h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Factorial 70 (that is, 70 x 69 x 68 x ... x 3 x 2 x 1) is

a) Over a million

b) Over a trillion trillion trillion

c) Over a googol

Despite the title, I'll try and work it out without a calculator.

The term trillion is slightly ambiguous, but let's say you mean the current meaning of a 1 with twelve zeroes after it. So 1 trillion trillion trillion means a 1 with 36 zeroes after it.

So the three questions are:

a) 1 with 6 zeroes
b) 1 with 36 zeroes
c) 1 with 100 zeroes

Well, factorial 70 is 70 times 69 times etc etc all the way down to 10 by factorial 9. Now 61 of those are numbers greater than 10, so the answer will have to have more than 61 zeroes. So it will certainly be greater than a) and b).

But what about the googol? Seems likely, but I'm not sure.

My calculator goes up to about 10^150, and indeed 70! is 1.197857166997e+100

So the answer to the question would be, all of the above ...

... only it wasn't a question, but a true statement.

Here's a question. If you enter 7 on your calculator, how many times must you press x-squared to reach a googol?

Well, let's see. Press it once to get 49, which is roughly 50, which is 100/2.

So press it again to get 10000/4, three times to get 1000000/8. Press it n times to get (100^n)/(2^n). Press it 50 times to get 100^50/2&50. But 100^50 is a googol. So pressing the squared button 50 times, we're still less than a googol. Now 2^50 is big, but it's not that big. 2^10 is about a thousand, so 2^50 will be roughly a 1 with 15 noughts. So a googol over 2^50 will have roughly 85 noughts. Pressing the button a 51st time will give a number much bigger than a googol.

Goes off to find a calculator to see if I'm right.

No, I was totally wrong. It's far less than that. The two glasses of wine have impaired my mathematical brain.

Good of you to try estimating it! I found it to be seven presses: hope I'm right.

Here's a dive into several googols of layers http://www.youtube.com/watch?v=0jGaio87u3A

Wow, there's some serious psychadelia there! I wasn't aware of the cross shapes in the Mandelbrot set before. Good video.

More numerical thoughts:

The Irish national lottery is more favourable than many; you only have to choose six numbers from 45. Most (apparently) get you to choose six from 49 numbers.

That means that your chances of hitting the winning combination in the Irish lotto are one in a little over eight million (if you call 145,060 a little).

If I was told that they were going to line up eight million people and shoot one of them at random, I would feel pretty safe to go along, even to go along every week.