## Posts tagged ‘number’

### Another Bad Email Math Puzzle…

It happened again. I received another email with a number trick that makes the ubiquitous claim, “This will work for everyone!” Sadly, it won’t, but it was kind of cool:

Calculate 39 × (your age) × 259.

The email said, “The result will surprise you.”

It didn’t. I suspected what the value of 39 × 259 would be, so I predicted the result. But if you don’t know the value of that product, then maybe you’ll be surprised.

The trick works well enough if you have a double-digit age. But my friend Ferdinand is 107 years old. His result was 1,080,807, and that just looks like a mess. The results for my six-year-old sons were better, albeit rather unsatisfying.

Ha-rumph. So much for the Internet providing mathematical inspiration.

Here are some similarly uninteresting puzzles that I created:

Calculate 7,373 × (your age) × 137.

Calculate 9,091 × (your age) × 11,111.

Calculate 101 × (your age) × 1,000,100,010,001.

To create more puzzles like this, enter **factor(101010…10101)** into Wolfram Alpha.

For your centenarian friends, try these:

Calculate 101,101 × (your age) × 9,901.

Calculate 3.3 × (your age) × 33.67.

Let’s not forget the little people whose age is still in the single digits:

Calculate 3 × (your age) × 37.

Calculate 41 × (your age) × 271.

And a math joke (or is it?) about age:

I’ve been good with numbers my whole life. When I turned 2, I realized that my age had doubled in one year. This concerned me… at that rate, I’d be 32 in four more years!

And another:

What goes up but never comes down?

Your age.

### A Muppet You Can Count On

A big MJ4MF thanks to Lindsey Witcosky, who directed me to a wonderful BBC article about one of my favorite Sesame Street characters, the Count! The link to the article is provided below, but first a quiz based on some trivia in the article.

**1.** What was the Count’s full name?

**2.** What was the Count’s favorite number?

**3.** Who was the voice of the Count from 1970 until 2011?

Answers are below, but you can also find them in the BBC article:

http://www.bbc.co.uk/news/magazine-19409960

The Count’s favorite number is equal to 187^{2}, and BBC Radio asked listeners of the show *More or Less* to speculate why. One listener noted that 187 = 94^{2} ‑ 93^{2} and, of course, 187 = 94 + 93. The BBC article referred to this coincidence as, “An embarrassment of riches!” But I prefer to think of it as, “An embarassment of algebra!”

Algebra can be used to show why this is true. The *n*th square number is equal to the sum of the first *n* positive odd integers. That is,

*n*^{2} = 1 + 3 + 5 + 7 + … + (2*n* ‑ 1)

From this it follows that

94^{2} = 1 + 3 + 5 + 7 + … + (2 × 94 – 1)

and

93^{2} = 1 + 3 + 5 + 7 + … + (2 × 93 – 1)

so of course

94^{2} – 93^{2} = 2 × 94 – 1 = 187

Moreover, the difference of two squares is equal to the product of the sum and difference of the two numbers. That is,

a^{2} ‑ b^{2} = (a + b)(a ‑ b)

Consequently,

187 = 94^{2} ‑ 93^{2} = (94 + 93)(94 ‑ 93) = (94 + 93)(1)

So, saying that 187 = 94^{2} ‑ 93^{2} = 94 + 93 is kind of like saying the same thing twice, just in different ways.

**Answers**

**1.** Count von Count

**2.** 34,969

**3.** Jerry Nelson, who passed away on August 23. R. I. P.

### Random Number Generation

While playing Nurikabe, my sons completed the following puzzle:

The puzzle itself isn’t very interesting, but did you notice the Puzzle ID? Exactly 1,000,000. The boys thought this was pretty cool, and I did, too. Yeah, yeah, I know, the occurrence of 1,000,000 shouldn’t impress me more than the appearance of, say, 8,398,176 or 3,763,985. But there are just under 10,000,000 unique 5 × 5 puzzles on the site, and only nine of them contain six 0’s. How lucky were we to get that random number?

Generating random numbers can be a difficult proposition, especially for a computer. This article from WIRED magazine — which describes a pattern that inadvertently appeared on lottery tickets, making it possible to predict winning tickets before they were scratched — shows how difficult it can be to generate numbers that appear to be random. (The article really is worth a read, especially for math geeks. Truth be known, WIRED is the only magazine that I read cover-to-cover every month.)

Robert Coveyou, a mathematician who worked on the Manhattan project, was an expert in pseudo-random number generators. He is most famously remembered for the following quote:

The generation of random numbers is too important to be left to chance.

Of course, Randall Munroe at xkcd has a foolproof method for generating a random number:

I would hate for you to need a random number and then have difficulty generating one. I’m here to help, so I present the…

**MJ4MF Random Number Generator (PDF)**

Creating the MJ4MF RNG is quite simple. Just follow these steps:

- Download and print the PDF from the link above.
- Cut out all six squares, one for each number 1-6.
- For each square, make two folds: first, fold the paper to the center vertically; then, fold the paper to the center horizontally. The result of these two folds is shown, below left.
- When all six pieces are folded, interlace them to form a cube. This is shown, below middle. The assembled cube is shown, below right.

Finally, a joke about random numbers.

A student is asked for the probability that a random number chosen between 0 and 1 will be greater than 2/3. The student answers 1/3. The teacher says, “Great! Can you explain to the class how you arrived at your answer?” The student says, “There are three possibilities: the number is either less than, equal to, or greater than 2/3, so the probability is 1/3!”

### My Favorite Game, Social Media Style

Inspired by Planet Money’s Pick A Number contest, and buoyed by a story about how NCTM President Mike Shaughnessy recently used my favorite game with a group of students at Albuquerque Academy, I’ve decided to conduct an online experiment using a Google Docs form.

If you’ve got a minute and are willing to participate, read on.

The rules for my favorite game are as follows:

- On a piece of paper, everyone playing writes down a positive integer.
- Show your number to a neighbor (for verification purposes only).
- The winner is the person who wrote down
*the smallest integer not written by anyone else*.

In order for this psychological math strategy game to be any fun, you need one important piece of information — how many people are playing. If played as a solitaire game, you *should* win every time. But if played with a group of 50, well, some real thought will need to go into your choice. Consequently, I’m going to limit the game to 100 players. (Well, sort of. What I’m actually gonna do is break the total number of responses into groups of 100, and I’ll consider each set as a separate game. So it’s not exactly the same, but this should allow you to play using the same strategy as if you were playing with just 99 other people.)

For this online version, the second step of the rules — show your number to a neighbor — is unnecessary. So all you need to do to play is enter your number. (I’ve also asked for your name and email address, too, just so I can give you proper credit and contact you if you win. But those are optional. If you do supply your email address, cross my heart, there will be no spam or third‑party solicitations.)

**[Update]** This game was originally run for one week, Nov 28 – Dec 5, 2011. The results of that initial trial (based on 1,042 entries) are available at the link given below. That said, I see no reason to prevent others from participating and, from time to time, I will update the results page to reflect new data.

**https://mathjokes4mathyfolks.wordpress.com/2011/12/05/results-for-my-favorite-game/**

If you have difficulty accessing the form below, click this link.

### More Number Picking

In a previous post, I mentioned the Pick-a-Number game that the folks at NPR’s Planet Money were running:

Pick a number between 0 and 100. The goal is to pick the number that’s closest to

halfthe average of all guesses. For example, if the average of all guesses were 80, the winning number would be 40.

If everyone picked randomly, you would expect the mean to be approximately 50, in which case the winning number would be 25. So, you’d choose 25, right? But if everyone uses that same logic, then the mean would be 25, and the winning number would be 12.5. So, you’d choose 12.5, right? But if everyone used that same logic…

Well, you get the point.

When making your choice, it starts to feel like a game against Vizzini, the Sicilian from *Princess Bride*.

Only a great fool would reach for what he was given. I am not a great fool, so I can clearly not choose half the expected mean. But you must have known I was not a great fool, so I can clearly not choose half of half the expected mean…

Well, the results are in, and you can view them (and an explanation) here.

I take a minimal level of pride in receiving one of 772 honorable mentions for my guess of 12. (Don’t look for my name in the list, though. I used my son’s name as a pseudonym.)

Here’s a very simple pick-a-number game:

Pick a number between 12 and 5.

Make your pick before reading the next paragraph.

Did you pick 7? Most people do. My theory is that the magnitude and order of the numbers matters. Because the larger number is given first, and because the difference between the numbers falls within the appropriate range (12 – 5 = 7), it’s the “obvious” choice.

The trick would probably work equally well if the set-up were, “Pick a number between 19 and 6.” I suspect the most common choice would be 13.

Of course, this is just pop math psychology.

Speaking of “picking” and “numbers,” here’s a line a friend of mine used on an attractive waitress:

How can it be it that I’ve memorized the first 100 digits of π, yet I don’t know the 7 digits in your phone number?

For the record, I condone neither hitting on a waitress nor using that line.

### 1 2 Find a Gr8 Name?

While listening to a recent episode of NPR’s *You Bet Your Garden*, host Mike McGrath said that 10-10-10 fertilizer is a marketing ploy. “No plants want nitrogen, phosphate, and potash in equal proportions,” McGrath said.

I’m not much of a gardener, despite my love of rose (curves), stems and leaves, (square) roots, and (factor) trees. But it struck me as numerically interesting that fertilizer manufacturers sell a product that has the wrong mixture of nutrients. Why would they do that?

Well, money, for one. Products with nice, round numbers tend to be purchased more than others, according to marketing researchers Dan King and Chris Janiszewski. A product with a name like 10-10-10 is more appealing to an average consumer than, say, 9-12-15 or 5-12-13, even though the latter might be more appealing to Pythagoreans.

Consumers will more often choose brands whose names contain likable numbers, of which there are several types:

- Small numbers, such as 1, 2, 3, …, 9.
- Round numbers, like 1, 10, or 1,000.
- Numbers that are frequent sums or products, such as 10 or 24.

It’s easy enough to recognize numbers of the first two types. The third category is a bit loosey-goosey, though, so I would improve the definition as follows: likable numbers of the third type can be represented as a product in more than two ways. For instance, 44 is a likable number because it can be represented in three different ways: 1 × 44, 2 × 22, and 4 × 11; but, 57 is not because it can only be represented in two ways, 1 × 57 and 3 × 19.

King and Janiszewski go on to say that consumers are further influenced if the operands of the number are included in advertisements. In their paper *The Sources and Consequences of the Fluent Processing of Numbers*, they state,

“…not only is a Volvo S12 more liked than a Volvo S29, but liking is further enhanced when an advertisement for a Volvo S12 includes a license plate with the numbers 2 and 6. The operands 2 and 6 make 12 more familiar because they encourage the subconscious generation of the number 12.”

Though some of it sounds like hooey to me, this theory of number relevance is appealing, mainly because it implies that humans are hard-wired for mathematics. (It also makes me think that I chose a good name for my book.)

Upon hearing about likable numbers in products, I tried to think of a well-known product for each likable number up to 100. As you can see from the list below, I had limited success. (Note that I relied entirely on memory. Sure, I could have used Google to find companies like Take 2 Interactive or products like *32 Poems Magazine*, but if likable numbers make a brand more attractive, then shouldn’t I be able to remember the name?)

1:One-a-Day, Mobil 1, A-1

2:Intel Core 2 Duo, Dos Equis

3:3M, Three Musketeers

4:Number 4 Hair Care, 4-H

5:5-Hour Energy, Five Alive, Chanel No. 5

6:Motel 6, Six Flags

7:7-11, Monistat 7, 7-Up

8:Super 8, V-8, Sulfur 8

9:9 West, 9 Lives

10:Tanqueray 10, Oxy 10, Pac 10

12:K12, Big 12

16:16 Handles

18:

20:Mad Dog 20/20, Commodore Vic 20

24:24-Hour Fitness, Claritin 24

25:

28:

30:30 Rock

32:

36:

40:WD-40

42:

44:Vicks Formula 44

45:Colt 45

48:

50:

52:

54:

56:

60:

63:

64:Commodore 64

66:

68:

70:

72:

75:

76:

78:

80:

81:

84:

88:88 Rice Bowl

90:P90X

92:

96:

98:

99:99 Designs

100:100 Grand Bar

I was also able to think of a few product names that include likable numbers greater than 100:

- RU-486
- Saab 900
- 2000 Flushes
- Atari 2600

And of course, there are many successful products whose names contain numbers that are not likable, too:

- Thirteen (WNET, New York City)
- X-14
- Product 19
- Select 55 Beer
- Heinz 57
- Vat 69
- Bacardi 151
- Formula 409
- Levi 501

If you can fill in any of the gaps from the likable numbers product list, please leave a comment. Or if you can think of any other products with numbers in the name, likable or not, feel free to leave a comment for those, too.

### Pick a Number

I’ve told you about my favorite game before.

Planet Money from National Public Radio is currently conducting an experiment using a similar game.

Pick a number between 0 and 100. The goal is to pick the number that’s closest to

halfthe average of all guesses. For example, if the average of all guesses were 80, the winning number would be 40.

You can be part of the experiment until 11:59pm ET on Monday, October 10. The winner and an explanation will be posted on the Planet Money blog on Tuesday, October 11.

### What is Your Favorite Number?

The WordPress Post-A-Week Challenge sends me a daily topic idea to consider for blog posts. Often, the prompts are not appropriate for a math jokes blog. For instance, some recent prompts have been:

- Grab the nearest book (or website) to you right now. Jump to paragraph 3, second sentence. Write it in a post.
- How do you find your muse?
- If you could bring one fictional character to life for a day, who would you choose?

But today’s prompt landed in my wheelhouse:

What is your favorite number, and why?

When Art Benjamin appeared on the Colbert Report, he said that 2,520 was his favorite number when he was a kid. When Stephen Colbert asked him why, he replied, “It was the smallest number that was divisible by all the numbers from 1 through 10.”

Tonight, I asked my twin sons Alex and Eli what their favorite numbers are.

Eli: 5, 15, 55, because my favorite number is really 5, but 15 and 55 are triangular numbers that have 5’s in them.

Alex: 21, because my favorite numbers used to be 1 and 2, and because it’s the number of cards you deal when we play Uno (3 players, 7 cards each).

My favorite number is 153, for lots of reasons:

- It is the smallest non-trivial Armstrong (or narcissistic) number — that is, it is an
*n*‑digit number that is equal to the sum of the*n*th powers of its digits: 1^{3}+ 5^{3}+ 3^{3}= 153. - Its prime factors are 3 and 17, and my birthday is 3/17.
- It is a triangular number. (Consequently, it’s the sum of 1 + 2 + 3 + … + 17.) As 351 is also a triangular number, 153 is also a reversible triangular number.
- It is the sum of the first five factorials: 1! + 2! + 3! + 4! + 5! = 153.
- The sum of its digits is 9, and the sum of its proper divisors is 9
^{2}. - It is one of only six known truncated triangular numbers, which means that 1, 15 and 153 are all triangular numbers.

Mathematician John Baez claims that his favorite numbers are 5, 8, and 24.

Got a favorite number? Share it, as well as the reason it’s your favorite, in the comments.

### 2011 is Prime Time

It’s been eight years coming, but finally, a year that’s a prime number. This hasn’t happened since 2003, but 2011 is prime.

Express 2011 as a sum of consecutive prime numbers.

That can be done in several different ways.

As it turns out, 2011 is extra cool because it can be written as a sum of a *prime number* of consecutive prime numbers.

When will that happen again?

And finally…

What is the next year that will be a prime number and also be a sum of a prime number of consecutive prime numbers? (Wow, that’s a mouthful, ain’t it?)