Doh!
Of course my very long explanation can be very simply summarised as ! (factorial) as Ravvie showed so succinctly!!
I never did get why 0! is equal to 1 though (nor why anything to the power 0 is 1).
1 Like
Well done Steve and Ravvie.
I was going to give you 10 out of 10 anyway Steve, because you could clearly see how to generate all the numbers following 64.
I was going to give Ravvie a bonus point for being able to generate all the numbers, including that mysterious double one at the start.
Well done Ravvie for recognising the factorial progression and well done Steve for recalling 0! once Ravvie had made his very thought-provoking post.
I’ll post later my version of the “Proof” that 0! = 1. It’s nothing special, and can easily be found on t’internet
Cheers
Don
“Proof” or explanation 
n! = n x (n - 1)!
1! = 1 x (1 - 1)!
1! = 1 x (0)!
1! = 1 x 0!
we know that 1! = 1 hence
1 = 1 x 0!
1 = 0! (this is the line that most people have difficulty with
0! = 1
Just as long as you don’t ask for 52 numbers in the sequence!
1 Like
I can’t work out how to post it, but it’s 87 upside down.
To be honest I wouldn’t have seen it so quickly, but the “8” gave it away with the smaller loop on the bottom.
4 Likes
I remember my maths A’level teacher showing us that proof of 0!=1.
Whilst I understand the “proof”, the resuIt still simply looks intuitively wrong.
0 is ignored in all factorial calculations, and rightly so, as every number factorial would result in 0. I have much less problem accepting that all factorial calculations start and finish at 1 (and that 0 is ignored) than I do accepting that 0! = 1 just to appear to make everything else work. Anyway, that’s just my view however naive it may appear.
1 Like
Yes, I thought that the shape of the 8s might be a bit of a giveaway. Nevertheless, well done.
I have the same reservations as you do Steve regarding both 0! = 1 and also that x^0 = 1. I don’t fully understand it all, but I can usually remember the rules !
Let just assume we are each in good company
The way I understand factorials is that they represent the number of ways of ordering a set of n items. For example, if you had 4 shirts in a closet you could arrange them in 24 ways (what a fun afternoon that would be). If you have 1 shirt there is only 1 way. If you have 0 shirts then you still have only 1 way of ordering, that is you have 1 empty closet.
1 Like
Thanks Ravvie. I have seen similar explanations ie using objects other than shirts !
It’s a very useful aid-memoir.
However, when I have used that explanation, most people follow it down from, say, 4 objects (I like the shirts !) to 1 object, but have difficulty accepting the now empty space as an “object” in itself.
But, as I say, I find it really helpful. Many thanks.
It took a long while for mathematicians to invent the number 0. After all, if you didn’t have any objects then you didn’t need to count them so there was no need for such a number. So 0 can be a bit abstract.
Douglas Adams makes fun of the anomaly in Hitchhikers Guide to the Galaxy:
Zaphod: “How many escape capsules are there?”
Ford: “None”
Zaphod: “Did you count them?”
Ford: “Yes, twice”
1 Like
But if I were to charge you £100 per shirt for putting your shirts in order, then I could also charge you £100 every time you had no shirts. Sorry, but it’s still nonsense to me.
And there was me thinking that all the atoms in the Earth dealing a deck of cards every second since the beginning of time was nonsense!
Yes, intuitively anything with 0 in it usually ends up as 0, so it does seem a bit strange.
Hi Don,
in situations like this I can’t stop myself from wondering if I missed some hidden meaning. I don’t translate the text into my native language, just as my english teacher advised us to do, but some uneasiness remains.
Hi Steve, I fluctuate between ‘difficult’ and ‘nonsense’.
I seem to recall from somewhere, that the “real” mathematicians (eg Andrew Wiles etc but probably not him specifically), added a few new “axioms” (I think that’s the word I want) just to maintain some semblance of mathematical law and order to our concept of difficult numbers such as 0!
I have difficulty remembering some of these rules, let alone understanding them !
I can’t even remember how to generate “minus b, plus or minus the sq rt of b squared minus four a c …” I feel fortunate that I can still remember it and use it
Hi Mulberry, your English is so natural that I nearly always forget that it’s not your native language.
I posted the 1, 1, 2, 6, 24, … teaser half expecting a few people to question why there were two number ones at the start. but hoping that others would recognise the 0!, 1!, 2!, 3! etc sequence.
However, I recognise that some of our brain teasers do have a hidden meaning, and some are more like a riddle rather than seeking a justifiable logical pattern. Sometimes this adds to the fun, other times it doesn’t. I hope for the former, and apologise for the latter.
You always seem to come up with good solutions, so I hope you find most of them fun, and a worthwhile challenge. Especially given the necessary translation you have to add to the challenge !!
Cheers
Don
1 Like
Hi Don,
I usually manage to avoid tranlating things I read here, including the teasers, by thinking in english. One of my english teachers said thar reading a problem in english, translating it to german, solving it in german and translating the answer back into english would take too much time to complete the upcoming exams. While I can’t say if would have failed my exam otherwise, I did pass it this way 
2 Likes
Ok. Not really a brain teaser, just a bit of quiet fun.
Do the arithmetic one line at a time, in your head - or better still, get someone to read it out to you !
Jot your initial answer down on a piece of paper (you didn’t cheat, did you !) then do it again with pencil and paper and/or a calculator.
Very clever that!
I am ashamed to admit I got 5,000 at first, albeit very quickly and after a couple of pints 
1 Like