Math Miscellany


M, Heart (underlined), 8
What symbol comes next?

Source: I first saw the M-Heart-8 puzzle in print, perhaps in the 1980s or 1990s, but I don't remember where. It was popularized in The Simpsons on TV (1998, episode title "Lisa the Simpson") and in The Oxford Murders book (2003) and movie (2008). If you know a source earlier than 1998, please let me know.

Logic Puzzle Grids

Three logic puzzle grids,
Logic puzzle grids

Logic puzzles are solved with the aid of logic grids. A standard completed grid is shown in figure A. I can visualize the relationships better and solve the puzzle faster if I draw curved lanes connecting the matching rows and columns of the subgrids (figure B) . I also find it helpful to color the "true" markers that go together because they are in the same rows and columns (figure C).

Interesting Digit Patterns

You may have seen this:

1/81 = 0.012345679012345679...

But you may not have seen this:

1/243 = 0.004115226337448559...

How does the pattern continue? What is special about the number 243? (Hint: find its factors.) What causes the pattern? Are there analogous numbers in other bases?

Source: I found this in Surely, You're Joking Mr. Feynmann by Richard Feynmann.

Imaginary Powers

The concept of imaginary powers is very strange, even if you are comfortable with imaginary numbers (ex., √-1), negative powers (ex., x-1 = 1/x), and fractional powers (ex., x1/2 = √x).

You may have seen this famous, beautiful, and strange equation that relates the most important transcendental numbers, π (3.14159...) and e (2.71828...), with i = √-1, the imaginary square root of -1:

e= -1

This is a special case (with x = π) of Euler's formula:

eix = cos(x) + i sin(x)

Here's another not-as-famous strange equation involving an imaginary power:

ii = e-π/2 = 0.207879576...

This is a special case (with n = 0) of this formula:

ii = e(-π/2 + 2πn)

According to this formula, ii has an infinite number of values! For example, another value (with n = 1) is:

ii = e3π/2= 111.317778...

Source: Euler's identity is famous. I first saw ii in Mathematics: The New Golden Age by Keith Devlin.