October's snippets are ready for your perusal!

Today, we'll use math videos to take you into surprising and bizarre aspects of mathematics. Don't worry, you're safe on this side of the play button.

• Let's start with a classic tale, which teaches the surprising nature of exponential growth:



How many grains of wheat are on a given square? On the first square, we have 1 grain, or 2⁰. On the second square, we have 2 grains, or 2¹, and on the third square, we have 4 grains, or 2². It's not hard to see that on the nth square, we'll have 2^n-1 grains of wheat on the chessboard.

That's only the amount of grains on a given square, however. What about the total number of grains? With the first square, we have 1 grain of wheat. On the first two squares, we have 3 grains of wheat. On the first 3 squares, we have 7 (1+2+4) grains of wheat, and so on. Notice that 2¹=2, 2²=4, and 2³=8, each just 1 more than the total number of grains on each of the corresponding squares. In other words, the total grains of wheat on the first n squares will be (2ⁿ)-1.

With a little help from Wolfram|Alpha, we can follow the story more closely. After the 17th square, the inventor would have 131,071 grains of wheat. After the 26th square, there would already be 67,108,863 grains of wheat. What about the 42nd square? More than 4 trillion grains of wheat (4,398,046,511,103 to be exact) had been given to the inventor by the surprised king!

The video implies that the process had to be ended there, but were it continued to the 64th square, the total number of wheat grains would come to 18,446,744,073,709,551,615 (read as 18 quintillion, 446 quadrillion, 744 trillion, 73 billion, 709 million, 551 thousand and 615)!

Just for fun, if we tell Wolfram|Alpha we're actually talking about grains of wheat, we can discover stranger things about so much wheat, like the total nutrition information and physical properties, that the king could never have known.

• Back in May of last year, I wrote about John Conway's game of life. Since it was first described in 1970, there have been many ingenious variations, such as Life For Two. When you search for conway's game of life on Google, Google even runs a sample game of it!

The classic version is built around integers (whole numbers) and grid squares, giving the game its classic blocky look. There's a new version called Smoothlife that's instead built around floating point numbers and distance measurements, giving the game of life a much more organic look and feel:



You can learn more about this project at the SmoothLife link above, as well as watching the videos on the SmoothLife playlist. There are even sample runs of SmoothLife in 3D later in the playlist, so take the time to explore!

• I imagine all this talk about life, death, and grains of wheat is probably making you hungry. Even if the above topics didn't, James Grime's and Numberphile's trip to McDonald's to discuss McNugget numbers will probably do it:



As mentioned, the largest number of McNuggets (or other objects) that cannot be made with given sets is known as the Frobenius number of the set. More colloquially, the problem is known as the coin problem. It was Henri Picciotto who first drew attention to McNugget numbers back in the 1980s. He still discusses McNugget numbers on his website.

Wolfram|Alpha has a built in function for finding the Frobenius number of any given set. Here's the classic McNugget problem set, as well as the modified McNugget problem set, including the boxes of 4.

If you enjoy the McNugget numbers problem, you might also the semi-related world of Golomb rulers. Golomb rulers are measuring rulers optimized so as to use the fewest markings possible that still allow you to measure any whole distance from 1 unit up to a given limit (say, 12 units).

That's all for now, so have fun exploring!