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]

5 Hilarious Discoveries from the 2019 Ig Nobel Prize Winners

andriano_cz/iStock via Getty Images
andriano_cz/iStock via Getty Images

Each September, the Ig Nobel Prizes (a play on the word ignoble) are given out to scientists who have wowed the world with their eccentric, imaginative achievements. Though the experiments are usually scientifically sound and the results are sometimes truly illuminating, that doesn’t make them any less hilarious. From postal workers’ scrotal temperatures to cube-shaped poop, here are our top five takeaways from this year’s award-winning studies.

1. Left and right scrota often differ in temperature, whether you’re naked or not.

Roger Mieusset and Bourras Bengoudifa were awarded the anatomy prize for testing the scrotum temperatures in clothed and naked men in various positions. They found that in some postal workers, bus drivers, and other clothed civilians, the left scrotum is warmer than the right, while in some naked civilians, the opposite is true. They suggest that this discrepancy may contribute to asymmetry in the shape and size of male external genitalia.

2. 5-year-old children produce about half a liter of saliva per day.

Shigeru Watanabe and his team nabbed the chemistry prize for tracking the eating and sleeping habits of 15 boys and 15 girls to discover that, regardless of gender, they each produce about 500 milliliters of spit per day. Children have lower salivary flow rates than adults, and they also sleep longer (we produce virtually no saliva when we sleep), so it seems like they may generate much less saliva than adults. However, since children also spend more time eating than adults (when the most saliva is produced), the average daily levels are about even—at least, according to one of Watanabe’s previous studies on adult saliva.

3. Scratching an ankle itch feels even better than scratching other itches.

Ghada A. bin Saif, A.D.P. Papoiu, and their colleagues used cowhage (a plant known to make people itchy) to induce itches on the forearms, ankles, and backs of 18 participants, whom they then asked to rate both the intensity of the itch and the pleasure derived from scratching it. Subjects felt ankle and back itches more intensely than those on their forearms, and they also rated ankle and back scratches higher on the pleasure scale. While pleasure levels dropped off for back and forearm itches as they were scratched, the same wasn’t true for ankle itches—participants still rated pleasurability higher even while the itchy feeling subsided. Perhaps because there’s no peace quite like that of scratching a good itch, the scientists won the Ig Nobel peace prize for their work.

4. Elastic intestines help wombats create their famous cubed poop.

In the final 8 percent of a wombat’s intestine, feces transform from a liquid-like state into a series of small, solid cubes. Patricia Yang, David Hu, and their team inflated the intestines of two dead wombats with long balloons to discover that this formation is caused by the elastic quality of the intestinal wall, which stretches at certain angles to form cubes. For solving the mystery, Yang and Hu took home the physics award for the second time—they also won in 2015 for testing the theory that all mammals can empty their bladders in about 21 seconds.

5. Romanian money grows bacteria better than other money.

Habip Gedik and father-and-son pair Timothy and Andreas Voss earned the economics prize by growing drug-resistant bacteria on the euro, U.S. dollar, Canadian dollar, Croatian luna, Romanian leu, Moroccan dirham, and Indian rupee. The Romanian leu was the only one to yield all three types of bacteria tested—Staphylococcus aureus, Escherichia coli, and Vancomycin-resistant Enterococci. The Croatian luna produced none, and the other banknotes each produced one. The results suggest that the Romanian leu was most susceptible to bacteria growth because it was the only banknote in the experiment made from polymers rather than textile-based fibers.

Visit Any National Park for Free on September 28—or Volunteer to Help Maintain Them

Yosemite National Park
Yosemite National Park
Nick Hanauer/iStock via Getty Images

By the end of September—which always seems especially busy, even if you’re not a student anymore—you might be ready for a small break from the hustle and bustle. On Saturday, September 28, you can bask in the tranquility of any national park for free, as part of National Public Lands Day.

According to the National Park Service, the holiday has been held on the fourth Saturday of every September since 1994, and it’s also the nation’s largest single-day volunteer effort. It’s up to you whether you’d like to partake in the service side or simply go for a stroll, but there is an added incentive to volunteer: You’ll get a one-day park pass that you can use for free park entry on a different day. Opportunities for volunteering include trail restoration, invasive plant removal, park cleanups, and more; you can see the details and filter by park, state, and/or type of event here.

If you’re not sure how you should celebrate National Public Lands Day, the National Park Service has created a handy flowchart to help you choose the best course of action for you—which might be as simple as sharing your favorite outdoor activity on social media with the hashtag #NPLD.

National public lands day celebration flowchart
National Park Service

There are more than 400 areas run by the National Park Service across the U.S., and many of them aren’t parks in the traditional sense of the word; the Statue of Liberty, Alcatraz Island, and countless other monuments and historical sites are also run by the NPS. Wondering if there might be one closer than you thought? Explore parks in your area on this interactive map.

For those of you who can’t take advantage of the free admission on September 28, the National Park Service will also waive all entrance fees for Veteran’s Day on November 11.

And, if you’re wishing a free-admission day existed for museums, you’re in luck—more than 1500 museums will be free to visit on Museum Day, which happens to be this Saturday.