## Four, or F**k You?

If you asked a student, “How many sides does a quadrilaterals have?” and you received the following response…

…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?

## 2017 KenKen International Championship

If you like puzzles and ping pong, then Pleasantville, NY, was the place to be on December 17.

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.

## Four Score and Seven Dwarfs Ago…

*Snow White and the Seven Dwarfs* was released 80 years ago today. It was America’s first feature length animated film, the first to be produced in English, and the first in Technicolor. It was the most successful film of 1938, and, when adjusted for inflation, is the tenth highest-grossing film of all time.

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.

## Red + Green = Christmas, and 62 Other M&M Color Combinations

‘Tis the holiday season, so every grocery store, pharmacy, and convenience store is now stocking the M&M^{®} Christmas Blend, a joyful combination of red and green button-shaped chocolate candies. It’s unclear whether this mixture actually helps to imbue the holiday spirit, but the consumption of these tasty morsels will make you look just a little more like St. Nick.

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.

## Fowl Formulae for Thanksgiving

Okay, I know you’re going to find this hard to believe, but **there is disagreement on the internet**. And I don’t mean about some insignificant topic like gun control or taxes or health care or the value of 6 ÷ 2(1 + 2). This is big. This is important.

We’re talking turkey. Literally.

According to the **British Turkey Information Service** — yes, there really is such an agency — the amount of time you should cook your turkey at 375° F can be found with the following formula:

where *t* is the cooking time in minutes and *w* is the weight of the turkey in kilograms.

If you’d rather not do the math yourself, try the British Turkey Cooking Calculator, which will not only give you the cooking time but also the defrosting time and the size of turkey to buy for a given number of servings.

By comparison, the Meat Chart provided by **FoodSafety.org** says that turkey should be cooked at 325° F for 30 minutes per pound.

But the cooking website** allrecipes.com** says that only 20 minutes per pound is sufficient if you bake the bird at 350° F.

Whereas the good folks at **delish** offer the following guidelines:

which translates to the lovely formula

but requires that you interpolate if your bird weighs an odd number of pounds. (Like 86 pounds, the world record for heaviest turkey ever raised. Even though the units digit is 6, you’d agree that 86 is an odd number of pounds for the weight of a turkey, no?)

As you might suspect, **Wolfram Alpha** has a more mathematically sophisticated formula:

where *t* is the cooking time in hours, *w* is the weight in pounds, and *T* is a coefficient to account for cooking environment. For normal conditions, *T* = 4.5, and the equation reduces to

if you use minutes instead of hours for the unit of time.

But this feels a little like a math joke; below the formula, Wolfram offers the following:

using the heat equation for

a spherical turkeyin a 325° F oven

Falling into the wrong hands, that idea could lead to an horrendous modification of the spherical cow joke…

The turkeys at a farm were not gaining sufficient weight in the weeks leading up to Thanksgiving, so the farmer approached a local university to ask for help. A theoretical physicist was intrigued by the problem and offered his assistance. He spent several weeks at the farm, examining the turkeys and filling his notebook with equation after equation. Finally, he approached the farmer and said, “I have found a solution.”

“Oh, that’s excellent!” said the farmer.

“Yes,” said the physicist. “Unfortunately, it only works for spherical turkeys in a vacuum.”

The Wolfram formula is very similar to one suggested by physicist **Pief Palofsky**, who apparently dabbled in poultry when not winning the National Medal of Science.

and when converted to minutes instead of hours, this becomes

According to *Turkey for the Holidays*, the average weight of a turkey purchased at Thanksgiving is 15 pounds. The cooking times for a 15-pound bird, based on the formulae above, appear to have been chosen by a random number generator.

Recommender |
Time (min) |
Temp (° F) |

British Turkey Information Service | 226 minutes | 375 |

Foodsafety.org | 450 minutes | 325 |

allrecipes.com | 300 minutes | 350 |

delish.com | 233 minutes | 325 |

Wolfram Alpha | 223 minutes | 325 |

Pief Palofsky | 243 minutes | 325 |

Even if you limit consideration to those who suggest a cooking temp of 325° F, the **range of times still varies from just under 2¾ hours to a staggering 7½ hours**. Wow.

With Thanksgiving just around the corner, where does all of this contradictory information leave us?

A number of sites on the internet claim that the only way to adequately check the doneness of a turkey is with a meat thermometer.

The folks at recipetips.com claim that a turkey can be removed when the temperature is at least 170° F for the breast and 180° F for the thigh. Yet on the very same page, they claim, “Turkey must reach an internal temperature of 185° F.”

On the other hand, the folks at the Food Lab claim a turkey can be safely removed when the breast temperature reaches 150° F, because after resting 15‑20 minutes before carving, the amount of remaining bacteria will be minimal. They explain, “What the USDA is really looking for is a 7.0 log_{10} relative reduction in bacteria,” particularly Salmonella, which means that only 1 out of every 10,000,000 bacteria that were on the turkey to start with will survive the cooking process. And according to the USDA guidelines, a turkey that maintains a temperature above 150° F for 3.8 minutes or longer will reach that threshold for safety.

Which has to make you wonder — if 3.8 minutes at 150° F is supposedly adequate, why then does the USDA Food Safety and Inspection Service recommend that the minimum internal temperature of the turkey in the thigh, wing, and breast should be at least 165° F? Who knows. I suspect it’s typical government over-engineering to remove all doubt.

So, how long should you cook your turkey? Hard to say. But if you put your turkey in the oven right now, it should be done by November 23.

When the turkey is finally ready, here are a few math jokes you can tell around the Thanksgiving table.

What do math teachers do on Thanksgiving?

Count their blessings!What does a math teacher serve for dessert on Thanksgiving?

Pumpkin Pi.How do you keep private messages secure on Thanksgiving?

Public turkey cryptography.Thanksgiving dinners take 18 hours to prepare. They are consumed in 12 minutes. Halftimes take 12 minutes. This is not coincidence.

~ Erma Bombeck

Gobble, gobble!

## The Virginia Governor’s Race and MP.6

Hotly contested. Particularly nasty. Ugly. Uncivilized. Controversial. Disgusting. Dirty.

Those are just some of the words and phrases used to describe this year’s race for governor in my home state of Virginia. It’s part of the reason that we had the highest voter turnout for a gubernatorial election in 20 years. It’s also why every citizen in the Old Dominion was anxiously awaiting the results.

I was no different. At 8:30 p.m. EST, I strolled on over to POLITICO, where I was presented with more information that I could handle:

It was surprising that Northam had a 5.8% lead, since some recent polls suggested that his lead had dwindled to as little as 3.3%. It was surprising that only 90 minutes after the polls had closed, many news organizations were already declaring Northam the winner. But it was outright astonishing that POLITICO was displaying the percent of precints reporting as:

72.68121590023384%

WTF?

On Feburary 13, 2011, in a post titled Statistically Speaking…, I presented the following joke:

69.8724% of all statistics reflect an unjustified level of precision.

Three years later, in a post titled Sound Smart with Math Words, I presented another version of that same joke, though this time the percentage was expressed to the millionths:

An unprecedented 69.846743% of all statistics reflect an unjustified level of precision.

Did I think the additional precision would make it more obvious that the sentence was actually a joke? Or did I just think it would make it funnier? I’m not sure.

But I do know that it would never have occurred to me to take the level of precision to 14 — count them, 14! — digits of accuracy beyond the decimal point.

But POLITICO thought it was necessary.

That’s right. They calculated and displayed the percent of precincts reporting **to the hundred-trillionths place**. Hundred. Trillionths.

That’s like stating the winning time for the Tour de France to the nearest millisecond.

Or estimating the weight of the Earth to the nearest gram.

In fairness to POLITICO, though, the percentages that they were reporting not only reflected an unjustified level of precision. They were also wrong.

According to the Virginia Department of Elections, there are 2,567 precincts in the commonwealth. If 1,865 precincts had submitted results, the percent of precincts reporting could have been displayed as:

72.6529%

If 1,866 precincts had turned in their results, the percent of precincts reporting could have been displayed as:

72.6918%

But there is no number of precincts for which the percent could have been reported as:

72.68121590023384%

So, either POLITICO was using an incorrect denominator, or their algorithm was incorrectly calculating percentages.

Oh, well. At least they got the election results correct. (I hope.)

In their defense, they did finally make a correction. When I checked the results at 9:24 p.m. EST, this is what was presented:

The percentage of precincts was displayed as a more reasonable 97.74%. From this, I can surmise that 2,509 precincts had reported their results (since 2,509 / 2,567 = 0.9774) and that POLITICO had finally found someone who was nimble with a slide rule.

## Interactions with Fractions

This math factoid, compliments of Learn Fun Facts, is just too good not to share…

Ah, but maybe you missed it. Did you notice that each digit 1-9 appears exactly once in all three fractions? Pretty cool. But I have to say that this is still my favorite fraction equation:

Simple. Beautiful.

On the other hand, I’m not sure I have a favorite fraction joke. I mean, how do you pick just one? The number of fraction jokes is a lot like

.

That’s right. There’s no limit.

5 out of 4 people have trouble with fractions.

I will express polynomials as partial fractions. I will compute the value of continued fractions. I will even find a least common denominator. But I draw the line between the numerator and denominator.

What is one-fifth of a foot?

A toe.How many tents can fit in a campground?

Ten, because ten tents (tenths) make a whole!

Before you go, here’s a fun little fraction problem for you:

**What is 1/2 of 2/3 of 3/4 of 4/5 of 500?**