Source: https://www.facebook.com/AmerMathMonthly/photos/a.250425975006394.53155.241224542593204/755598191155834/?type=1&theater

Square any number up to 1000 without a calculator

The Mathematical Association of America has an excellent series of 10-minute lectures on various topics in mathematics that are nevertheless accessible to the general public, including gifted elementary school students. From the YouTube description:

Mathemagician Art Benjamin [professor of mathematics at Harvey Mudd College] demonstrates and explains the mathematics underlying a mental arithmetic technique for quickly squaring numbers.

The minimal possible length of a doctoral dissertation in mathematics

Source: https://www.facebook.com/AmerMathMonthly/photos/a.250425975006394.53155.241224542593204/737394819642838/?type=1&theater

Sets, Planets, and Comets

From the article “Sets, Planets, and Comets” that recently appeared in College Mathematics Journal.

Set is an enjoyable—even addictive—card game that challenges players to identify

certain visual patterns. A mathematically rich game, it provides ample opportunity for

students and teachers to ponder combinatorial, algebraic, and geometric questions. Part

of Set’s appeal is that once the fundamentals of the game are understood, it is nearly

impossible to resist investigating its structure, whatever one’s background. We con-

centrate on the geometry, introducing interesting objects we call planets and comets,

which lead to an elegant variation on the game.

The full article can be found here: http://www.maa.org/sites/default/files/pdf/pubs/SetsPlanetsAndComets.pdf

Symphonic Equations: Waves and Tubes

This excellent and engaging video describes how sine and cosine functions can be applied to music.

Another proof of the Pythagorean theorem

Source: https://www.facebook.com/AmerMathMonthly/photos/a.250425975006394.53155.241224542593204/737392972976356/?type=1&theater

What to Expect When You’re Expecting to Win the Lottery

I can’t think of a better way to tease this video than its YouTube description:

Recounting one of the stories included in his book How Not to Be Wrong: The Power of Mathematical Thinking, Jordan Ellenberg (University of Wisconsin-Madison) tells how a group of MIT students exploited a loophole in the Massachusetts State Lottery to win game after game, eventually pocketing more than $3 million.

A personal note: though I haven’t talked with him in years, Dr. Ellenberg and I were actually in the same calculus class about 30 years ago.

Proof without words: The difference of consecutive cubes

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For a more conventional algebraic proof, notice that

$(n+1)^3 - n^3 = n^3 + 3n^2 + 3n + 1 - n^3 = 3n(n+1) + 1$

The product $n(n+1)$ is always an even number times an odd number: if $n$ is even, then $n+1$ is odd, but if $n$ is odd, then $n(n+1)$ . So $n(n+1)$ is a multiple of 2, and so $3n(n+1)$ is a multiple of 6. Therefore, $3n(n+1)+1$ is one more than a multiple of 6, proving the theorem.