42 was the last remaining number below 100 which could not be expressed as the sum of three cubes - until now

Since 1954, scientists have been using computers to search for three numbers that could be cubed and summed to equal 42. Every other possible number below 100 already had a known solution, but the solution for 42 wasn't found until recently. Here's Numberphile on the significance of this discovery.

Previously: 42 is the sum of three cubes Read the rest

This trig problem kept me up too late last night

My daughter is taking a precalc summer school course. Last night she was doing her homework, which was about verifying trigonometric identities. Out of the 25 homework problems, there was one that she got stuck on. I decided to give it a try and spent two hours on it without solving it.

Here it is. Verify the identity:

(sec x - tan x)² = (1 - sin x )/(1 + sin x )

You don't need to know anything about trigonometry to solve this. All you need to know are the fundamental trigonometric identities, which are:

My daughter is in class now and she texted me the answer. There's not too many steps involved. Let's see how fast you can solve it. Read the rest

A mathematician explores the cat-and-mouse problem

A swimming mouse is in a circular pond. A non-swimming cat on the perimeter of the pond can run four times as fast as the mouse can swim and will always run in the most optimal way around the pond to catch the mouse. The mouse can run faster than the cat. The question: can the mouse get away from the cat? Mathematician Ben Sparks explores different methods the mouse can try.

Image: Numberphile/YouTube Read the rest

Three interesting infinite series

Numberphile takes a look at three interesting infinite series. The first is 1 + 1/2 + 1/4 + 1/8 ... which equals 2. The second is 1 + 1/2 + 1/3 + 1/4 + 1/5 ... which equals infinity. The third is 1 + (1/2)2 + (1/3)2 + (1/4)2... which π2/6 Read the rest

Professor Stewart's Incredible Numbers is a delightful book of recreational math

When I was a kid one of my favorite books was George Gamow's One Two Three . . . Infinity: Facts and Speculations of Science. (It's a pity that the current edition has such a crappy cover. Here's the cover to the copy I own, which is much cooler looking). This book taught me about huge numbers, infinity, and the fourth dimension. I loved Gamow's hand-drawn illustrations, too. If you don't have this book, I suspect you will enjoy it.

Professor Stewart's Incredible Numbers, by Ian Stewart, reminds me of One Two Three . . . Infinity. It's missing the charming hand-drawn illustrations, but it has many of the same topics in Gamow's book (like the Towers of Hanoi, and the Four Color Map Theory), plus quite a few other fun number-related items that Gamow didn't cover, such as fractals, the Birthday Paradox, and the Sausage Conjecture.

When I told my 12-year-old about the Four Color Map Theory, she immediately went to work with colored pencils and paper to prove me wrong. I can't find the fantastically complex maps she came up with, but if I locate them, I'll post them here. She eventually came up with an intuitive understanding of why any map you draw only needs four colors to ensure no two bordering shapes have the same color.

UPDATE: I found Jane's maps:

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