The Triple Crown

I once heard the hosts on a sports talk radio show ask rapper and music exec Rick Ross to confirm the rumor that he mows his own lawn, because he gets his best ideas while cutting the grass. Ross replied, “Fact! Shout out to John Deere!”

More recently, I learned that John Deere and Co. posted worldwide revenue of $52B in FY22, a record for the farm equipment company.

Well, as the Romans said with just three little words, omne trium perfectum, so here goes…

If you don’t like geometry, then you don’t want to talk to that guy in the John Deere hat. He’s nice enough, but he’s pro-tractor.

The manure spreader is the only piece of equipment that John Deere won’t stand behind.

Did you hear about the woman who left her husband for a tractor salesman?
She wrote him a John Deere letter.

And here are my three favorite math signs:

And, finally, some random math jokes that involve the number three:

There are three kinds of people: those who are good at math, and those who aren’t.

Why do teenagers travel in groups of three?
Because they can’t even.

What’s a mathematician’s favorite candy bar?
n Musketeers, where n = 3!

How many ears did Alexandre Dumas have?
Five. A left ear, a right ear, and three musketeers.

If you like things that come in threes, you may appreciate Rob Eastaway and Jeremy Wyndham’s book, Why Do Buses Come in Threes? (This is not an affiliate link, just an unsolicited suggestion. Enjoy if you so choose.)

Sidhu Moose Wala Would Love This Post

[Sidhu Moose Wala – 47]

The following is a modified version of a problem that appeared in Concept Quests, a collection of problems originally written by Kim Markworth and soon to be offered by The Math Learning Center as a supplement to Bridges in Mathematic Second Edition.

I was today years old when I realized that I’ve been alive for 47 years, 47 months, 47 weeks, and 47 days. How old will I be on my next birthday?

Incidentally, the number 47 was also featured in this week’s challenge on the NPR Sunday Puzzle:

Take this equation: 14 + 116 + 68 = 47. Clearly this doesn‘t work mathematically. But it does work in a nonmathematical way. Please explain.

The number 47 is very interesting, especially if you’re a member of The 47 Society. But even if you aren’t, the number 47 has some noteworthy credentials:

• The number 47 is a safe prime, meaning it has the form 2q + 1, where q is also prime.
• The greatest number of cubes (not necessarily the same size) that cannot be rearranged to form a larger cube is 47. That is, it’s not possible to make a cube from 47 smaller, not-necessarily-the-same-size cubes, but it is possible to make a cube from 48, 49, 50, 51, or any greater number of cubes.
• 47 = 2⁵ + 5² – 2 × 5
• Lord Asano Takumi’s followers swore to avenge his death — a suicide he was obliged to commit for drawing his sword in the palace — and survives in a story known as the Legend of the 47 Ronin.
• Mexican revolutionary Pancho Villa was killed by 47 bullets.
• It takes 47 divisions of one cell to produce the number of cells in the human body; that is, there are approximately 2⁴⁷ cells in a human.
• Proposition 47 of Euclid’s Elements is the Pythagorean theorem.
• The Bible credits Jesus with 47 miracles.
• The Declaration of Independence has 47 sentences.
• There are 47 strings on a concert harp.
• The tropics of Cancer and Capricorn are located 47 degrees apart.

This fascination with the number 47 seems to have started a long time ago.

In the summer of 1964, two students at Pomona College — Laurens “Laurie” Mets and Bruce Elgin — began an extracurricular experiment to determine how often the number 47 occurred in nature. While this may seem like an odd thing to do, they weren’t the only ones; a number of Pomona students were involved in a summer program sponsored by the NIH, and many of them conducted similar experiments about the occurrence of random numbers, questioning the existence of patterns in nature. However, it was Mets’s and Elgin’s search for 47 that took on a life of its own and developed a cult following.

The experiment was to determine if 47 showed up in nature more often than other numbers. As a first step, they hypothesized that 47 would appear in two percent of all California license plates, which would be higher than would occur by random chance. According to Mets, a funny thing happened: when they looked at a bunch of license plates and counted the occurrence of 47, they found their hypothesis to be true. According to Elgin, however, the license plate experiment was a failure; but, within the next week, he saw a Rolaids commercial claiming that Rolaids absorbs up to 47% of its weight in excess acid. That sparked a desire to see where else 47 might occur… and before long, people all over campus were counting everything to look for 47s.

As part of that same summer program, statistician Donald Bentley gave a talk in which he stated that any number could be shown to be equal to any other number. In his (invalid) proof, he chose to show that every number was equal to — you guessed it — 47. How did he prove it?

Consider an isosceles right triangle for which the base is 47 units, and the congruent legs are each 37 units. Now, connect the midpoints of the congruent sides to the midpoint of the base, as shown in Figure 1. The combined lengths of the red segments in Figure 1 is the same as the sum of the lengths of the congruent sides, 74 units. Then repeat, connecting the midpoints of these smaller triangles, as shown in Figure 2. Again, the combined lengths of the red segments is 74 units. Then do this again and again and again — all the way to infinity — and eventually the triangles will get so small that the red segments will appear to be a straight-line segment that overlaps the base of the triangle, as shown in Figure 3. The length of the red segment is 74 units, and length of the base is 47 units, so 74 = 47. By similar (incorrect) reasoning, it would be possible to show that any number is equal to 47.

Although Dr. Bentley’s proof was meant as a classroom lesson on statistical computing, leave it to the 47-hunters to accept the proof and believe that all numbers equal 47.

(If you’d like to know more about this sordid history of 47, you can read The Mystery of 47 at the Pomona College website.)

Anyway, where was I? Oh, yes… I’ve been alive for 47 years, 47 months, 47 weeks, and 47 days.

Mathematically, what I find most interesting is the general case. That is, if I’ve been alive for n years, n months, n weeks, and n days, how old will I be on my next birthday?

For the problem I presented at the top of this post, n = 47, and the answer is 52 years; that is, I’ll turn 52 on my next birthday. But if n = 48, I won’t turn 53 on my next birthday; I’ll turn 54. That’s right, it skips a year. This happens periodically, and with a little help from the Date Calculator at www.timeanddate.com, I was able to generate this beautiful — albeit incomplete — graph:

 If you’re interested in seeing the data, check it out at https://www.desmos.com/calculator/u43gz6ahyi.

Something really wacky happens when n = 38. I was born on March 17, 1971, and the date 38 years, 38 months, 38 weeks, and 38 days from my date of birth was exactly 42 years later: March 17, 2013. Whoa.

Who knows what any of this means? Perhaps this implies that 38, not 47, is a magical number. In any case, I hope you have fun playing with this scenario and seeing what you can discover.

If that’s not your thing, though, here are some time-and-date jokes for you:

How many seconds are in a year?
Twelve. January 2nd, February 2nd, March 2nd, …

Did you hear about the hungry clock?
It went back four seconds.

Traditional calendars are for people who are week-minded.

I was fired from my job at the calendar factory. My boss was mad that I took a few days off.

Math Problems for 2023

Happy New Year!

I’m feeling lucky about 2023… but perhaps that’s because 7⁷ mod 7! = 2023.

But I’m also feeling lucky because there are many interesting problems that involve the number 2023. I’m a little late in getting this post out, but all of the following problems attempt to use the number 2023 in some interesting way. Thanks to Professor Harold Reiter from UNC‑Charlotte, who supplied the ideas for the first two problems; the rest are MJ4MF originals. Enjoy!

• For 2023, the sum of the digits of the year, times the square of the sum of the squares of the digits, is equal to the year itself. Yeah, that’s a mouthful; more concisely, (2 + 0 + 2 + 3) × (2² + 0² + 2² + 3²)² = 2023. Can you find the only other four-digit number abcd with the property that (a + b + c + d) × (a² + b² + c² + d²)² = abcd?
• If you place one dot inside an equilateral triangle and connect it to each vertex, you get three non-overlapping triangles. Similarly, if you place two dots in an equilateral triangle — and no subset contains three collinear dots — you can connect the dots to form five non-overlapping triangles. How many non-collinear dots must you place inside the triangle to get 2023 non-overlapping triangles?
• The sum of the digits of 2023 is 2 + 0 + 2 + 3 = 7, and 7 is a factor of 2023. For how many numbers in the 2020s is the sum of the digits a factor of the number?
• In how many ways can a 4 x 4 grid be covered with monominos and L-shaped triominos? One such covering is shown below.

• What is the sum of 1 + 3 + 5 + 7 + ··· + 2023?
• How long would it take you to count to 2023?
• Using only common mathematical symbols and operations and the digits 2, 0, 2, and 3, 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 3 are formed and arranged in ascending order. What is the first number in the list?
• Place addition or subtraction symbols between the cubes below to create a true equation:

9³     8³     7³     6³     5³     4³     3³     2³     1³ = 2023

• Find a fraction with the following decimal equivalent.

• How many positive integer factors does 2023 have?
• Create a 4 × 4 magic square in which the sum of each row, column, and diagonal is 2023.
• What is the units digit of 2023²⁰²³?
• What is the value of the following expression, if x + 1/x = 2?

• Each dimension of a rectangular box is an integer number of inches. The volume of the box is 2023 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 2023?

Year-End Puzzle

There’s a math puzzle that’s been making its rounds on the web with the following proclamation:

82 beers minus your age plus $40 is the year you were born!

Of course, it gets way more interesting on the web, where you’ll find images of beer:

Don’t even get me started about the grammar and punctuation. (Why is “Beers” capitalized? And “your”? Ugh.)

You’ll also find pretty colors and clip art of beer mugs:

I’m also going to ignore the nonsensical nature of misaligned units — beers, years, and dollars? WTF?

But I won’t ignore the math. When the puzzle is shared online, it’s often accompanied by expressions of amazement — “OMG! This is amazing!” — or statements about its veracity — “This always works!” And it’s the latter claim that’s worth investigating.

To begin, I’ll show that this equation does, in fact, hold for me. I was born in 1971, and I’m 51 years old:

82 – 51 + 40 = 71

So far, so good.

But my friend Mike turns 51 on Friday. So he tried it:

82 – 50 + 40 = 72

Uh-oh. He was born on December 23, 1971, not 1972. It didn’t work for Mike; in fact, it won’t work for anyone who hasn’t had their birthday this year. Luckily, it’s late December, so this only excludes about 3% of the population; but in the US alone, that’s about 9 million people.

Separately, a version of this trick was texted to my sons, who are 15‎‑year‑old twins. They tried it:

82 – 15 + 40 = 107

Nope! They were born in 2007. Granted, the result of 107 means 107 years after 1900. But that’s inconsistent with the result being the last two digits of the year, so I’m not accepting it. In fact, this trick won’t work for anyone who wasn’t born in the 1900s. The graph of age distribution below — females on the left, males on the right — shows that approximately 90 million people were born in the year 2000 or later (age 22 or younger on the graph below). Of course, about 3% of them (3 million) were already counted above — those are the people who haven’t had birthdays this year and are too young for the trick to work. Which means that this trick doesn’t work for about 13 + 90 – 3 = 100 million people. And that’s just in the US.

Population Distribution of the United States [USA Facts]

So let’s say you know someone who was born in the 1900s and who has celebrated a birthday this year. Well, you could share this trick with them… but, you better do it soon! This trick won’t work once the calendar flips to 2023. (You could, of course, modify and start the trick with 83 beers or subtract 39, and then it’ll be fine. Though it’ll still only apply to people born in the 1900s who’ve already had their birthdays, and those people will be a little harder to find in January.)

This trick requires so many caveats, it hardly seems worth sharing, right? Well, not if you’re a math teacher and especially if you’re a secondary math teacher. Above, we dabbled in a little set theory with the principle of inclusion and exclusion while considering

(people born in the 1900s) ∪ (people who’ve had birthdays this year).

It can also lead to a good discussion of domain and range. The function y(a), where y is the year based on your age, a, can be simplified as:

y(a) = 82 – a + 40 = 122 – a

A restriction on the domain of 22 < a < 122 allows the trick to work just fine. And from a practical standpoint, the restriction that a < 122 is probably unnecessary. Lucille Randon is currently the oldest person in the world at 118 years of age, so unless you plan to share the trick with a giant tortoise or a bristlecone pine tree, you’re probably okay.

But I don’t want to leave you like that, so here’s a math trick that will always work.

1. In the grid below, circle any number. Then, cross out the other numbers in the same row or column. (For instance, if you circled the 204 in the upper left cell, you’d then cross out 506, 402, and 305 in the top row as well as 505, 406, and 307 in the left column.)
2. Circle one of the remaining nine numbers, and again cross out the other numbers in the same row or column.
3. Circle one of the remaining four numbers, then cross out the other numbers in the same row or column.
4. This should leave just one number. Circle it.
5. At this point, one number should be circled in each row and column. If that’s not the case, start over at Step 1. If that is the case, though, good job! Find the sum of the four circled numbers.

I obviously have no idea which four numbers you selected. But I do know that the sum you obtained was 2023. (Now would be an appropriate time for applause. But that may be a little awkward if you’re reading this at work and folks in neighboring cubicles are within earshot.)

I know, it feels like magic. But it’s not; it’s just math.

Happy holidays, and happy new year!

Push-Up Percentages

Hands down, push-ups are my favorite exercise!

If you’re friends with me on Facebook, then you know I’m raising money for St. Jude Children’s Hospital by participating in the St. Jude Push-Up Challenge. And if not, then let me tell you: I’m challenging myself to do 3,000 6,000 push-ups during the month of November. As of November 20, I was just 660 push-ups away from reaching my goal; you can see all of my progress on this Google sheet.

What’s been most fun about this challenge, though, is the math of it all. (Isn’t that always the case?)

Arithmetic. How many push-ups will I have to average each day to reach my goal? November has 30 days, and 6000 ÷ 30 = 200.

Logistics. How should I break up those 200 push‑ups each day? A regimented person might do 10 push‑ups a minute for 20 minutes, or maybe 25 push‑ups every hour for 8 hours. For the record, I’m not that routinized. Most days, I do anywhere from 8‑12 sets, with anywhere from 10 to 50 push‑ups per set. Though I did have one day — November 13 — where I set a timer and did 10 push‑ups every 45 seconds for 90 minutes; I completed 1,235 push‑ups that afternoon, a personal record for push‑ups in day.

Physics. When doing a push‑up, how much of your bodyweight are you actually moving? I know that I’m not pushing up all 200 pounds of my weight when doing a push‑up. And I’m sure the percentage decreases more as I elevate my hands. This is only a hypothesis, but incline push‑ups sure seem easier — I can do more of them before reaching failure — than regular push‑ups, and decline push‑ups seem harder. As it turns out, Jane Reaction explained exactly why this is in a Physics of Fitness Friday blog post, and her conclusions about the percent bodyweight supported by the hands when doing a push‑up were as follows:

• Regular Push-Up: 64%
• Incline Push-Up, Hands Raised 12″: 55%
• Decline Push-Up, Feet Raised 12″: 70%
• Decline Push-Up, Feet Raised 24″: 74%
• Handstand Push-Up: 100%

These, of course, are estimates, and Jane used her body for the calculations. The exact percentage will differ person to person, depending on your weight, the length of your arms, your body shape, the location of your center of mass, and other factors. For a reasonable estimate, Jane suggests placing your hands on a scale while in push‑up position, then dividing by your standing weight (i.e., your weight when standing on a scale). For instance, if your scale shows 128 pounds for your hands in push‑up position and 200 pounds when you stand on it — which is exactly what it showed for me — then your hands are supporting 128 ÷ 200 = 0.64, or 64%, of your bodyweight when doing a push‑up.

The following graph, from Zatsiorsky’s Science and Practice of Strength Training, shows the percent body weight based on the height that your feet or hands are elevated:

There were two reasons I was so curious about this estimate. First, math is cool. Second, I get bored easily, and I was wanting to substitute some bench presses for push-ups occasionally. Based on what’s above, I figured that if I did bench press with 125 pounds, then one rep would be equivalent to one push‑up. So on two of the days, I did a chest workout that included bench press, incline press, cable flyes… and then some push‑ups. And, yes, I was sore as hell the next day! But I was also rejuventated since I got to incorporate some activity other than just push‑ups.

I’m proud that, at 51 years of age, I’m going to complete this challenge. But I’m even more proud that I’m raising money for a good cause. If you’d like to contribute to St. Jude, visit my Donate page on Facebook. And if not, no worries; here’s one more joke for you before you go…

A man in a bar offers $100 to anyone who can do 100 push‑ups. Another patron leaves for a few minutes, then returns and says, “I’ll take that bet!” He drops to the ground and does 30 easily but then starts slowing down around 40 and collapses before he reaches 50. “I don’t understand,” he says. “I just did 150 outside!”

IDK Puzzles

I like logic, and I like beer, so it’s no surprise that this is one of my favorite online comics:

Not sure why that’s funny? There’s an explanation at www.beingamathematician.org.

Logic puzzles in which a protagonist states, “I don’t know!” are ubiquitous. Borrowing from texting culture, I’ve taken to calling these IDK Puzzles.

Every math person remembers the first non-routine problem they solved and, more importantly, the feeling they experienced when solving it. The first time I had that feeling occurred after solving a logic puzzle about three children’s ages that I discovered in a Martin Gardner book; now many years later, I don’t remember the title of the book, and the following is my best recollection of the puzzle:

Two neighbors are speaking. One asks the other, “I know you have three children, but how old are they?”

The other says, “The product of their ages is 72.”

The first neighbor says, “I still don’t know their ages.”

“Well,” says the other, “the sum of their ages is equal to our street address.”

The first neighbor again replies, “I still don’t know their ages.”

“I’m sorry,” says the other, “I can’t talk anymore, because I have to take my oldest child to the dentist,” and then leaves.

While saying good-bye, the first neighbor thinks, “Ah, now I know their ages.”

This puzzle is typical of the genre, in that it appears there is insufficient information, but those who persist will be rewarded. Can you figure out the three children’s ages?

A slightly different IDK Puzzle involves geometric shapes.

Two people are shown the following five shapes:

They are told that a prize has been placed under one of the shapes. One of the people is told the color, and the other is told the shape, but they are not allowed to share their information with each other.

They are asked, “Do either of you know where the prize is hidden?”

Both of them reply, “I don’t know.”

They are asked a second time, “Do either of you know where the prize is hidden now?”

Again, they both reply, “I don’t know.”

They are asked a third time, “What about now?”

They both reply, “Yes!”

Under what shape has the prize been hidden?

Enjoy solving those puzzles. Staying with the theme, let’s end this post with a logic joke of sorts…

Sam comes home from the grocery store with twelve gallons of milk. Pam asks, “Why’d you buy so much milk?”

“Because before I left, you told me to buy a gallon of milk, and then you said, ‘If they have eggs, buy a dozen.’ And they had eggs.”

Pam shakes her head at Sam’s response. But then she notices he hasn’t bought anything else and asks, “Where are the rest of the things we needed?”

“Remember how you told me to put ketchup on the list?” replies Sam.

“Yeah. So?”

“So I put ketchup on the list, but then I couldn’t read the other items!” Sam says. “But I remembered the eggs!”

They’re Moving Second Base

When I first heard that baseball is moving second base, my first thought was, “My, goodness! Isn’t it enough that we’re dealing with a global pandemic, a Russian tyrant invading a neighboring country, a humanitarian crisis in Nicaragua, food insecurity in Somalia, Haiti, and Madagascar, an ever-widening wealth gap, an uptick in calls from unknown numbers, paper cuts, and excessively long lines at the Starbucks drive-thru? I mean, when’s it gonna stop?”

But my second thought was, “This is going to wreak havoc on the secondary textbook publishing industry.” Just look at all the problems that exploit the baseball context:

• Location of the pitcher’s mound (Q2)
• Throw from catcher to second base
• Throw from catcher to second base
• Distance from first base to third base (Q4)
• Runner’s speed in relation to second base

All of those problems are predicated on a consistent distance between bases. Won’t the relocation of second base cause inconsistency?

Well, actually, it won’t.

According to the official rules of baseball, one vertex from first base, third base, and home plate are to be coincident with three vertices of the infield square; but, the center of second base is to be coincident with the fourth vertex. With the rule change, second base will be moved so that one of its vertices will be coincident with the fourth vertex of the infield square, finally bringing a state of geometric consistency to the game that I, for one, believe is long overdue. The image above shows the new (white) and old (gray) locations of second base.

The question all fans should be asking isn’t why are they changing the layout of the infield. The more pertinent question is, what took so damn long?

As it turns out, second base doesn’t get all the credit for the previous configuration issues. To the contrary, it was the movement of the other bases that resulted in a problem. In the 1860s, it was generally agreed that all four bases should be positioned with their centers at the vertices of the infield square. And by “generally agreed,” I mean that there was consensus about this, but it wasn’t officially stated in the rules until 1874. Then in 1877, the rules changed so that the back corner of home plate — at the time, home plate was still a square, not a pentagon like today — coincided with the vertex of the infield square, positioning all of home plate in fair territory. A decade later, first base and third base were moved to be entirely within fair territory, too; but most folks didn’t even notice, because that same year (1887) a number of other rules changes garnered more attention:

• Pitchers were limited to just one step when delivering a pitch; previously, they could take a running start
• Batters were prohibited from requesting a high or low ball from the pitcher, as they had been allowed in the past
• The pitcher’s mound was moved back five feet (from 50′ to 55′)
• Five balls were required for a walk, reduced from six
• Four strikes were required for a strikeout, increased from three

With so many drastic rules changes happening simultaneously, it’s hardly a surprise that first and third were repositioned in relative obscurity while second was left floundering in geometric misalignment.

Just so you know, the rules change will only occur in the minor leagues this year. If it pans out, you can bet you’ll see it in the MLB in a year or two.

But why stop there? Here are some other rules changes in sports that should probably be implemented.

Scoring system in football. I mean, you can score 1, 2, 3, 6, 7, or 8 points depending on what you do. Isn’t that a little excessive? While we’re at it, let’s change the width of the field, too — who the hell thought 53⅓ yards was an appropriate dimension?

College basketball uniforms. Bring back 6, 7, 8, and 9. You may not have known that those digits are not allowed, because each of them requires two hands. Referees indicate the player who committed a foul using their fingers — for instance, holding up two fingers on the right hand and three fingers on the left to indicate that an infraction was committed by number 23 — and the digits 6‑9 would require more than five fingers.

Jan 4, 2020; Philadelphia, PA — Penn State basketball players pose in front of the student section, but do you see a 6, 7, 8, or 9 on a single jersey?
Photo Credit: Bill Streicher-USA TODAY Sports

Frames in bowling. Two balls ain’t enough. Give everyone three attempts to knock down all ten pins.

Cheerleader weigh-ins. Really, folks? The 15th century called, and they want their misogyny back. One anonymous NFL cheerleader wrote that she was banned from performing because she weighed more than 122 pounds. While we’re at it, ban weigh-ins for jockeys, too. The Kentucky Derby — which apparently has one of the more liberal weight allowances — caps the weight at 126 pounds; that includes 7 pounds for the jockey’s gear, so the jockey can’t tip the scale at more than 119 pounds.

Taunting. Allow it everywhere. In college football, the rule is just stupid. Admittedly, one player shouldn’t be allowed to stand over another player while making insulting comments about their mother; but “taunting” according to the NCAA Rule Book includes spinning or spiking the ball, choreographed acts, and the player altering stride when approaching the end zone. C’mon! Further, I’d like to see taunting encouraged a bit more at some events, such as math competitions. Wouldn’t it be great if one participant walked up to another and said, “You can’t even spell Q.E.D.!”

The Great Puzzle Hunt

A woman after my own heart, Millie Johnson loves both puzzles and jokes. She regularly posts to Facebook, and these two gems appeared recently in her feed:

• If all whole numbers between 1 and 1000 were spelled out and arranged in alphabetical order, which number would be next-to-last?

• Arrange the digits 0‑9 into a ten-digit number such that the leftmost n digits comprise a number divisible by n. For example, if the number is ABCDEFGHJK, the three-digit number ABC must be divisible by 3, the five-digit number ABCDE must be divisible by 5, and so on.

The answers to both can be found with an online search, and the latter can be computed in milliseconds by writing some code, but you’ll have more fun if you find the answers using that big lump of gray matter in your skull.

She also recently shared this joke, which I just adore:

Millie is also the founder and puzzle creator for the Great Puzzle Hunt, a free, fun, full-day, team puzzle-solving event. While folks in and around Bellingham, WA, on April 9 will be lucky enough to participate in the face-to-face event on the Western Washington University campus, the rest of the world is invited to participate virtually. There are divisions for secondary (middle and high school), open (any age), WWU students, and WWU alumni. And in case you missed the subtle mention above, I’ll say it again: registration for a team of up to six participants is FREE!

Participants will engage with four puzzles; then, the answers to those puzzles will be used to form a fifth and final meta-puzzle. How hard are the puzzles? See for yourself; the puzzle below, My Life Is In Ruins!, is from the 2021 Great Puzzle Hunt:

GPH2021_LifeInRuinsDownload

The Great Puzzle Hunt is a full-day event, so pack a lunch! But my sons and I participated last year and had a phenomenal time. By the end, we were mentally exhausted but invigorated and intellectually satisfied. If you like to solve puzzles, head on over to https://www.greatpuzzlehunt.com/register and register today!

What an Amazing Date!

There are lots of good dates — meeting at a bookstore coffee shop for, well, perusing books and sipping coffee; spending hours playing Super Mario Bros. and Pac-Man at a retro video game arcade; and, of course, going to an open-mic comedy show where one of the performers tells nothing but math jokes.

But great dates? Well, those are pretty rare. My first date with my wife — where I took her to a hotel and “whispered” to her from the couch across the lobby — is an example, though the stimulating conversation and her perfect laugh may have contributed more than the elliptical ceiling. (Maybe.)

Few dates, however, can compare to today’s date:

12/3/21

Look at that beautiful symmetry! Marvel at its palindromic magnificence! The way it rises then falls, like a Shostakovich melody.

But wait… there’s more! Consider the following pattern:

1 × 1 = 1

11 × 11 = 121

111 × 111 = ?

That’s right! The number 12,321 is a perfect square! And not only that, its square root contains only 1s.

Moreover, check this out:

1 + 2 + 3 + 2 + 1 = 9

That’s right! It’s a square number, and the sum of its digits is also a square number!

Finally, here’s a KenKen puzzle that makes use of the number, though it’s not unique unless one of the digits is already filled in:

No matter how you choose to celebrate, here’s hoping your day is as great as the date!

Better Multiple-Choice

If I were a K-12 student right now, I’d want to live in San Diego. In 2020, San Diego Unified School District introduced a new district-wide math test that contained no multiple-choice questions. The district was allowed to use their own internal test instead of the state test last year, due to the pandemic, and they’re apparently allowed to use it again this year. Supposedly, the test moves away from a reliance on computational ability and instead measures three dimensions: students’ knowledge of mathematics (concepts and formulas); their application of that knowledge; and, their ability to communicate mathematically.

The optimist in me says, “It’s about time!” But the pessimist in me thinks, “Don’t they know that it’s a lot more expensive, and harder to ensure reliable and replicable results, when using humans to do the scoring instead of machines?” I’m old enough to remember the controversy and eventual dissolution of the Maryland State Performance Assessment Program (MSPAP) exams, which required extended answers and contained no multiple-choice questions. FairTest called the MSPAP test “perhaps the single best state exam,” but it was criticized for providing school-level but not individual student scores. Though generally agreed to have been a catalyst for improved teaching, it was replaced by an entirely multiple-choice test to meet the requirements of the Elementary and Secondary Education Act (ESEA).

Ah, the good old days. Reminiscing sure ain’t what it used to be. But, I digress.

To determine if multiple-choice questions are valid tools, I made a list of pros and cons regarding their use in educational assessment. 

Pros	Cons
They can be scored quickly.	The correct answer can be guessed.
They can be scored objectively and without bias.	The correct answer can be found by process of elimination.
	They encourage students to think like the test creators instead of like themselves.
	They provide no information about the student’s solution strategy.
	They are required to have only one answer.
	They exacerbate test anxiety.
	They don’t prepare students for college or the work force.
	Incorrect answer choices expose students to misinformation, which can influence future recall and thinking.

Seems a bit lopsided.

In recent years, multiple-choice questions have gotten a bit of a makeover. Those in the educational assessment industry now call them “selected-response items” because, well, students get to select a response.

But this is just semantics. Referring to a pig as a mud wrestler may sound nicer, but the pig won’t be any less dirty.

It’s the advertising trope of…

New look, same great taste!

Or as a pretentious coffee brand said when they changed the label…

Innovative presentation, but consistent quality.

Truth is, selected-response items look like they’ve always looked, typically with a really boring prompt and even more boring answer choices. As one example, the following item is from a PACE (Packet of Accelerated Christian Education), which “integrate Godly character-building lessons into the academic content.”

Mr. Louis Pasteur did experiments with milk. Mr. Louis Pasteur was…

1. a glass bottle
2. an airplane
3. a scientist

Despite your religious beliefs, you have to admit that this question is rather absurd. Would any student ever think that a glass bottle or an airplane would be referred to as “mister”? To be fair, this question appeared in a PACE packet in 2013, so it’s quite possible that it’s since been updated. Still, 2013 wasn’t that long ago, and there’s no time in history when those answer choices wouldn’t have been ridiculous.

And here’s one that was presented during a session at an NCTM regional conference:

To convert to radians, multiply by…

1. π/180
2. 180/π
3. 225π/180
4. π/40,500

Ignoring the fact that this question attempts to assess something that your calculator knows so you don’t have to, this question is fine. But in the reading passage directly above the question, it stated, “To convert an angle from degrees to radians, multiply by π/180.”

Well, that will just never do.

I’m not convinced that a great multiple-choice question actually exists. That said, some are better than others, so I offer you the following seven multiple-choice — or selected-response, or objective-response, or whatever-you-want-to-call-them — items.

What is the probability that you will randomly choose the correct answer to this question?

1. 25%
2. 50%
3. 0%
4. 25%

At any given time, the number of people in the air — that is, those who are flying in motorized aircraft, and not counting those who were recently launched by catapults or who have bounced on a trampoline — is closest to the population of…

1. Flint, Michigan
2. Seattle, Washington
3. New York, New York

The approximate volume of an average chicken egg is…

1. 7 cm³
2. 70 cm³
3. 700 cm³
4. 7,000 cm³

The polar (north-to-south) diameter of the Earth is about…

1. 1,000,000 inches
2. 20,000,000 inches
3. 500,000,000 inches
4. 1,000,000,000 inches

One million one-dollar bills weigh about as much as…

1. a three-toed sloth
2. a giant panda
3. Chris Christie
4. a grizzly bear
5. a black rhinoceros

The total number of calories in all the hot dogs consumed at Yankee Stadium during one season of Major League Baseball is closest to…

1. the number of five-card poker hands
2. the number of possible license plates in Indiana
3. the number of combinations in the Powerball lottery
4. the number of humans on Earth
5. the number of stars in the Milky Way

If the residents of New Mexico joined hands and stood in a straight line, they could reach from one side to the other of…

1. New Mexico
2. Rhode Island
3. Texas
4. Alaska

The answers to these questions will not be provided, though each question absolutely has a best answer among the choices. In lieu of an answer key, enjoy the following joke:

How do you keep a fool in suspense?