Buzz
Happy Pi Day! For years, math enthusiasts have been celebrating this fundamental irrational number on March 14 (or 3/14, the first three digits of the circle's circumference-to-diameter ratio). In fact, the U.S. House of Representatives even passed a non-binding resolution in 2009 to officially recognize the day. Join in the fun by tackling (or simply pondering) these problems curated by pi lovers from various fields.
PI IN SPACE
Stars with a blue hue in a distant galaxy. | iStockPi plays a crucial role for NASA engineers, aiding them in calculating everything from spacecraft trajectories to the densities of celestial objects. NASA's Jet Propulsion Laboratory (JPL) in Pasadena, California, has been marking Pi Day with the Pi in the Sky challenge for several years, offering non-rocket scientists a glimpse into the kinds of problems engineers tackle daily. The following problems come from Pi in the Sky 3, and you can explore detailed solutions and tips there. JPL also presents new challenges for this year's Pi in the Sky 5.
1. MYSTERIOUS HALO
This undated NASA image shows Saturn's moon Titan in ultraviolet and infrared wavelengths, captured by the Cassini spacecraft during its mission to study Saturn, its rings, atmosphere, and moons. The different colors represent the diverse atmospheric components of Titan. | NASA, Getty ImagesGiven that Titan, Saturn's moon, has a radius of 2575 kilometers and is enveloped by a 600-kilometer thick atmosphere, what fraction of Titan's total volume is made up of atmospheric haze? Additionally, if scientists aim to generate a comprehensive global map of Titan's surface, what is the total surface area a future spacecraft would need to cover?
[Answer: 47 percent; 83,322,891 square kilometers]
2. SPHERICAL SURVEY
NASA's Hubble Space Telescope, orbiting Earth, captured this image of Mars on June 26, 2003. | NASA, Getty ImagesGiven that Mars has a polar diameter of 6752 kilometers and that the Mars Reconnaissance Orbiter (MRO) reaches a minimum distance of 255 kilometers from the planet’s south pole and 320 kilometers from the north pole, how much distance does the MRO cover in one complete orbit? (JPL notes, "MRO’s orbit is nearly circular, so circle formulas can be applied.")
[Answer: 23,018 km]
3. SOLAR SHIELD
This image provided by NASA shows Mercury in silhouette, positioned at the lower left as it crosses the Sun on May 9, 2016, as seen from Boyertown, Pennsylvania. Mercury passes between Earth and the Sun roughly 13 times every century, with the last transit occurring in 2006. | NASA/Bill Ingalls, Getty ImagesIf 1360.8 w/m^2 of solar energy reaches Earth's atmosphere, how many fewer watts are received by Earth when Mercury (diameter = 12 seconds) transits the Sun (diameter = 1909 seconds)?
[Answer: 0.05 w/m^2]
PI AND PIZZA CONNECTION
Pizza on a wooden table | iStockWhile many celebrate Pi Day with a slice of pie, the definition of a "pie" can vary. For Pizza Hut, their signature dishes are considered pies, and in 2016, they embraced Pi Day by inviting customers to tackle several math puzzles crafted by John Conway, the renowned mathematician from Princeton. The reward? Free pizza for the winners, lasting 3.14 years! While these puzzles are still challenging, unfortunately, the chance to win free pizza has long since passed. Below are two of Conway's most perplexing problems.
4. 10-DIGIT PUZZLE
Floating blue numbers | iStockI'm thinking of a 10-digit number where each digit is unique. The interesting part is that the number formed by the first n digits is divisible by n for every n from 1 to 10. What is the number I'm thinking of?
[Answer: 3,816,547,290]
5. PUZZLE SOCIETY
Old door | iStockEvery Friday after school, our school's puzzle club gathers in one of the classrooms.
Last Friday, one member remarked, "I’ve hidden a set of numbers in this envelope that total the room number." A girl responded, "That’s clearly not enough to figure out the room number. Would telling us the count of numbers in the envelope and their product be sufficient to solve it?"
He (after writing for a while): "No." She (after continuing to write for a bit longer): "Well, at least I’ve figured out their product."
What is the number of the classroom where we gather?
[Answer: Room #12 (The numbers in the envelope are either: 6222 or 4431, both adding up to 12 with a product of 48.)]
COM-PI-TITIVE MATH
Blackboard filled with math and science equations | iStockPo-Shen Loh led the U.S. Mathematical Olympiad team to consecutive victories in 2015 and 2016. These wins were especially remarkable given that Team USA had not secured a gold at the International Mathematical Olympiad (IMO) in 21 years. Outside of coaching, Loh serves as an associate math professor at Carnegie Mellon University. His platform, Expii, provides weekly challenges across a variety of topics. Expii has celebrated Pi Day annually, and this year, it even released a video featuring an actual pie to help us better understand pi. The following problems are from previous Expii challenges.
6. PIE ESTIMATION
Pi symbol on a blackboard | iStockPi is widely recognized as one of the most essential constants in mathematics. However, because it is an irrational number, it cannot be exactly represented as a fraction, and its decimal expansion is infinite. Over time, we have relied on various approximations of π. Which of the following is the most accurate approximation?
A) 3 B) 3.14 C) 22/7 D) 4 E) Square root of 10
[Answer: C]
7. TELEPHONE TAG
Yellow rotary phone. | iStockWhen Expii's founding team established the organization in the United States, they needed to pick a phone number. Being math lovers, they decided to choose pi as their seven-digit number within the new 844 toll-free area code. What is Expii's phone number? (Exclude the area code.)
[Answer: 314-1593; in case you forget to round, you get their FAX number!]
8. PI CONNECTION
Metal pentagon | iStockPi is defined as the ratio of a circle's circumference to its diameter. We all know that the area of a circle is pir^2. Is it just a coincidence that both formulas use pi, even though one pertains to the circumference and the other to the area? Not at all!
Now, let's explore this with a regular pentagon. For an appropriately defined "diameter" of a regular pentagon, if we define theta as the ratio of the pentagon's perimeter to its diameter, then the pentagon's area will always be thetar^2, where r is half of the diameter. For this to be accurate, 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]
9. WHAT'S IN A NAME?
Globe on chalkboard | iStockThe name "Expii" evokes a range of pleasant words such as "experience," "explore," "explain," "expand," "express," and others. However, the true origin of the name comes from what is considered 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)]
GETTING EXCITED FOR PI DAY
Calculator and blocks that read | iStockFounded in 1915, the Mathematical Association of America works to promote and celebrate mathematics in all its forms. With thousands of members from a variety of fields, including mathematicians, educators, and enthusiasts, Pi Day is always a major celebration. The first two challenges come from Gary Gordon, a professor at Lafayette College, while the remaining problems have been posed to over 300,000 middle and high school students who participate in the annual American Mathematics Competitions. The top scorers often go on to represent the U.S. on Team USA at the International Mathematical Olympiad (IMO), supported by the MAA.
10. FLIPPING A COIN
Thumb flipping a coin. | iStockAlice and Bob each have a coin. Alice flips hers 1000 times, and Bob flips his 999 times. What is the probability that Alice will flip more heads than Bob?
[Answer: 50 percent. Alice must end up with either more heads or more tails than Bob because she has one extra flip. Since the outcomes are symmetric, each possibility has a 50 percent chance.]
11. CUTTING CHEESE
Wheel of gouda | iStockYou have a cube of cheese (or tofu, for our vegan friends) and a sharp knife. What is the greatest number of pieces you can create with n straight cuts? The pieces cannot be rearranged between cuts!
[Answer: ((n^3)+5n+6)/6). The key is that the sequence begins with 1, 2, 4, 8, 15, so stopping after the fourth cut will give a misleading result.]
12. BUYING SOCKS
Socks hanging on a clothesline | iStockRalph went to the store and purchased 12 pairs of socks for a total of $24. Some of the pairs cost $1 each, some $3 each, and others $4 each. If he bought at least one pair of each type, how many pairs of $1 socks did Ralph purchase?
A) 4 B) 5 C) 6 D) 7 E) 8
[Answer: D]
13. THE COLOR OF MARBLES
Blue and red marbles. | iStockIn a bag containing marbles, 3/5 of them are blue, and the remaining ones are red. If the number of red marbles is doubled while the number of blue marbles remains unchanged, what fraction of the total marbles will be red?
A) 2/5 B) 3/7 C) 4/7 D) 3/5 E) 4/5
[Answer: C]
14. SODA CANS
Tops of soda cans. | iStockIf one can contains 12 fluid ounces of soda, what is the least number of cans needed to make up a gallon (128 ounces) of soda?
[Answer: 11 (since you can't use a fraction of a can)]
15. CARPET COVERAGE
Feet on a pink rug | iStockHow many square yards of carpet would be needed to cover a rectangular floor that measures 12 feet in length and 9 feet in width?
A) 12 B) 36 C) 108 D) 324 E) 972
[Answer: A]
