not just go ahead and ask for nickels on the dime?
Phillip Carlyle:
15.
P. T. Barnum:
I’ll do 8.
Phillip Carlyle:
12.
P. T. Barnum:
9.
Phillip Carlyle:
10!
Though never formally stated, the implication is that they settle on 10%. For some reason, this negotiation always bothered me, and I finally realized why.
Barnum’s numbers progress arithmetically by consecutive positive integers: 7, 8, 9, 10. As it should be.
But Carlyle’s numbers descend thusly: 18, 15, 12, 10. No, no, no, no, no, no, NO! Back-to-back decreases of three, then suddenly a switch to a decrease of two? Carlyle, what are you thinking???
Luckily, “Remember the Name” by Fort Minor is also on our family playlist, with the following chorus:
This is ten percent luck, twenty percent skill,
Fifteen percent concentrated power of will.
Five percent pleasure, fifty percent pain;
And a hundred percent reason to remember the name.
It adds up: 10 + 20 + 15 + 5 + 50 = 100.
Proper.
Math Jokes from Comedians
It’s hard to believe that it’s been 13 years since I originally shared the Math Humor of Mike Vecchione. But since so many great things happen on 13‑year cycles — like the 13‑year cicada, the orbital period of the P/2010 T2 comet, and… well, that’s about it, I guess — it seems fitting that I should once again share this bit of mathematical brilliance:
I got an F in algebra. I took it home. My mom was furious. She’s like, “You failed algebra?” I’m like, “That’s not what that means. That’s a F. It’s a variable, an unknown. We don’t know what F means. It could mean I failed, but it probably just means I’m fantastic.”
Mike Vecchione isn’t the only comedian to poke fun at math. Even Euclid, in one of the first Netflix comedy specials that’s now lost to posterity, expounded, “Without geometry, life would be pointless!”
But many comedians have tapped the fertile ground of mathematics for comedic purposes for many years. Here are a few of my favorites…
George Carlin
There are 400,000 words in the English language, and there are seven of them you can’t say on television. What a ratio that is: 399,993 to 7. They must re-e-e-eally be ba-a-ad…
Maria Bamford
I really think before giving me a credit card, they should’ve given me a math test. Like, a series of story problems.
Question #1: If Maria works as a comedian for $100 a week but spends $20 a day on hair scrunchies, how many years will it take her to pay off a Taco Bell gordita she bought in 1992?
Question #2: If Maria’s boyfriend is in a folk band, and he only smokes pot every other day, what percentage of the rent will he be able to contribute? I thought 50%, but the answer is 0.
Question #3: If Maria moved to Hollywood at the age of 22 in order to become a TV superstar, and by the age of 35 has only succeeded in getting her cable television turned off, what fraction of her dream will have died?
Stewart Francis
I was horrible in school. I failed math so many times, I can’t even count.
Sammy Obeid
I know a lot of people hate math, but half of us are good at it. You know, half the world is technically above the 50th percentile, right? And if you don’t know what that means, you’re below it. Sorry, that’s a mean joke. It’s about averages.
Katie Hughes
[After a long day at work, Barbie] gets home to that big pink house, heads straight for the liquor cabinet, and pours herself four fingers of vodka. And who can blame her? It’s her only unit of measurement.
Do you have a favorite math bit from a comedian? Post the joke or a link in the comments!
Problems for the 100th Day of School, Part II
In early 2021, I posted 100 problems for the 100th day of school in which every problem has an answer of 100:
Problems with 100 as the Anwer
I’m reposting the link again as we approach the 100th day of school in the 2024‑25 school year. Of course, which day will be the 100th in your school depends on when you started, the holidays you celebrate, and the lengths of your fall and winter breaks. Students at Chula Vista Elementary School District (CA) started school about as early as anyone in the country; their first day was Wednesday, July 24. Consequently, they’ll reach their 100th day on Friday, January 24. Students in Portland Public Schools (OR), on the other hand, didn’t start school until Tuesday, August 27; their 100th day will occur on Thursday, February 13.
Regardless of when you reach your 100th day, I hope you’ll enjoy the problems in that collection, as well as some extra problems from my book, One Hundred Problems Involving the Number 100 (NCTM, Amazon), and a few new ones, below.
- Use the digits 2, 0, 2, and 5 to make an expression equal to 100. Better yet, use the digits in order!
- Draw a polygon whose perimeter is 100 units and whose area is 100 square units.
- Would you rather have 100 lb of dimes or 100 lb of quarters?
- Let A = 1, B = 2, C = 3, …, Z = 26. The letter product of a word is the product of the values of all of its letters. For instance, the letter product of CAT is C × A × T = 3 × 1 × 20 = 60. How many common English words can you find that have a letter product of 100?
- Write every possible fraction from 0 to 1, in order, with every possible denominator less than or equal to 100. What is the 100th fraction in the list?
- For the complete grid in Figure I, if movement is restricted to only north or east, there are 12 possible paths from A to B. For the incomplete grid in Figure II — in which one segment has been removed — there are only 6 possible paths from C to D. Construct an incomplete grid in which there are exactly 100 possible paths from the lower-left to upper-right vertices.
Math Problems for 2025
The calendar is about to change again, and another year of possibilities lies before us. And for you, the possibility of solving some problems involving the number 2025. Before you dive into those 24 chestnuts, though, how about a piece of trivia?
If you separate 2025 into two parts and add them, you get 20 + 25 = 45, and 452 = 2025. Cool, huh? What’s more, without significant advancements in medical science, this is the only time that any of us will see such an occurrence in our lifetimes. (See the first problem below for more details!)
Happy New Year, and enjoy!
- If you separate 2025 into two two-digit numbers and add them, you get 20 + 25 = 45, and 452 = 2025. What is the sum of all other four-digit years ABCD with the property that (AB + CD)2 = ABCD?
- If n2 = 2025, what is the value of n?
- For what positive integers a and b is a2 + b2 = 2025?
- Madhar and Mikenna love M&Ms. They bought 2025 M&Ms to share with friends at their party on Saturday. But Madhar couldn’t control himself, and he ate 1/5 of the M&Ms on Tuesday. On Wednesday, Mikenna ate 1/4 of what was left. On Thursday, Madhar ate 1/3 of what was left. On Friday, Mikenna was back at it and ate 1/2 of what was left. How many M&Ms were left for the party on Saturday? (Bonus: During the week, who ate more M&Ms, Madhar or Mikenna?)
- A number n is said to be amenable if there exists a multiset of n integers such that the sum of those numbers is equal to the product of those numbers. For instance, the number 9 is amenable, because the multiset {-1, -1, 1, 1, 1, 1, 1, 3, 3} contains 9 elements such that the sum of the elements and the product of the elements are equal. Find (or describe) a set of 2025 integers that have the same sum and product.
- A “multiplicative date” is one in which the product of the month and date is equal to the (two-digit) year. How many multiplicative dates will there be in 2025?
- The sum of the digits of 2025 is 2 + 0 + 2 + 5 = 9, and 9 is a factor of 2025. (In fact, we know that 9 is a factor of 2025 because the sum of the digits is 9, thanks to divisibility rules.) For how many numbers in the 2020s is the sum of the digits a factor of the number?
- A rectangular garden has integer side lengths and an area of 2025 square meters. What is the least possible perimeter? What is the greatest possible perimeter?
- What is the sum of 1 + 2 + 3 + 4 + ··· + 2025?
- What is the sum of 15 + 30 + 45 + ··· + 2025?
- For what value of n does 1 + 3 + 5 + ··· + (2n + 1) = 2025?
- For what value of n does 13 + 23 + 33 + ··· + n3 = 2025?
- How long would it take you to count to 2025?
- Using only common mathematical symbols and operations and the digits 2, 0, 2, and 5, make an expression that is exactly equal to 100. (Bonus: make an expression using the four digits in order.)
- All possible four-digit numbers that can be made with the digits 2, 0, 2, and 5 are formed and arranged in ascending order. What is the median of those numbers?
- What fraction is equivalent to 0.2025?
- How many positive integer factors does 2025 have?
- Create a 3 × 3 magic square in which the sum of each row, column, and diagonal is 2025.
- Create a 5 × 5 magic square in which the sum of each row, column, and diagonal is 2025.
- Why can’t you create a 4 × 4 magic square in which the sum of each row, column, and diagonal is 2025?
- What is the units digit of 20252025?
- Each dimension of a rectangular box is an integer number of inches. The volume of the box is 2025 in3. What is the minimum possible surface area of the box?
- What is the maximum possible product for a set of positive integers that have a sum of 2025?
Multiplicative Dates
If you read the previous post containing Math Problems for 2024, then you already know what a multiplicative date is. But in case you missed it, here’s the definition again:
A multiplicative date is one in which the product of the month and date is equal to the (two-digit) year. For instance, today — January 24, 2024 — is a multiplicative date, because 1 × 24 = 24.
The question I asked in that previous post was, how many multiplicative dates will there be in 2024?
But I’m revisiting this topic again today, because today is the first multiplicative date of a year that has more multiplicative dates than any other year this century. That’s right, no year 20XX has more multiplicative dates than 2024.
Of course, there’s no reason to stop there. There are quite a few interesting questions related to multiplicative dates:
- How many years this century have no multiplicative dates?
- Which two consecutive years have the most multiplicative dates combined?
Math Problems for 2024
The calendar is about to change again, and another year of possibilities lies before us. While there are many things that you might do this year, I beg you, please, do not find the sum of all odd positive divisors of 2024. It’s two gross!
There are many interesting problems that involve the number 2024. Enjoy!
- A “multiplicative date” is one in which the product of the month and date is equal to the (two-digit) year. For instance, August 3 is a multiplicative date in 2024, because 8 × 3 = 24. How many multiplicative dates will there be in 2024?
- Create a polygon whose perimeter is 2024 units and whose area is 2024 square units.
- The sum of the digits of 2024 is 2 + 0 + 2 + 4 = 8, and 8 is a factor of 2024. For how many numbers in the 2020s is the sum of the digits a factor of the number?
- The number 2024 is an iban number, because the English name for the number “two-thousand twenty-four” never contains the letter i. (It has no relationship to the banking term IBAN, which stands for “International Bank Account Number.” It also only applies to the standard English names for numbers, not special names like googol or Kaprekar’s constant.) What is the largest iban number?
- What is the sum of 1 + 2 + 3 + 4 + ··· + 2024?
- What is the sum of 2 + 4 + 6 + 8 + ··· + 2024?
- What is the sum of 13 + 26 + 39 + ··· + 2024?
- How long would it take you to count to 2024?
- Using only common mathematical symbols and operations and the digits 2, 0, 2, and 4, make an expression that is exactly equal to 100. (Bonus: make an expression using the four digits in order.)
- All possible four-digit numbers that can be made with the digits 2, 0, 2, and 4 are formed and arranged in ascending order. What is the median of those numbers?
- What fraction is equivalent to 0.2024?
- How many positive integer factors does 2024 have?
- Create a 4 × 4 magic square in which the sum of each row, column, and diagonal is 2024.
- What is the units digit of 20242024?
- Each dimension of a rectangular box is an integer number of inches. The volume of the box is 2024 in3. What is the minimum possible surface area of the box?
- What is the maximum possible product for a set of positive integers that have a sum of 2024?
- A polynomial p has non-negative integer coefficients and satisfies p(1) = 8, p(-1) = 0, and p(3/2) = 13.75. What is p(10)?
- Let S(n) denote the sum of the digits of integer n. For example, S(123) = 6. Find a number n such that n + S(n) + S(S(n)) + S(S(S(n))) = 2024.
Special thanks to Professor Harold Reiter for supplying the last two problems in the list.
[Update 12/31/23] Too beautiful not to share, this truth about the year was posted in the Wolfram Community by Ed Pegg of Wolfram Research:
23 + 33 + 43 + 53 + 63 + 73 + 83 + 93 = 2024
And this representation appeared in a post about math beauties of the new year at Math 1089:
2024 = 1 − 2 + 3 × (4 + 5) × (6 + 78 − 9)
Awesome.
A Tent for n Persons
Over 58 million households went camping in 2022, and almost half of them camped 3 or more times during the year. With the number of campers and camping nights increasing as a result of the pandemic, it’s no surprise that campgrounds are full in 2023. The good news is, it’s relatively easy to determine the capacity of a campground.
How many tents can fit in a campground?
Ten. Because ten tents make one whole!
To make that joke work, you have to mispronounce “tenths” as “tents,” so it’s helpful if you speak like My Cousin Vinny (“the two yutes”).
What’s more difficult, though, is determining the capacity of a tent.
Sometimes, the listed area is very generous. For instance, the Pacific Pass Two‑Person Family Dome Tent offers a spacious 24 square feet to each inhabitant. By comparison, campers in the North Face Wawona 6 Tent are bound to feel a little cramped with just 14 square feet per person.
Then there’s the ambiguous Moon Lence 4-5 Person Family Camping Tent that gives a range of 4‑5 people in the name, lists an occupant capacity of five people in the specifications, and says it’s a “roomy four-person tent” in the description. WTF? Moreover, the floor has an area of just 58 square feet, so each person would get less than 15 square feet if there were four people and less than 12 square feet if inhabited by five people; and, with its cool but impractical pentagonal shape, two of them would need to sleep in the fetal position.
So, what is the right amount of space per person? It’s hard to say. According to manufacturers, though, it seems to be about 15 square feet. That estimate is based on data from 26 tents that yield the following line of best fit:
With a coefficient of 15 associated with the independent variable, the regression equation implies that each camper needs about 15 square feet of sleeping area.
I’m not buying it, though. Tents never seem to have enough area for the number of people they claim to fit. Theoretically, our family tent will comfortably sleep three people, but in practice it’s plenty crowded with just two, and sometimes there shouldn’t be more than one — especially if that one suffers from sleep apnea or had burritos for dinner.
Should you need a tent, I recommend you wait till after camping season. You’re more likely to find a good deal when there’s snow on the ground, during the winter of discount tents. If the camp store offers you insurance, however, you should decline it. Were someone to steal your tent while camping, you wouldn’t be covered. And sometimes, the camp store will put tent pegs on the highest shelf; sometimes, the stakes are just too high. Should a bunch of crows accompany you on a camping trip, that would be murder within tent.
Finally, I have to ask:
Is a tent two-and-a-half-times as large as a fort?
Math Jokes 4 Mathy Folks 