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!

Mathy Birthday Problems

Last week, I turned 42. Here’s a math problem related to that number.

Take 27 cubes, numbered consecutively from 1 to 27. Arrange them into a magic cube so that every row, column, corridor, and space diagonal has a sum of 42.

If that’s too much for ya, try this problem instead. It’s a slight modification of a math problem that appeared on the birthday card given to me by colleagues.

Two men my age go out for drinks at 10 o’clock on a Saturday night. One of them drinks six 12-ounce beers, each of which is 8% ABV. The other drinks four Lynchburg lemonades, each of which contains one ounce of 80-proof Jack Daniels and one ounce of 60-proof triple sec. Assuming the men are the same size, which one gets more drunk?

The answer to the first question can be found at Math Palette.

The answer to the second one? Trick question. Men my age don’t go out after 10 o’clock.

It’s Back to Prime Time

On Saturday, I turned 41 years old. I’ve been looking forward to this for a while. It’s a prime year, and its twin prime is two years away. In between, I’ll be a number of years that is “the answer to life, the universe, and everything.”

Forty-one is also cool because f(x) = x² + x + 41 is a prime-generating function. That is, f(1) = 43, f(2) = 47, f(3) = 53, and so on.

What is the first value of x for which x² + x + 41 is not prime?

The following image might help you answer that question. The number 41 appears in the center, and consecutive positive integers then proceed in a spiral. Notice that all of the numbers highlighted in yellow are prime. A pattern of primes continues along the diagonal — at least for a little while.

It also turns out that 41 is the smallest number whose cube is the sum of three cube numbers in two different ways:

41³ = 2³ + 17³ + 40³ = 6³ + 32³ + 33³ = 68,921

And 41 is the sum of the first six prime numbers:

2 + 3 + 5 + 7 + 11 + 13 = 41

At 41, I still feel young. But you know you’re an old mathematician when…

• You report your age in hexadecimal. (I’m only 29!)
• You’re not dead, but you’ve lost most of your functions.
• The distance you walked to school as a kid is directly proportional to your age.
• Your age can be described as “countably infinite.”
• You regularly go off on tangents.
• The phrase “pulling an all nighter” means not getting up to pee.
• When asked your age, you reply, “I’m in the 99th percentile.”
• You use the term surd, and you know how to calculate its value on a slide rule.

Dude, You’re Such a Cube

Today’s a good day to work in a cube farm, to hang out with a dorky friend so square that he’s a cube, or to cube out your glass with some lemonade and enjoy one last, lazy summer afternoon. We’re 53 years from the aweseomely cubic date of 8/27/64, but it’s still pretty cool that today’s month is 2³ and the date is 3³.

If you do enjoy a glass of lemonade today, be sure to keep it cold with an (ice)³.

Here’s a tree that’s appropriate for today:

If you’re in the path of Hurricane Irene, please don’t stand under this tree! (Seriously, be safe today.)