numbers or expression of a number

Our man in Havana

There was a puzzle on tv last night basically one of those matchstick quizzes where you move 2 matches to create the largest number possible.

3895 was the start number, Answer was 151321 which is 389 squared.

Is 389 squared a natural number or merely an expression of 151321?

It is a very devious question and I would appreciate if anyone could assist.

Fremen

Sounds like a bit of a cheat to me. I guess it's up to how you interpret "create the largest number possible". It's technically correct, yes, but most people would interpret "create the largest number possible" as being restricted to an expression of the digits, so I'd say it's unfair.

I doubt that 389 squared is even the largest number you can make, though I'd have to see the layout of the matches to be sure. If powers are allowed, presumably you can use Knuth's up-arrow notation too, which would allow waaaay bigger numbers.

I'd say there are plenty of other things you could do, too, like forming a 2 below the number so it becomes something like 2^389, which is much bigger than 389 squared.

(Edit: would have been nice if you had posted that up as a puzzle for us )

Our man in Havana

The only way I can see how it was done was by taking 2 matches off the 5 and making a superscript 2.

This was on one of those late night phone in quizzes.

The matches were arranged the same as the digits on a calculator.

Fremen

Yes, I see what they've done. They represent the 5 using 5 matches, then move the bottom two to the top to make a two.

Pity they didn't realise you can do the same trick with the 3 to make 2^895, or in fact 9^895.

Our man in Havana

Very interesting stuff. Thanks.

gillo_100

Fremen wrote: »

Pity they didn't realise you can do the same trick with the 3 to make 2^895, or in fact 9^895.

2^895 yeah, 9^895 not possible if same as calculator as 9 has 6 matches 3 only has 5, there is a bottom one on the 9.

Our man in Havana

Discussion of the puzzles is allowed, comments on the running of the show are not allowed.

red_fox

gillo_100 wrote: »

2^895 yeah, 9^895 not possible if same as calculator as 9 has 6 matches 3 only has 5, there is a bottom one on the 9.

You can also get 3^895. You could ask similar, what is the smallest number possible starting with 1874 and moving three matches (a, allowing exponents. b, not allowing exponents)

Fremen

By smallest do you mean least in absolute value, or least overall?

Edit: for smallest in absolute value, the best I can do is

18^-6

red_fox

I was thinking absolute value, I should have said. I think that's the smallest alright. Although I just thought if you're a little sloppy with falling factorial notation then you can get zero, but that's pushing things.

Fremen

red_fox wrote: »

if you're a little sloppy with falling factorial notation then you can get zero

Hm, How'd you manage that?

red_fox

Falling factorials are usually written (n)_k = n(n-1)...(n-k+1)

But if you drop the brackets you can get 1_919, the second term of the expansion will be zero. (although if you insist that n>k then of course this won't work)

MathsManiac

Or if you use the "square corner" factorial notation, you can get (7895)!