15 Pi Day Math Problems to Solve


Happy Pi Day! For decades, math lovers have been honoring this crucial irrational constant on March 14 (or 3/14, the first three digits of the ratio of a circle's circumference to its diameter) every year. The U.S. House of Representatives even passed a non-binding resolution in 2009 to recognize the date. Join the celebration by solving (or at least puzzling over) these problems from a varied collection of pi enthusiasts.


Blue-tinted stars in galaxy.

Pi is a vital number for NASA engineers, who use it to calculate everything from trajectories of spacecraft to densities of space objects. NASA's Jet Propulsion Laboratory, located in Pasadena, California, has celebrated Pi Day for a few years with a Pi in the Sky challenge, which gives non rocket engineers a chance to solve the problems they solve every day. The following problems are from Pi in the Sky 3 (and you can find more thorough solutions and tips there). JPL has brand-new problems for this year's event, Pi in the Sky 5.


Saturn's moon, Titan.
This undated NASA handout shows Saturn's moon, Titan, in ultraviolet and infrared wavelengths. The Cassini spacecraft took the image while on its mission to gather information on Saturn, its rings, atmosphere and moons. The different colors represent various atmospheric content on Titan.
NASA, Getty Images

Given that Saturn's moon Titan has a radius of 2575 kilometers, which is covered by a 600-kilometer atmosphere, what percentage of the moon's volume is atmospheric haze? Also, if scientists hope to create a global map of Titan's surface, what is the surface area that a future spacecraft would have to map?


[Answer: 47 percent; 83,322,891 square kilometers]


NASA's Earth-orbiting Hubble Space Telescope took this picture June 26, 2003 of Mars.
NASA's Earth-orbiting Hubble Space Telescope took this picture June 26, 2003 of Mars.
NASA, Getty Images

Given that Mars has a polar diameter of 6752 kilometers, and the Mars Reconnaissance Orbiter comes as close to the planet as 255 kilometers at the south pole and 320 kilometers at the north pole, how far does MRO travel in one orbit? (JPL advises, "MRO's orbit is near enough to circular that the formulas for circles can be used.")


[Answer: 23,018 km]


Mercury is seen in silhouette, lower left of image, as it transits across the face of the sun.
In this handout provided by NASA, the planet Mercury is seen in silhouette, lower left of image, as it transits across the face of the sun on May 9, 2016 as viewed from Boyertown, Pennsylvania. Mercury passes between Earth and the sun only about 13 times a century, with the previous transit taking place in 2006.
NASA/Bill Ingalls, Getty Images

If 1360.8 w/m^2 of solar energy reaches the top of Earth's atmosphere, how many fewer watts reach Earth when Mercury (diameter = 12 seconds) transits the Sun (diameter = 1909 seconds)?


[Answer: 0.05 w/m^2]


Pizza on wooden table

People often celebrate Pi Day by eating pie, but what is considered a "pie" is subjective. Pizza Hut considers its main offerings pies, and got into the spirit of Pi Day in 2016 by asking their customers to solve several math problems from English mathematician and Princeton professor John Conway, with promises of free pizza for winners for 3.14 years. Below are two of his fiendishly tricky problems. Unfortunately, even if you solve them, your chance at free pizza is long gone.


Floating blue numbers

I'm thinking of a 10-digit integer whose digits are all distinct. It happens that the number formed by the first n of them is divisible by n for each n from 1 to 10. What is my number?


[Answer: 3,816,547,290]


Old door

Our school's puzzle club meets in one of the classrooms every Friday after school.

Last Friday, one of the members said, "I've hidden a list of numbers in this envelope that add up to the number of this room." A girl said, "That's obviously not enough information to determine the number of the room. If you told us the number of numbers in the envelope and their product, would that be enough to work them all out?"

He (after scribbling for some time): "No." She (after scribbling for some more time): "Well, at least I've worked out their product."

What is the number of the school room we meet in?


[Answer: Room #12 (The numbers in the envelope are either: 6222 or 4431, which both add up to 12 and the product is 48.)]


Blackboard with math and science equations on it

Po-Shen Loh coached the U.S. Mathematical Olympiad team to victory in 2015 and 2016. The back-to-back win was particularly impressive considering Team USA had not won the International Mathematical Olympiad (or IMO) in 21 years. When not coaching, Loh is an associate math professor at Carnegie Mellon University. His website, Expii, challenges readers weekly with a large range of problems. Expii has celebrated Pi Day for several years now—this year it published a video that uses an actual pie to help us visualize pi better—and the following problems are from its past challenges.


Pi on blackboard

Pi has long been noted as one of the most useful mathematical constants. Yet, due to the fact that it is an irrational number, it can never be expressed exactly as a fraction, and its decimal representation never ends. We have come to estimate π often, and all of these have been used as approximations to π in the past. Which is the closest one?

A) 3
B) 3.14
C) 22/7
D) 4
E) Square root of 10


[Answer: C]


Yellow rotary phone.

When Expii's founding team registered the organization in the United States, they needed to select a telephone number. As math enthusiasts, they claimed pi in the new 844 toll-free area code. What is Expii's seven-digit telephone number? (Excluding the area code.)


[Answer: 314-1593; in case you forget to round, you get their FAX number!]


Metal pentagon

The number pi is defined to be the ratio circumference/diameter for any circle. We also all know that the area of a circle is pir^2. Is it a sheer coincidence that they are both the same pi, even though one concerns the circumference and one concerns the area? No!

Let's do it for a regular pentagon. It turns out that for the appropriate definition of the "diameter" of a regular pentagon, if we define the number theta to be the ratio of the perimeter/diameter of any regular pentagon, then its area is always thetar^2, where r is half of the diameter. For this to be true, what should be the "diameter" of a regular pentagon?

A) The distance between the farthest corners of the pentagon.
B) The diameter of the largest circle that fits inside the pentagon.
C) The diameter of the smallest circle that fits around the pentagon.
D) The distance from the base to the opposite corner of the pentagon.
E) Other, not easy to describe.
F) It's a trick question.


[Answer: B]


Globe on chalkboard

"Expii" brings to mind a number of nice words like "experience," "explore," "explain," "expand," "express," and more. The truth behind the name, however, is based on the most beautiful equation in mathematics:

e^pii + 1 = 0

What is (-1)^-i/pi?

Round your answer to the nearest thousandth.


[Answer: Euler's number, also known as e, or 2.718 (rounded off)]


Calculator and blocks that read

The Mathematical Association of America was founded in 1915 to promote and celebrate all things mathematical. It has thousands of members, including mathematicians, math educators, and math enthusiasts, and of course they always celebrate Pi Day. The first two problems are by Lafayette College professor Gary Gordon, while the following four have been sprung on the 300,000+ middle and high school students who participate in the association's annual American Mathematics Competitions. Top scorers in these competitions will sometimes go on to compete on the MAA-sponsored Team USA at the IMO.


Thumb flipping a coin.

Alice and Bob each have a coin. Suppose Alice flips hers 1000 times, and Bob flips his 999 times. What is the probability that the number of heads Alice flips will be greater than the number Bob flips?


[Answer: 50 percent. Alice must have either more heads or more tails than Bob (since she has one additional flip), but not both. These two possibilities are symmetric, so each has a 50 percent probability.]


Wheel of gouda

You are given a cube of cheese (or tofu, for our vegan readers) and a sharp knife. What is the largest number of pieces one may decompose the cube using n straight cuts? You may not rearrange the pieces between cuts!


[Answer: ((n^3)+5n+6)/6). The trick is that the sequence starts 1, 2, 4, 8, 15, so stopping before the fourth cut will give the wrong impression.]


Socks hanging on a line

Ralph went to the store and bought 12 pairs of socks for a total of $24. Some of the socks he bought cost $1 a pair, some $3 a pair, and some $4 a pair. If he bought at least one pair of each kind, how many pairs of $1 socks did Ralph buy?

A) 4
B) 5
C) 6
D) 7
E) 8


[Answer: D]


Blue and red marbles.

In a bag of marbles, 3/5 of the marbles are blue, and the rest are red. If the number of red marbles is doubled, and the number of blue marbles stays the same, what fraction of the marbles will be red?

A) 2/5
B) 3/7
C) 4/7
D) 3/5
E) 4/5


[Answer: C]


Tops of soda cans.

If one can holds 12 fluid ounces of soda, what's the minimum number of cans required to provide a gallon (128 ounces) of soda?


[Answer: 11 (you can't have a fraction of a can)]


Feet on a pink rug

How many square yards of carpet are required to cover a rectangular floor that is 12 feet long and 9 feet wide?

A) 12
B) 36
C) 108
D) 324
E) 972


[Answer: A]

A Handy Map of All the Royal Residences in the UK

Frogmore House, Prince Harry and Meghan Markle's primary estate on the grounds of Windsor Castle.
Frogmore House, Prince Harry and Meghan Markle's primary estate on the grounds of Windsor Castle.

Somewhere along the way, you probably learned that Buckingham Palace is home to the ruler of the United Kingdom and many unflinching, fancily clad guards. And, if you watch The Crown or keep a close eye on royal family news, you might recognize the names of other estates like Windsor Castle and Kensington Palace.

But what about Gatcombe Park, Llwynywermod, or any of the other royal residences? To fill in the gaps of your knowledge, UK-based money-lending site QuickQuid created a map and corresponding illustrations of all 20 properties, and compiled the need-to-know details about each place.

quickquid map of royal family residences

Queen Elizabeth II and Prince Philip kept eight estates for themselves, and divvied up the rest among their children and grandchildren, some of whom have purchased their own properties, too. Though Buckingham Palace is still considered the official residence of the Queen, she now splits most of her time between Windsor Castle and other holiday homes like Balmoral Castle in Scotland and Sandringham House, which Prince Philip is responsible for maintaining.

quickquid illustration of royal family residences

Windsor shares its grounds with two other properties: Prince Harry and Meghan Markle’s home, Frogmore House, and the Royal Lodge, where Prince Andrew (the Queen’s second youngest child) lives.

illustration of frogmore house

Southwest of Windsor is Highgrove House, Prince Charles’s official family home with wife Camilla, Duchess of Cornwall. They also own Birkhall in Scotland, Clarence House in London, Tamarisk House on the Isles of Scilly, and the aforementioned Llwynywermod in Wales. Much like the Queen herself does, Charles and Camilla basically have a different house for each region they visit.

illustration of highgrove house

In 2011, the Queen gave Anmer Hall—which is on the grounds of Sandringham House—to Prince William and Kate Middleton as a wedding gift, but they’ve recently relocated to Kensington Palace so Prince George could attend school in London.

Queen Elizabeth and Prince Philip’s only daughter, Anne, resides in Gatcombe Park with her daughter, Zara Tindall. Anne also owns St. James’s Palace in London, where her niece (Princess Beatrice of York) and her mother’s cousin (Princess Alexandra) sometimes live.

Lastly there's Edward, Elizabeth and Philip's youngest son, who lives with his wife in Bagshot Park, which architectural historian Nikolaus Pevsner called “bad, purposeless, [and] ugly.”

illustration of bagshot park

If you’re feeling particularly cramped in your tiny one-bedroom apartment (or even regular-sized house) after reading about the royal family’s overabundance of real estate, take solace in the knowledge that at least you’ll never have to follow their strict fashion rules.

A Book Fair for Grown-Ups Is Coming to New York

seb_ra/iStock via Getty Images
seb_ra/iStock via Getty Images

Amid all the prepubescent drama and uncertainty of elementary school was one glimmering spot of hope and happiness: the Scholastic Book Fair. Getting to take just a few minutes out of your regular school day to wander the temporary bookshelves seemed about as enchanting as walking through the wardrobe into Narnia.

For folks who’ve been chasing that particular brand of ecstasy well into their adult lives, we have some big news. Next month, Penguin Random House is hosting a book fair for grown-ups. The Pop Insider reports that the event will take place at Lightbox in New York on Saturday, November 23, and you must be at least 21 years old to attend.

It’s not intended to be an exact replica of the book fair from your own school days, but rather a full-fledged recreation of your entire grade-school experience. The electronic invitation promises pop culture trivia, Mad Libs, an “awkward school photo booth,” spin art, snap bracelets, Mr. Sketch markers, cubbies, and “severe middle school flashbacks.”

There will also, of course, be books for sale, though it’s not clear if the inventory will include throwback series like Junie B. Jones and The Magic Treehouse, or just books for adults.

In addition to tsunami-sized waves of nostalgia, the event will feature appearances from some of Penguin Random House’s beloved authors. The list hasn’t been revealed in full, but Viking Books, an imprint of Penguin Random House, tweeted that its author John Hodgman will be there to promote his new book, Medallion Status.

Tickets are $25 for a one-hour time slot, or you can pay $50 to stay for the whole five hours. And your afternoon of embracing your inner kid will benefit actual kids—Penguin Random House will donate a portion of ticket sales to Read Ahead, a non-profit that uses reading to help students learn life-long social and emotional skills.

While the Scholastic Book Fair is still going strong in schools today, the same can’t be said for card catalogs, dodgeball, or these other things.

[h/t The Pop Insider]