The four fours maths puzzle

[Evariste Euler Gauss]

I've been trying the four fours challenge in dull moments (typically during tedious lectures at conferences) ever since I first heard of it at school. Given 30 minutes or so, I normally manage an unbroken sequence from 1 to 112 but have never managed to solve 113.

So when I looked up the Wiki entry on the puzzle and followed the links, I was (a) immensely pleased to see that it appears to be impossible to solve 113 without using the gamma function (which I'd never heard of before today) and (b) gobsmacked that, with the help of the gamma function, it seems to be possible to solve all numbers up to 1,000, according to a Mr Wheeler.

I have enough new capacity now to amuse myself in tedious conference lectures for weeks on end!

[Wyatt Earp]

The four fours maths puzzle

By the look of it, you need to know a fair bit about the gamma function though, including its behaviour for non-integer n.

[evilC]

The four fours maths puzzle

4/4 is the best rhythm.

Coincidence? I think not!

;-)

[Evariste Euler Gauss]

The four fours maths puzzle

Yes, I can see it's way beyond me. Also, the gamma function seems not to be the only notational/functional tool I'd been deprived of. I've no idea what that upward pointing arrow used, for example, at 510 to 514 means!

[Gangster Octopus]

The four fours maths puzzle

Ok, I'll bite. What is the four fours maths puzzle?

[Rogin the Armchair fan]

The four fours maths puzzle

What the feck is the gamma function? Is this some kind of Star Trek Voyager thing?

[Evariste Euler Gauss]

The four fours maths puzzle

GO, look up "four fours" on Wiki. Basically you have to express every whole number from 1 onwards by four fours, no more no fewer, and using no other digits. So, for example 1 can be expressed a 44/44, and ninety as "(four over point four) times (four over point four recurring)".

Rogin, the gamma function is the factorial of one less than the number in question (says he with all the authority of someone who's known that for 25 minutes already)

Actually, my last post was incorrect - I can see that the upward arrow thing just means "to the power of". What I definitely don't get is how that guy's arithmetic works. I mean, take his solution for 644 for example. WTF???

[Rogin the Armchair fan]

The four fours maths puzzle

So basically there's a mathematical function that means "this number minus one".

That's a bit of cheat in this context, isn't it?

How to get to nine with 2 fours. Oh, easy, it's gamma 4 times gamma 4.

What's the point of that?

[Evariste Euler Gauss]

The four fours maths puzzle

No, it's the FACTORIAL of (n-1). At least, that's how it can be defined with whole numbers. For non-whole numbers it's all, like, really complicated.

Factorial means the product of n and all integers below n. The notation for it is an exclamation mark. So, for example:

1! = 1
2! = 2
3! = 6
4! = 24
5! = 120

and so on.

Edit: and as to what the point of it is, outside trivial mathematical games, I refer you to the Wikipedia entry "Gamma Function", which I expect has a bit on uses of the function, though I must confess I didn't read that far.

[Jimski]

The four fours maths puzzle

What I definitely don't get is how that guy's arithmetic works. I mean, take his solution for 644 for example. WTF???

Looks ok to me:
4^4 = 256
256/.4 = 640
640 + 4 = 644

[Evariste Euler Gauss]

The four fours maths puzzle

Thanks Jimski, that's much clearer. My problem was that I had no idea in which order to apply the three operations of powering, dividing and adding, without the help of brackets or a more two-dimensional layout. I'd just tried a few random orders and not really been methodical about it.

Edit: clearly, I overlooked the obvious option of simply applying them from left to right.

[Evariste Euler Gauss]

The four fours maths puzzle

I'm loving his solution for 123:

sqrt(sqrt(sqrt((sqrt(4)/.4)^4!)))-sqrt(4)

[West End Zuma]

The four fours maths puzzle

Is this mathematical code for "Sting is a cunt"?

[Wyatt Earp]

The four fours maths puzzle

Lord Mauleverer wrote:

Yes, I can see it's way beyond me. Also, the gamma function seems not to be the only notational/functional tool I'd been deprived of. I've no idea what that upward pointing arrow used, for example, at 510 to 514 means!

"To the power".

[Wyatt Earp]

The four fours maths puzzle

Lord Mauleverer wrote:

Edit: clearly, I overlooked the obvious option of simply applying them from left to right.

No, there's a standard order for them: powers, times/divide, then plus/subtract.

[Hofzinser]

The four fours maths puzzle

"BODMAS" we called it in school.

[Ek weet nie]

The four fours maths puzzle

We called it "BOMDAS".

[Wyatt Earp]

The four fours maths puzzle

Yeah, the m/d order doesn't matter.

4/2*3

and

4*3/2

are the same.

[Gerontophile]

The four fours maths puzzle

yes, but if you do the 2*3 bit first, the answer becomes 0.66r not 6

[Wyatt Earp]

The four fours maths puzzle

Yeah, that's different.

[Hofzinser]

The four fours maths puzzle

You need to always have the same numbers above and below the line, as it were (I'm sure there's a better way of putting that).

[Wyatt Earp]

The four fours maths puzzle

The point is that our standard priority of operations reflects a kind of "natural ranking", and according to that natural ranking, divide and times are equal. (The way we prioritise is arbitrary, but in some loose sense it minimises the need for brackets.)

[Evariste Euler Gauss]

The four fours maths puzzle

The choice of acronym matters though.

"BOMDAS" sounds like a name for a trendy new Indian restaurant.

"BODMAS" sounds either like something fitness trainers waffle on about, or alternatively a Satanic festival from Cornwall.