## Posts tagged ‘time’

### 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:

$t = \begin{cases} 20w + 70 & w < 4; \\ 20w + 90 & w \geq 4, \end{cases}$

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:

Cooking Times at 325° F from delish.com

which translates to the lovely formula

$t = \begin{cases} 5w + 125 & w \leq 10; \\ 15w & w = 12; \\ 7.5w + 120 & w \geq 14, \end{cases}$

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:

$\displaystyle t = T \times \left( \frac{w}{20} \right)^\frac{2}{3}$

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

$\displaystyle t = 36.64w^\frac{2}{3}$

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 turkey in 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.

$\displaystyle t = \frac{2}{3} w^\frac{2}{3}$

and when converted to minutes instead of hours, this becomes

$\displaystyle t = 40w^\frac{2}{3}.$

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 log10 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!

### AWOKK, Day 3: KenKen Times

Today is Day 3 in MJ4MF’s A Week of KenKen series. In case you missed the fun we’ve had previously…

Yesterday, I introduced you to the KENtathlon.

While completing a KENtathlon, my goal is to complete a 6 × 6 puzzle in less than 2 minutes; a 5 × 5 puzzle in less than 1 minute; and a 4 × 4 puzzle in less than 20 seconds. Even though the sum of those times for all three puzzles is 3 minutes, 20 seconds, my goal is a combined time of 3 minutes. It’s good to have goals.

 Puzzle Size Goal Time Personal Best 4 × 4 0:20 0:12 5 × 5 1:00 0:27 6 × 6 2:00 1:29 KENtathlon 3:00 2:32

I don’t always perform well enough to meet those goals. And when I don’t, I repeat the same size puzzle again… and again… and again… for as many attempts as it takes to complete each puzzle in the allotted time. And when I’ve met the time goal for each puzzle individually, if the combined time isn’t satisfactory, then I start the whole thing over.

To say that I’m slightly obsessive would be like saying that the Pope is a little bit Catholic.

As you may have noticed in the table, I once finished a 4 × 4 puzzle in 12 seconds. The key word there is once. The stars were in alignment that day — it was an easy puzzle, and the dexterity of my thumbs and fingers was at an all-time high. Though I’ve attained 13 a handful of times, I’ve never replicated that 12-second feat.

My fastest puzzle ever,
not Photoshopped.

That said, I regularly complete 4 × 4 puzzles in 14 or 15 seconds. With that being the case, you have to wonder if the 20-second goal is really a challenge. And what about the goal times for 5 × 5 and 6 × 6 puzzles?

Admittedly, my time goals are arbitrary, though not random. When I chose those goals, I had completed enough KenKen puzzles that I intuitively knew what felt right. Still, it wasn’t based on hard data… and if you’ve read this blog long enough, you know that that bothered me. A lot.

But what’s a boy to do?

I suppose a well-adjusted human might do nothing, think it’s not worth the trouble, and just let the whole thing go. But an obsessive numbers guy? Well, he’d painstakingly solve 132 KenKen puzzles, collect data on the amount of time each one took to complete, meticulously record the data in an Excel spreadsheet, and perform a thorough analysis. You may think that undertaking such a project is ludicrous; but to me, it was absolutely essential.

The graph below shows the results. The circular dots represent my median time for each puzzle size, and the square dots represent the upper and lower quartiles. For instance, the median time for 6 × 6 puzzles was 217 seconds, while the interquartile range for 6 × 6 puzzles extended from 163 to 284 seconds.

What this reveals is that my intuition wasn’t perfect, but not bad.

• I completed 49% of 4 × 4 puzzles in less than the goal time of 20 seconds.
• I completed 58% of 5 × 5 puzzles in less than the goal time of 1 minute.
• But, I completed only 14% of 6 × 6 puzzles in less than the goal time of 2 minutes.

Further analysis revealed that I completed 40% of the 6×6 puzzles under 3 minutes, and that seems a bit more reasonable, so my new goal time for 6 × 6 puzzles is 3 minutes.

Now, I know you thought this analysis was completely unnecessary, but the proof is in the pudding. The results were invaluable. By considering the data, interpreting the results, and revising my goal time for 6 × 6 puzzles, the probability that I can now complete each size puzzle in the allotted time on the first or second try has increased from 16% to 39%. Or said another way, Remy’s morning walks now last an average of just 15 minutes, whereas some of them used to take an hour-and-a-half.

### Mathematically Unconscious

Both of my sons sleepwalk. At least once a week, one of them will wake up an hour after bedtime, walk down the stairs, and start speaking gibberish. They have no idea what they’re saying, because they aren’t awake — even though their eyes are open. (Freaky!)

During a recent somnambulation, Alex stood at the top of the stairs. He appeared frustrated. Finally, he said:

I just need to find the numbers. It shouldn’t take long.

As you might well imagine, it’s a little scary to have your son walking and talking while asleep. The only solace is that his subconscious thoughts are about math.

I don’t sleepwalk. But I recently had a dream in which I attended a cocktail party and asked the other attendees a most unusual question:

I suspect that my 7 years as an editor and 4 years as a question writer for MathCounts are to blame, but that doesn’t make it any easier to accept.

I vividly remember a dream I had in college, on the night prior to my Linear Algebra midterm. Feeling unprepared for the exam, my nightmare consisted of two brackets pinching my head like a vice, while numbers floated past.

I awoke in a cold sweat at 5 a.m., and proceeded to a study carrel for more test prep.

I was happy to learn that other folks dream about math, too. While subscribed to a listserve for former instructors of the Center for Talented Youth, I received a message from Mark Jason Dominus that read, “I dreamt of the following problem while I was sleeping last night. When I woke up, I convinced myself that it was a good problem, so I’ve decided to share it.”

The volume of a 3 × 3 × 3 cube is 27 cubic units, and the volume of a 2 × 2 × 1 rectangular prism is 4 cubic units. Theoretically, six prisms should be able to fit inside the cube, with three cubic units empty. But can you arrange six 2 × 2 × 1 prisms so they fit inside a 3 × 3 × 3 cube?

Good luck, and sweet dreams!

### Eat, Sleep, Do Math!

The Golden Rule of Food Shopping:

Never shop for groceries when you’re hungry.

Corollary for Mattress Shopping:

Never shop for a mattress when you’re tired.

When buying a mattress, Consumer Reports recommends that you lie down on “lots of mattresses” in the store and spend at least 15 minutes on each mattress — five minutes lying on each side, and another five minutes on your front or back, depending on your sleeping preference. I’m not certain what number is implied by “lots of mattresses,” and I’ve never been very good at math, but if you try out 6-8 different mattresses for 15 minutes each, plus some chit-chat and the requisite haggling with a salesperson, you’re trip to the mattress store is going to last at least an entire afternoon, maybe more.

This is a mattress, but
mathy folks sleep on matrices.

They also recommend that you wear loose-fitting clothes, so I donned a smoking robe and slippers. Our family then headed to Sleepy’s.

The first mattress I tested was too firm. It took far less than 15 minutes to eliminate it as a possibility.

The second mattress I tested, however, was damn near perfect. I rested on my left side for five minutes, and it felt very good. I then rolled over to my right side… and I fell asleep. Not sure what to do, my wife did what any dedicated wife would do — she left. She and the kids walked to the grocery store, and when they returned 35 minutes later, I was rousing from my slumber.

“This is the one,” I said.

“Yeah,” she said. “No shit.”

I did not need to test any more. That said, the one I liked was far from the cheapest one on the showroom floor. Consequently, haggling ensued. As I was asking for a 25% discount and the salesperson was countering with, “How ’bout I throw in a free pillow?” my sons were inspecting a poster in the store:

The intent of the poster, of course, is to show that Americans spend 1/3 of their lives in bed. (And, implicitly, to suggest that price should not be a consideration for something you use so often.) But it caused some slight bewilderment for my sons.

Only 21.8 hours are accounted for.

If there had been no category called “Other,” it might not have been so odd. But couldn’t they have included the missing 2.2 hours in “Other”? Unless that time is spent doing something other than “Other,” but I have no idea what that might be.

If this information were represented as a pie chart, it might look like this:

The source of the statistics, according to a footnote on the poster, is the Bureau of Labor Statistics. But that doesn’t seem true. At the top of this data table from BLS, the sum total of all activities is 24.00 hours.

My job is done here. I’m off to enjoy my new mattress. Good night.

### Exponentially Smarter, Literally

To show my sons what Siri can do, I asked her (it?) the following question:

What is 6 + 4?

Siri told me, “The answer is 10.” But she also provided a bunch of other information pulled from Wolfram Alpha, including the following data:

This data appears to be taken from dissertation research by B. A. Fierman which was furthered by psychologist Mark H. Ashcraft. What it shows is that we get exponentially smarter — or at least faster at calculating — as we get older.

According to Excel, this data can be modeled exponentially by y = 8.36 · e–0.129x, though this model has obvious limitations. For example, it implies that a one-year-old would be able compute this sum in 7.35 seconds, yet I know no one-year-old who understands addition. Further, it claims that it would take me 0.03 seconds to compute the sum, but I would argue first that I don’t compute the sum, I merely recall it; and second, my reaction time when asked for the sum would be greater than 0.03 seconds.

Playing around with the generic function y = abx + c using the world’s best graphing calculator from Desmos, I found a model that may approximate the data a little better:

y = 57 · 0.65x + 0.9

With this model, it would take a one-year-old 37.95 seconds to compute sum. That’s still not reasonable for any one-year-old that I know, but at least the model says it would take me 0.9 seconds to recall the fact, a far more reasonable estimate than the 0.03 seconds given by the Excel model above.

Interestingly, How To Geek claims that Siri uses Wolfram Alpha for 25% of its searches. Yet if you ask Siri, “What is the meaning of life?” it will respond,

I can’t answer that right now, but give me some very long time to write a play in which nothing happens.

or

Try and be nice to people. Avoid eating fat. Read a good book every now and then, get some walking in, and try to live together in peace and harmony with people of all creeds and nations.

On the other hand, if you ask Wolfram Alpha, “What is the meaning of life?” it will respond,

42.

Proper.

All this talk of exponentials reminds me of a joke.

Q: How do you know that your dentist studied algebra?

A: She tells you that candy will lead to exponential decay.

Perhaps the most famous joke about exponentials is not one of which I’m terribly fond. I share it here only to honor my mission of providing math jokes to the world, not because I think any of you will enjoy it.

Several functions are sitting in a bar, bragging about how fast they go to zero at infinity. Suddenly, one hollers, “Look out! Derivation is coming!” All of the functions immediately cower under the table, but the exponential function sits calmly on the chair.

The derivation comes in, sees the exponential function, and says, “Don’t you fear me?”

“No, I’m ex,” says the exponential confidently.

“That’s all well and good,” replies the derivation, “but who says I differentiate with respect to x?”

### What Time Is It?

Here’s a math puzzle that is rather easy. Or is it?

You look in a mirror and see the reflection of a clock. In the reflection, the clock appears to show a time of 11:51. What is the real time?

Before I share the solution, how ’bout some clock jokes?

A hungry clock goes back four seconds.

I spent 35 minutes fixing a broken clock yesterday.
At least, I think it was 35 minutes…

What’s the difference between a man and a broken clock?
At least a broken clock is right twice a day.

It took me four hours to eat a dozen clocks.
It was very time consuming.

My clock stopped at 8:23 a.m. I’m going to have a day of morning.

Puzzle Solution

The puzzle above is based on a math trivia question I found at Trivia Cafe.

The answer could be 12:09, if the reflection in the mirror looks like this…

Then again, the answer could be 12:11, if the reflection in the mirror looks like this…

And of course, there are all the silly possibilities — for instance, if the clock is broken, it doesn’t matter what time shows in the reflection, regardless if it’s analog or digital.

### Excuses Are Like Graphing Calculators…

You may have noticed that there haven’t been very many new posts on this blog recently. I apologize for that. The following flowchart — an idea blatantly stolen from Brewster Rocket — provides my excuse.

Since starting a new job in March, I’ve been working 60-80 hours per week. I’m also serving as the chair of the MathCounts Question Writing Committee. Mix in the time demanded by two energetic, six-year-old boys, and, well, that doesn’t leave a lot of time for making math people laugh on the Internet. Don’t get me wrong — I’ve been funny as hell the past six months, both creating and delivering amazing one-liners. I just haven’t had time to type them up for all of you.

Not that you care about any of that. You come here for jokes, not excuses.

The scientist asks, “Why does it work?”
The engineer asks, “How does it work?”
The project manager asks, “How much will it cost?”
The novelist asks, “Do you want fries with that?”

And here are 11 excuses I could have used, but didn’t:

1. I created a great joke but then divided by zero, and the joke burst into flames.
2. It’s Isaac Newton’s birthday.
3. I could only get arbitrarily close to my computer. I couldn’t actually reach it.
4. I had a really funny joke to share, but this blog is too narrow to contain it.
5. I was watching the World Series and got tied up trying to prove that it converged.
6. I have a solar-powered laptop, and it was cloudy.
7. I wrote some jokes in a notebook and locked them in my trunk, but a four-dimensional dog got in and ate it.
8. I was typing up some jokes when my wife brought me a doughnut and a cup of coffee. I spent the rest of the night trying to figure out which was which.
9. I put some jokes in a Klein bottle, but then I couldn’t find them.
10. I was too busy celebrating the coincidence of Einstein’s birthday and Pi Day.
11. I was contemplating a formula for Phi Day, determining the first Friday the 13th in 2013, and wondering why Tau Day isn’t more popular than Pi Day.

The Math Jokes 4 Mathy Folks blog is an online extension to the book Math Jokes 4 Mathy Folks. The blog contains jokes submitted by readers, new jokes discovered by the author, details about speaking appearances and workshops, and other random bits of information that might be interesting to the strange folks who like math jokes.

## MJ4MF (offline version)

Math Jokes 4 Mathy Folks is available from Amazon, Borders, Barnes & Noble, NCTM, Robert D. Reed Publishers, and other purveyors of exceptional literature.