## Posts tagged ‘magic’

### 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.

### Rectangles for Mathemagicians

Depending who you ask, *mathemagician* has at least two different definitions:

- A person who enjoys both math and magic. (Wikipedia)
- A person who is so good at math that the answers to math problems seem to come to them magically. (Urban Dictionary)

When professor Art Benjamin told Stephen Colbert that he was a mathemagician, Colbert asked, “What does that mean? Were those two words by itself not nerdy enough?”

Below is a math puzzle involving magic. To be precise, magic rectangles. But first, a little warm-up…

What do you call a quadrilateral with four right angles that’s been in a car accident?

A wrecked angle.

For the last several years, I’ve had the pleasure of creating puzzles for the Daily Puzzle Challenge at the NCTM Annual Meeting. A new set of four or five puzzles appears in each day’s challenge. The following puzzle, which appeared on Friday’s Daily Puzzle Challenge, involves rectangles and is my favorite puzzle from this year’s meeting.

A magic rectangle is an *m* × *n* array of the positive integers from 1 to *m* × *n* such that the numbers in each row have a constant sum and the numbers in each column have a constant sum (although the row sum need not equal the column sum). Shown below is a 3 × 5 magic rectangle with the integers 1-15.

Below are three arrays that can be filled with the integers 1-24, but only two of them can be filled in such a way as to form a magic rectangle. Construct two magic rectangles below; for the array that cannot be used to construct a magic rectangle, can you explain why not? More generally, can you determine what types of rectangles can be used to construct magic rectangles and which cannot?

### C’mon, Have a (Magic) Heart

The Zen of Magic Squares, Circles and Stars by Clifford Pickover is chock full of magic arrangements. On page 55, Pickover discusses Dürer’s method for creating a 4 × 4 magic square:

- Starting with the upper left corner and proceeding horizontally to the right, number the squares of a 4 × 4 grid with the consecutive integers 1‑16.
- Starting with the lower right corner and proceeding horizontally to the left, number the squares of a different 4 × 4 grid with the consecutive integers 1‑16.
- From the first grid, keep the integers that occur on the main diagonals. From the second grid, keep the integers that do not occur on the main diagonals.

A visual representation of the process might help to clarify:

The result is a 4 × 4 magic square. In fact, it is a slightly modified version of the magic square that appears in Albrecht Dürer’s *Melencolia I*. (Note that if the other 8 numbers from each grid were combined in a similar fashion, they would form a magic square, too.)

Serendipitously, my sons and I recently completed an art project that can be combined with Dürer’s method to form a “magic square heart.” The project my sons completed is as follows:

- Draw a square with a semicircle on top. Repeat to create two of these figures, preferably on paper of two diffferent colors, and cut them out.
- Cut from the bottom of each figure to the diameter of the semicircle, to divide the squares into equal‑width strips.
- Finally, “weave” the strips to form a checkerboard pattern.

This idea can be combined with Dürer’s method to create a magic square heart. But instead of dividing the squares into equal‑width strips, divide them into three strips whose widths are in the ratio 1:2:1. Then, draw the outlines for 16 squares, and number the squares as described in Dürer’s method above. The two pieces will look like this:

Then, weave the three strips into a pseudo‑checkerboard pattern. When woven together, the result will be the following magic square heart:

To complete this project with students, you can use the template below.

**Magic Heart Template:**

http://mathjokes4mathyfolks.com/mj4mf-magicheart.pdf

That said, it’s my belief that students will have maximum mathematical fun if they are allowed to create the heart from scratch. It’s an exercise in geometric construction to draw a square with a semicircle on top; weaving the strips into the appropriate configuration can lead to a discussion of geoemtric symmetry; investigating the patterns formed by the numbers can lead to a discussion of numerical symmetry; and, investigating the square to find that the rows, columns, and diagonals have a constant sum may inspire young minds in the same way that it inspired Albrecht Dürer.