## Posts tagged ‘integer’

### Infinite Integer Triangles

Here’s an interesting question.

Given the side of a triangle with integer length, what is the set of all points in the plane for which the other two sides will also have integer lengths?

And by *interesting*, I mean that the answer wasn’t immediately obvious to me.

So I drew a segment 5 units long in Geometer’s SketchPad, created a bunch of concentric circles with integer radii and centers at the endpoints of the segment, identified the intersection points of those circles, and finally hid the circles. The result was the following beautiful image:

And by *beautiful*, I mean that the result is, well, beautiful. At least to a math dork. If this had been painted by Van Gogh, it would have been called **Triangle in a Starry Night**. (Okay, maybe not.)

The triangle indicated by the dashed lines is the famous 3-4-5 right triangle. The points in the upper right and upper left corners yield the less well known but similarly intoxicating 5-5-8 triangle. If the limitations of the web allowed this image to extend infinitely in all directions, the result would be infinite beauty. Alas, reality confines us.

I have an infinity of jokes that deal with triangles and circles, but I’ll only share a subset of them here.

What did the triangle say to the circle?

Your life is pointless.Why don’t circles hang out with ellipses?

Too eccentric.What did the hypotenuse say to the other two sides?

Nice legs!Where do circles and ellipses spend their vacations?

Coney Island.What’s a circle?

A round, straight line with a hole in the middle.What did the circle say to the tangent line?

Stop touching me!

### Depressing Expressions

Did you hear that Monday, January 23, 2012, was the most depressing day of the year? That’s according to Cliff Arnall, a British life coach who, for a little while, was a tutor at Cardiff University. He used the following formula to make his prediction:

In that expression, *W* = weather, *D* = debt, *d* = days until next payday, *T* = time since Christmas, *Q* = time since a failed quit attempt (such as abandoning a New Year’s resolution), *M* = motivation level, and *N _{a}* = need to take action.

When I heard that a psychologist was creating mathematical expressions, I had just one thought:

Why did the psychologist send the expression to a doctor?

Because he wasn’t being rational.

When I read the formula, my first thought was, “Wow, that’s an incredible bunch of rubbish!” (Funny, I don’t normally use the word *rubbish*. Maybe it happened because I read about the expression in a British newspaper?) Only *W* and *T* are universally measurable variables. While *D*, *d*, and *Q* are also measurable, they vary from person to person and shouldn’t be used to predict a global most depressing day. And what’s this nonsense about time from Christmas? Is that really a factor for Jews, Sikhs, and other non-Christians? (Note: Many online sources incorrectly state that *d* = monthly salary. But that would cause the formula to make even less sense.)

This expression has been used for several years to predict the most depressing day, and a similar expression has been used to predict the happiest day of the year. The happiest day expression, which is similarly unintelligible, regularly predicts a date in June. Interestingly, the “research” was sponsored by Wall’s Ice Cream. (Hmm, now why would an ice cream company have an interest in people being happy during the summer?)

It’s a little too early in the year to say that January 23 will be the most depressing day of 2012. In fact, January 23 isn’t even the most depressing day *so far* in January — that distinction belongs to Sunday, January 8.

As for crazy expressions, the following equation contains my favorite:

The value of *k* doesn’t matter, but the equation doesn’t hold if the placeholder variable *k* is not included.

Incidentally, this equation is related to the following problem: Raise *n* + 1 consecutive integers to the power *n*. Subtract the first from the second, the second from the third, and so on, until you’re left with a set of *n* integers. Then subtract the first from the second, the second from the third, and so on, until you’re left with a set of *n* ‑ 1 integers. Continue this process until you’re left with just one integer. Its value may surprise you.

### Results for My Favorite Game

Thanks to everyone who participated in the online version of my favorite game. Thanks, especially, to those people who helped to share it via Twitter, Facebook, and other blogs.

Though I am posting the results today, **I will continue to leave the form online**. Though I had only planned to let the contest last one week, entries continue to roll in, and I see no reason to forbid people from playing. From time to time, I’ll update this page… such updates will occur at the intersection of two events: when enough entries warrant an update, and when the muse hits me.

Without further adieu, here are the results.

With 1,042 entries divided into groups of 100, there were 10 complete games played. The charts below show the results for each game. (Sorry if they’re a little hard to read. Click on the images to view them full-size in a separate window. There are two images below — games 1-5 are shown in the top image, and games 6-10 are shown in the bottom image.)

The graphs above do not reflect all 100 entries for each game. In each game, many numbers greater than 35 were chosen. However, 84.1% of all selected entries were 35 or less.

The winning numbers, respectively, were **3, 3, 20, 4, 7, 2, 16, 4, 4, 2**.

If you’re interested in the raw data, download this Excel spreadsheet.

Congratulations to **Loïc Grobol** of Chécy, France! Loïc was the winner of the 4th game, and his name was randomly selected as the overall winner of the signed copy of *Math Jokes 4 Mathy Folks* and the specially-designed, one-of-a-kind random number generator. (There is beautiful symmetry that Loïc chose the number 4, that he was the winner of the 4th game, and that the prize has a 4 in the title.)

As shown in the graphs above:

- On average, the number 1 was chosen by 10.4% of entrants. (Edward Early of St. Edward’s College said, “I use that [game] as a bonus question on a test in every class I teach. I believe 1 has never been the winning number, even in a class with only 6 students.” I have played this game over 100 times with various size groups. In my experience, the number 1 has only won twice… by the same woman, who — in a show of incredible bravado — chose the number in consecutive rounds of the game.)
- The number 2 won twice, but it was chosen rather infrequently — less than 1/3 as often as 1. My suspicion is that people figure if you’re gonna go big, go REALLY big… why choose 2 when you can choose 1?
- Besides 1, the number most often chosen was 17, which was selected 5.5% of the time. This seems to corroborate numerous studies that found 17 to be the most commonly selected random number.
- The number 151 was the greatest number chosen more than once.

The chart below shows the frequency of the top nine guesses.

Among the most interesting entries were 666, 1012, 1337, 53,479, and 3,010,994.

The most amusing entry was 10, with the accompanying note, “I’ll choose the base later.”

And you may be wondering… if all 1,000 entries were considered as just one game, what number would have won? That distinction would have gone to 32.

By running this contest, I learned about two interesting uses of this game in classrooms.

Edward Early said that he uses the following version as a bonus question.

Write a positive integer in the blank: _______

How this will be graded: The least positive integer that is submitted by exactly one person will be worth 5 points. The next-smallest will be worth 4 points, and the next-smallest after that will be worth 3 points. All other positive integers submitted by exactly one person will be worth 2 points. Positive integers submitted by more than one person will be worth 1 point. Anything other than a positive integer will receive no credit. Do not ask me to explain this question.

Not surprisingly, with these modifications comes a change in strategy — Edward said that some students choose a large random number, just to ensure they receive 2 points.

Matt Skoss of Possum Educational Services and the Northern Territory Dept of Education and Training said that he’s used this game for years with his kids at school.

Pick the lowest prime number, composite number, surd, cube number or triangular number, etc., depending upon what I’d like the kids to think about.

What an excellent use of a simple game!

### Random Thoughts

The following was sent to me by my friend Pat Flynn, and it may enter my email signature soon.

The derivative of my enthusiasm for mathematics is positive for all values of the independent variable.

And here are some one-liners that don’t warrant their own posts, but they’re just too good not to share…

Heisenberg might have slept here.

Old mathematicians never die; they just lose some of their functions.

Whenever four mathematicians get together, you’ll likely find a fifth.

“Take a positive integer

n. No, wait,nis too large; take a positive integerk.”