1, 2, 4, 8, 16, __

and was dumbfounded to learn that the missing value was **31**, *not 32*, because the pattern was *not* meant to represent the powers of 2, but rather, the number of pieces into which a circle is divided if *n* points on its circumference are joined by chords. Known as Moser’s circle problem, it represents the inherent danger in making assumptions from a limited set of data.

Last night, my sons told me about the following problem, which they encountered on a recent math competition:

*What number should replace the question mark?*

Well, what say you? What number do you think should appear in the middle stick figure’s head?

Hold on, let me give you a hint. This problem appeared on a multiple-choice test, and these were the answer choices:

- 3
- 6
- 9
- 12

Now that you know one of those four numbers is *supposed* to be correct, does that change your answer? If you thought about it in the same way that the test designers intended it, then seeing the choices probably didn’t change your answer. But if you didn’t think about it that way and you put a little more effort into it, and you came up with something a bit more complicated — like I did — well, then, the answer choices may have thrown you for a loop, too, and made you slap your head and say, “WTF?”

For me, it was Moser’s circle problem all over again.

So, here’s where I need your help: **I’d like to identify various patterns that could make any of those answers seem reasonable.**

In addition, I’d also love to find a few other patterns that could make some answers other than the four given choices seem reasonable.

For instance, if the numbers in the limbs are *a*, *b*, *c*, and *d*, like this…

then the formula 8*a* – 4*d* gives 8 for the first and third figures’ heads and yields 8 × 6 – 4 × 9 = **12** as the answer, which happens to be one of the four answer choices.

Oh, wait… you’d don’t like that I didn’t use all four variables? Okay, that’s fair. So how about this instead: ‑3*a* + *b* + *c* – 2*d*, which also gives ‑3 × 6 + 7 + 5 + 2 × 9 = **12**.

Willing to help? **Post your pattern(s) in the comments.**

[**UPDATE (3/9/18):** I sent a note to the contest organizers about this problem, and I got the following response this afternoon: “Thanks for your overall evaluation comments on [our] problems, and specifically for your input on the Stick Figure Problem. After careful consideration, we decided to give credit to every student for this question. Therefore, scores will be adjusted automatically.”]

But no complaints. It was nice to wear only a t-shirt and no jacket, and it allowed me to use this joke at the start of my presentation:

What a beautiful day! When I was invited to this event, I suspected it’d be close to 20° outside. I just didn’t realize it’d be Celsius.

Horrible, ain’t it? But don’t worry… I’ve got more:

- Do math majors receive degrees or radians?
- If it’s 0° today, and it’s supposed to be twice as cold tomorrow, how cold is it going to be? (Stephen Wright)
- Are you a 45° angle? Because you’re acute-y.
- Why did the obtuse angle go to the beach? Because it was more than 90°.
- Are you cold? Go sit in a corner. It’s 90° over there.
- Why didn’t the circle go to college? It already had 360 degrees.
- I asked the trigonometer what the weather was like, and he said it was 15π/16 outside.
- The number you dialed is imaginary. Please rotate your phone 90° and try again.
- What brand of deodorant did the angle use?
*Degree*.

Patrick —

Thanks so much for sending me

… I’ve already annoyed my entire family, and I’m only 46/111 through the book. (I think your work here is done.)More JokesI can’t provide a formal proof, but empirically speaking, this is the best collection of math jokes known to exist. Honestly, I enjoy the blend of quips and jokes; almost everything I read translates well into a groaner I can splice into my physics classroom repertoire!

Hoping you don’t object to the informality of my “stationery” — thought you’d appreciate the utility of engineering paper.

^{1}Thanks again! Best,

RobP.S. I noticed — when I looked up your address (ed. note:

STALKER!) — that we only live about seven root two^{2}minutes apart. How would you feel about raising a pint some evening? I’d love to hear about your work and exchange a few quips!

This is one of the nicer letters I’ve received, and it was fun to be reminded that not all mail needs to be electronic. And you can bet that some of those compliments will be used on a promotional flyer soon!

^{1} Quarter-inch quadrille paper, if you must know.

^{2} He had correctly written the irrational number as

but since LaTeX looks like shit as inline text, I converted it to straight English.

]]>From Gene Weingarten’s recent column, “Rhymes Against Humanity,” in the January 28 edition of the *Washington Post Magazine*:

An infinite number of mathematicians

Walked into a bar on one recent night,

And, under the strangest of barroom conditions,

What followed quite nearly became a big fight.“I’ll have a pint,” said the first to the ’tender.

“I’ll have a half,” said the next fellow down.

“I’ll have a quarter,” said the third (no big spender).

“Give me an eighth,” said the next, like a clown.The bartender fumed and grew suddenly pale

Then, calmly, he turned and he went to the spout

Drew up two pints, set them down at the rail.

Said, “Enough of this nonsense — you all work it out.”

This is an MJ4MF original, though like Gene’s, it’s based on a stale, old joke:

With my head in an oven

And my feet on some ice,

I’d say that, on average,

I feel rather nice!

What other classic math jokes can be easily converted to poems? Or have already been?

]]>Uterine size in non-pregnant women varies in relation to age and gravidity [number of pregnancies]. The

mean length-to-width ratio conformed to the golden ratioat the age of 21, coinciding with peak fertility.

Claiming that a uterine golden ratio coincides with peak fertility is highly suspect. The good folks at Ava Women claim that, “Most women reach their peak fertility rates between the ages of 23 and 31.” Information at Later Baby states, “Female fertility and egg quality peak around the age of 27.” And WebMD says, “A woman’s peak fertility is in her early 20s.” So, there seems to be some debate about when peak fertility actually occurs. Consequently, this strikes me as retro-fitting, and it seems that Dr. Verguts and his colleagues may have played loose with the age of peak fertility in order to make a connection to the golden ratio.

In their defense, though, it’s not the first time that folks have gone uptown trying to find a connection to the golden ratio. A claim by The Golden Number states, “[The DNA molecule] measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral,” and 34/21 ≈ 1.6190476, which is approximately equal to φ, 1.6180339.

Though this guy — an honest-to-goodness biologist — seems to disagree:

I’ve also heard folks say that people are perceived as more beautiful if certain bodily proportions are in the golden ratio. The most extreme example of this that I’ve found involves the teeth:

…the most “beautiful” smiles are those in which central incisors are 1.618 wider than the lateral incisors, which are 1.618 wider than canines, and so on.

In a study of 4,572 extracted adult teeth, Dr. Julian Woelfel found the average width of the central incisor to be 8.6 mm. If the teeth in a beautiful smile follow the geometric progression described above, well, that would imply that the first molar would be just 8.6 × 0.618^{5} ≈ 0.8 mm wide, which isn’t reasonable and, moreover, is not even remotely close to the average width that Dr. Woelfel found for the first molar: approximately 10.4 mm.

But all of these claims involving the golden ratio are not even close to being the stupidest statistics I’ve heard in my life. Mary Anne Tebedo made a remark on the floor of the Colorado State Senate in 1995 that may hold that distinction:

Statistics show that teen pregnancy drops off significantly after age 25.

Of course, it’s hard to call that a *statistic*, since it’s completely nonsensical. Maybe it’s only the stupidest *statement* I’ve ever heard.

Then there’s this one, from the *New York Times* on August 8, 2016, which couldn’t be more useless:

No presidential candidate has secured a major party nomination after an FBI investigation into her use of a private email server.

Well, duh. Email didn’t even exist before the 1970’s. Moreover, besides Hillary Clinton, has *any* presidential candidate ever had their use of a private server investigated by the FBI? This is like saying, “No one has ever been named *People*‘s Sexiest Man Alive after writing a math joke book.” (Not yet, anyway.)

Randall Munroe made fun of these types of “no politician has ever…” claims in 2012 with his cartoon *Election Precedents*:

And it’s true:

But perhaps my all-time favorite is one that Frank Deford — may he rest in peace — included in his piece “The Stupidest Statistics in the Modern Era” on NPR’s Morning Edition:

He’s [Brandon Phillips] the first National League player to account for as many as 30 steals and 25 double plays in one season.

About this stat, Deford commented, “Steals and double plays together? This is like saying, ‘He’s the first archaeologist to find 23 dinosaur bones and 12 Spanish doubloons on the same hunt.'” (I sure am going to miss him.)

The preponderance of dumb stats shouldn’t come as a surprise, though. A recent study found that people deemed real news headlines to be accurate 83% of the time and fake news headlines to be accurate 75% of the time. So, if we can’t tell truth from fiction, how can we possibly distinguish useful statistics from inane?

If you’d like to test your ability to detect fake news, check out Factitious from American University.

]]>Which of the following numbers don’t belong: 64, 16, 36, 32, 8, 4?

What I liked about this puzzle was the answer posted by Danny Mittal, a sophomore at the Thomas Jefferson High School for Science and Technology. Danny wrote:

64 doesn’t belong, as it’s the only one that can’t be represented by fewer than 7 binary bits.

36 doesn’t belong, as it’s the only one that isn’t a power of 2.

32 doesn’t belong, as it’s the only one whose number of factors has more than one prime factor.

16 doesn’t belong, as it’s the only one that can be written in the form

x, where^{y}xis an integer andyis a number in the list.8 doesn’t belong, as it’s the only one that doesn’t share a digit with any other number in the list.

4 doesn’t belong, as it’s the only one that’s a factor of all other numbers in the list.

I suspect that Danny has visited Which One Doesn’t Belong or has read Christopher Danielson’s *Which One Doesn’t Belong*. Or maybe he’s just a math teacher groupie and trolls MTBoS.

But then Jim Simpson pointed out the use of “don’t” in the problem statement, which I had assumed was a grammatical error. Jim interpreted this to mean that there must be two or more numbers that don’t belong for the same reason, and with that interpretation, Jim suggested the answer was 32 and 8, since all of the others are square numbers.

Don’t get me wrong — I don’t think this is a great question. But I love that it was interpreted in many different ways. It could lead to a good classroom conversation, and it makes me consider all sorts of things, not the least of which is standardized assessments. How many times have students gotten the wrong answer for the right reason, because they interpreted an item on a state exam or the SAT differently than the author intended? And how many times have we bored students with antiseptic questions, only because we knew they’d be free from such alternate interpretations? Both scenarios make me sad.

]]>…well, you might be upset.

But perhaps the student learned to count in binary on her fingers, where the right thumb is the register for 1, the right index finger is the register for 2, the right middle finger is the register for 4, and so on. Then the response above would be appropriate, despite appearances.

If you then asked, “Into how many regions will a circle be divided if 6 points are placed randomly on a circle, and each point is connected to every other point?” the student might appear to wave at you — or, she may just be telling you (correctly) that 31 regions would be created by holding up all 5 fingers. (In binary counting, all five fingers add up to 1 + 2 + 4 + 8 + 16 = 31.)

My sons learned to count in binary when, at age 5, they asserted that the highest you can count on your fingers is 10. “Actually,” I told them, “You can count as high as 1,023 on your fingers. If you want, I can show you how.”

Of course, they wanted to learn, and I was happy to teach them. There are at least four good reasons for teaching students to count in other bases, and “Dr. Peterson” at the Math Forum had this to say:

I taught my son to multiply in binary before he really learned it in decimal, because it’s easier; you have only the algorithm (method) with no multiplication tables to learn.

Knowing how bases work helps to develop number sense while clarifying the concept of place value. And not understanding place value leads to things like this…

My former boss shared this video with me on Facebook recently, and he asked,

Does this work with other numbers?

I had a fun time playing with that question, so let me now give you a chance to think about it. Can you find another pair of numbers that produce analogous incorrect results when multiplying and dividing? And if you’re feeling really ambitious, can you generalize to determine what types of number pairs will always give these kinds of incorrect results?

]]>More than 200 Kenthusiasts — people who love KenKen puzzles — descended on Will Shortz’s Westchester Table Tennis Center for the 2017 KenKen International Championship (or the KKIC, for short). Participants followed 1.5 hours of solving KenKen puzzles with a pizza party and several hours of table tennis.

The competition consisted of three rounds, with the three puzzles in each round slightly larger and more difficult than those from the previous round. Consequently, competitors were given 15, 18, and 20 minutes to complete the puzzles in the first, second, and third rounds, respectively.

Competitors earned 1,000 points for each completely correct puzzle, and 0 points for an incomplete or incorrect puzzle. In addition, a bonus of 5 points was earned for every 10 seconds in which a puzzle was turned in before time was called. So, let’s say you got two of the three puzzles correct and handed in your answers with 30 seconds remaining in the round; then, your score for that round would be

The leader after the written portion was John Gilling, a data scientist from Brooklyn, whose total score was 10,195. And if you’ve been paying attention, then you know what that means — Gilling earned 9,000 points for completing all of the puzzles correctly, so his time bonus was 1,195 points… which is the amount you’d earn for turning in the puzzles 2,390 seconds (combined) before time was called. The implication? Gilling solved all 9 puzzles from the written rounds — which contained a mix of puzzles from size 5 × 5 to 8 × 8 — in just over 13 minutes.

Wow.

As a result, Gilling, the defending champion, earned a spot in the Championship Round against Tess Mandell, a math teacher from Boston; Ellie Grueskin, a high school senior at The Hackley School; and Michael Holman, a technology consultant. In the final round, each of them attempted a challenging 9 × 9 puzzle, which was displayed on an easel for the crowd to see. Solving a challenging 9 × 9 is tough enough; having to do it as 200 kenthusiasts follow your every move is even tougher.

So, how’d they do? See for yourself…

When the dust settled, Gilling had successfully defended his title. For his efforts, he received a check for $500. But more importantly, he retained bragging rights for one more year.

If you think you’ve got what it takes to compete with the best KenKen solvers, try your hand at the 9 × 9 puzzle that was used in the final round. In the video above, you saw how fast Gilling solved it to win the gold. But even the slowest of the four final-round participants finished in under 15 minutes.

Again, wow.

Finally, I’d be failing as a father if I didn’t mention that my sons Alex and Eli competed in the Delta (age 10 and under) division. Though bested by Aritro Chatterjee, a brilliant young man who earned a trip to the 2017 KKIC by winning the UAE KenKen Championship, Eli took the silver, and Alex brought home the bronze. They’re shown in the photos below with Bob Fuhrer, the president of Nextoy, LLC, the KenKen company and host of the KKIC.

#proudpapa

For more KenKen puzzles, check out www.kenken.com, or see my series of posts, A Week of KenKen.

]]>In honor of its anniversary, let’s start with a quiz.

What are the names of the Seven Dwarfs?

If you said Blick, Flick, Glick, Plick, Snick, Whick, and Quee, you’d be correct. What? Of course, those aren’t the names of the dwarfs in the Disney movie, but apparently those were the names used in a 1912 theater adaptation of the original Brothers Grimm fairy tale.

Okay, let’s be a little more fair.

What are the names of the dwarfs in the 1937 Disney movie Snow White and the Seven Dwarfs?

In the image above, left to right, the dwarfs are Bashful, Happy, Dopey, Sleepy, Doc, Grumpy, and Sneezy.

The number seven is ubiquitous, possibly even more popular than Cristiano Ronaldo or Kylie Jenner. Lots of things come in groups of seven, like dwarfs, samurai, games in the World Series, and — appropriate for this time of year — swans a-swimming.

So for your enjoyment, here you go: an entire quiz dedicated to groups of seven.

- What are the seven wonders of the world?
- What are the seven words you can’t say on TV, according to George Carlin? (NSFW)
- What are the Seven Seas?
- What are the Seven Habits of Highly Effective People?
- Sherwood Schwarz, creator of the TV show
*Gilligan’s Island*, said that each character on the island corresponds to one of the seven deadly sins. Can you name the characters, name the sins, and form a one-to-one correspondence between them?

‘Tis the season of social events. Feel free to use any of those trivia questions at a holiday party near you. And if the crowd at your gathering prefers jokes to quizzes, well, here are a few that involve dwarfs or the number seven:

Why is 6 afraid of 7? Because 7 8 9.

How do you make 7 even? Take away the ‘s’.

Bob has seven daughters, and each daughter has a brother. How many children does Bob have? Eight.

I got in a car crash the other day. A dwarf got out of the other car and said, “I’m not happy.” To which I replied, “Then which one are you?”

And if you need something just a bit more risque…

]]>Why did Happy get out of the hot tub? Because the other dwarfs were feeling happy.

As far as I’m concerned, the Christmas Blend — not to be confused with Holiday Mint, which uses a (disgusting) mint chocolate filling — is one of just a few acceptable color combinations. Why? Because it uses colors that can only be found in the original Plain M&M packs, which contain red, orange, yellow, green, blue, and brown.

The original packs didn’t contain white M&M’s — sorry, Freedom Blend (Fourth of July). The original packs didn’t contain pastel colors — hop on by, Easter Blend. And nowhere on God’s green Earth will it ever be acceptable to use white chocolate inside those delectable candy shells — hit the road, Carrot Cake M&M’s. (Yuck.)

As you can tell, I’m a purist, and I have fairly strong opinions about this.

To my knowledge, there are only two other blends produced by Mars, Inc., that satisfy my acceptability criteria:

- Harvest Blend: red, yellow, brown
- Birthday Cake: red, yellow, blue

So, where am I going with all this? Glad you asked.

The Christmas, Harvest, and Birthday Cake blends represent just three of the 63 possible color combinations that can be made from the original six colors. That leaves 60 combinations that are just begging for names.

(A little history. As you may know, I have a quirk. I eat M&M’s in pairs of the same color, so I can place one on each side of my mouth and feel “balanced.” But it’s atypical for a pack to contain an even number of every color. When I near the end, I’m often left with one to six unmatched M&M’s. And I’ve always thought that these various color combinations deserved a name.)

What would you call a combination of red, yellow, and green? Obviously, STOPLIGHT.

What might you call a combination of red, yellow, and blue? Based on the Man of Steel’s outfit, I like SUPERMAN. But Mars, Inc., has already applied the moniker BIRTHDAY CAKE.

What would you call a collection of just green M&M’s? I don’t know — QADDAFI, maybe? (Sorry, dated reference.)

What would you call a combination of orange, green, and brown? I have no idea.

And that’s where you come in.

Below is a Google poll where you can enter a color combination and suggest a name. In early January, for any color combinations that have more than one suggestion, we’ll vote on it. That’s right — crowdsourcing, baby!

But before you scroll and start clicking, let me lay out some ground rules:

- Keep it clean, please, no worse than PG-13.
- No sports teams! Why? Because the Pittsburgh Steelers, Pirates, and Penguins are black and gold… and although yellow is close to gold, there are no black M&M’s in the Plain M&M’s pack, so that combination is not possible. If M&M’s can’t be used to represent my team, then they can’t be used to represent any team. Sorry — my game, my rules. Not to mention, nearly every color combination corresponds to at least one sports team, so it also demonstrates a lack of creativity. Unless, of course, you pick the colors of a team from the Swedish Bandyliiga, but let’s be honest — were you really going to do that?

Some time ago, I tried to craft names for all the combinations on my own, but I failed miserably. You can see how far I got on **this Google sheet**. So you can tell that I really, really need your help.

Have at it, y’all!

If you can’t see the form below, click this link:

**https://goo.gl/forms/jiCEClAMSDTJtHGZ2**

Don’t want to goof around with a Google form? Fine. Place your thoughts in the comments.

]]>