Math Game: Combining 4 Fours...

[ultrafilter]

*Originally posted by The Weak Force *
**Every other time I’ve seen this problem, the allowable operations were all things that a middle schooler could reasonably be expected to know, and nothing that involved writing other numbers or letters (i.e., just notation using non-alphanumeric symbols).

Using this guideline, I’ve seen lists going all the way up to 112. I haven’t seen one for 113. **

113 = 112 + 4/4.

[Q.E.D]

*Originally posted by ultrafilter *
**113 = 112 + 4/4. **

No.

[MonkeyMensch]

So do I just misunderstand the term numberline?

[Q.E.D]

I think the OP meant the set of natural numbers.

[MonkeyMensch]

Thanks.

So can the 4x4 rule generate all rational munbers between 0 and 1?

[Urban_Ranger]

*Originally posted by Q.E.D. *
6 = 4+4+4/(sqrt(4)) (ok, i cheated here :-P)

6 = 4 + (4+4)/4

[Urban_Ranger]

The upper bound is 4[sup]4[sup]4[sup]4[/sup][/sup][/sup], but that’s probably cheating.

[Q.E.D]

*Originally posted by MonkeyMensch *
**Thanks.

So can the 4x4 rule generate all rational munbers between 0 and 1? **

Given that there’s an infinite number of rational numbers between 0 and 1, and only a finite number of operations that can be performed on 4 fours, I don’t see how.

[Q.E.D]

*Originally posted by Urban Ranger *
**The upper bound is 4[sup]4[sup]4[sup]4[/sup][/sup][/sup], but that’s probably cheating. **

Wouldn’t that be 4![sup]4![sup]4![sup]4![/sup][/sup][/sup]?

[RM_Mentock]

*Originally posted by Gyan9 *
**How do you define 4! without writing 1,2,3 ? **

How do you define 4 without 1, 2, and 3?

[Urban_Ranger]

I don’t think the factorial operator is legal. 4! = 432*1 but you are not allowed to use these other numbers.

[Urban_Ranger]

Oops.

Anyway, the exponential operation is probably also illegal, because 4[sup]4[/sup] is 444*4, and you are only allowed to use four 4’s.

[Q.E.D]

If you’re going to look at it that way, 4*4 is illegal too, since it boils down to 4+4+4+4. to my mind, as long as only 4 fours appear in the expression, it’s fine, regardless of what the operation symbolics represent.

[Urban_Ranger]

That’s true, though multiplication is taken as elementary for the most part.

[Sunspace]

“Cheat”? What is this “cheat”? If we’re gonna go big, let’s go big all the way.

Let me introduce tetration: [sup]4[/sup]4. You fans of Rudy Rucker may recognise this: it’s described in his “Infinity and the Mind” .

The expression [sup]4[/sup]4 means the same as Urban Ranger’s 4[sup]4[sup]4[sup]4[/sup][/sup][/sup]. It means “four exponentiated upon itself, four times”.

So… 4[sup]4[/sup] = 4 * 4 * 4 * 4 = 256.

4[sup]4[sup]4[/sup][/sup] = 4sup[/sup] = 4[sup]256[/sup] = 1.34078 *10[sup]154[/sup]. (The exponents associate from the top down to give the largest numbers.)

And, 4[sup]4[sup]4[sup]4[/sup][/sup][/sup] = 4sup[/sup] = 4sup[/sup] = 4[sup](1.34078 *10[sup]154[/sup])[/sup] = …um, a lot.

Now, 4[sup]4[sup]4[sup]4[/sup][/sup][/sup] = [sup]4[/sup]4. Tetration presumably associates downwards as well. What would [sup][sup][sup]4[/sup]4[/sup]4[/sup]4 be?

[Q.E.D]

Bottom line is the OP stated those operations were legal, so I’ll go with those stated ground rules.

[The_Weak_Force]

There is definitely no upper bound if factorial is allowed. Whatever you think the upper bound is, I’ll just tack a couple factorials on it

18 = 44/[symbol]Ö[/symbol]4 - 4

[Q.E.D]

*Originally posted by Sunspace *
**“Cheat”? What is this “cheat”? If we’re gonna go big, let’s go big all the way.

Tetration presumably associates downwards as well. What would [sup][sup][sup]4[/sup]4[/sup]4[/sup]4 be? **

I can top that yet again:

[sup][sup][sup]4![/sup]4![/sup]4![/sup]4!
I don’t know if I can calculate the actual value of either of these. I don’t think my old version of MathCad is up to it, but I’ll give it a shot…

[The_Weak_Force]

Sunspace, don’t make me break out the chained arrows. How big is 4 -> 4 -> 4 -> 4

[Sunspace]

MathCad? Hmm. Seeing the exact value of 4[sup]256[/sup] would be interesting for a start. Less than two hundred digits. Are there any indefinite-precision calculators on the web?