# Math Game: Combining 4 Fours... **URL:** \ **Category:** Factual Questions\ **Created:** [February 5, 2003, 3:37am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870 "2003-02-05T03:37:27Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [February 5, 2003, 4:36am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/21 "2003-02-05T04:36:18Z")
> [@](#): > > \*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. ---
**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\ **Post date:** [February 5, 2003, 4:41am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/22 "2003-02-05T04:41:36Z")
> [@](#): > > \*Originally posted by ultrafilter \* > \*\*113 = 112 + 4/4. \*\* No. ---
**Author:** ![MonkeyMensch](https://avatars.discourse-cdn.com/v4/letter/m/82dd89/32.png) [@MonkeyMensch](https://boards.straightdope.com/u/MonkeyMensch)\ **Post date:** [February 5, 2003, 4:52am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/23 "2003-02-05T04:52:19Z")
So do I just misunderstand the term numberline? ---
**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\ **Post date:** [February 5, 2003, 4:53am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/24 "2003-02-05T04:53:55Z")
I think the OP meant the set of natural numbers. ---
**Author:** ![MonkeyMensch](https://avatars.discourse-cdn.com/v4/letter/m/82dd89/32.png) [@MonkeyMensch](https://boards.straightdope.com/u/MonkeyMensch)\ **Post date:** [February 5, 2003, 5:10am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/25 "2003-02-05T05:10:06Z")
Thanks. So can the 4x4 rule generate all rational munbers between 0 and 1? ---
**Author:** ![Urban\_Ranger](https://avatars.discourse-cdn.com/v4/letter/u/e9c0ed/32.png) [@Urban\_Ranger](https://boards.straightdope.com/u/Urban_Ranger)\ **Post date:** [February 5, 2003, 5:15am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/26 "2003-02-05T05:15:50Z")
> [@](#): > > \*Originally posted by Q.E.D. \* > 6 = 4+4+4/(sqrt(4)) (ok, i cheated here :-P) 6 = 4 + (4+4)/4 ---
**Author:** ![Urban\_Ranger](https://avatars.discourse-cdn.com/v4/letter/u/e9c0ed/32.png) [@Urban\_Ranger](https://boards.straightdope.com/u/Urban_Ranger)\ **Post date:** [February 5, 2003, 5:20am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/27 "2003-02-05T05:20:27Z")
The upper bound is 4[sup]4[sup]4[sup]4[/sup][/sup][/sup], but that’s probably cheating. ---
**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\ **Post date:** [February 5, 2003, 5:22am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/28 "2003-02-05T05:22:26Z")
> [@](#): > > \*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. ---
**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\ **Post date:** [February 5, 2003, 5:25am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/29 "2003-02-05T05:25:58Z")
> [@](#): > > \*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]? ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [February 5, 2003, 5:42am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/30 "2003-02-05T05:42:06Z")
> [@](#): > > \*Originally posted by Gyan9 \* > \*\*How do you define 4! without writing 1,2,3 ? \*\* How do you define 4 without 1, 2, and 3? ---
**Author:** ![Urban\_Ranger](https://avatars.discourse-cdn.com/v4/letter/u/e9c0ed/32.png) [@Urban\_Ranger](https://boards.straightdope.com/u/Urban_Ranger)\ **Post date:** [February 5, 2003, 5:43am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/31 "2003-02-05T05:43:09Z")
I don’t think the factorial operator is legal. 4! = 4_3_2\*1 but you are not allowed to use these other numbers. ---
**Author:** ![Urban\_Ranger](https://avatars.discourse-cdn.com/v4/letter/u/e9c0ed/32.png) [@Urban\_Ranger](https://boards.straightdope.com/u/Urban_Ranger)\ **Post date:** [February 5, 2003, 5:44am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/32 "2003-02-05T05:44:34Z")
Oops. Anyway, the exponential operation is probably also illegal, because 4[sup]4[/sup] is 4_4_4\*4, and you are only allowed to use four 4’s. ---
**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\ **Post date:** [February 5, 2003, 5:56am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/33 "2003-02-05T05:56:42Z")
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. ---
**Author:** ![Urban\_Ranger](https://avatars.discourse-cdn.com/v4/letter/u/e9c0ed/32.png) [@Urban\_Ranger](https://boards.straightdope.com/u/Urban_Ranger)\ **Post date:** [February 5, 2003, 6:05am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/34 "2003-02-05T06:05:37Z")
That’s true, though multiplication is taken as elementary for the most part. ---
**Author:** ![Sunspace](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/sunspace/32/1250_2.png) [@Sunspace](https://boards.straightdope.com/u/Sunspace)\ **Post date:** [February 5, 2003, 6:11am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/35 "2003-02-05T06:11:47Z")
“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? 🙂 ---
**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\ **Post date:** [February 5, 2003, 6:12am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/36 "2003-02-05T06:12:06Z")
Bottom line is the OP stated those operations were legal, so I’ll go with those stated ground rules. ---
**Author:** ![The\_Weak\_Force](https://avatars.discourse-cdn.com/v4/letter/t/9dc877/32.png) [@The\_Weak\_Force](https://boards.straightdope.com/u/The_Weak_Force)\ **Post date:** [February 5, 2003, 6:15am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/37 "2003-02-05T06:15:34Z")
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 ---
**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\ **Post date:** [February 5, 2003, 6:16am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/38 "2003-02-05T06:16:50Z")
> [@](#): > > \*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… ---
**Author:** ![The\_Weak\_Force](https://avatars.discourse-cdn.com/v4/letter/t/9dc877/32.png) [@The\_Weak\_Force](https://boards.straightdope.com/u/The_Weak_Force)\ **Post date:** [February 5, 2003, 6:24am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/39 "2003-02-05T06:24:34Z")
Sunspace, don’t make me break out the chained arrows. How big is 4 -\> 4 -\> 4 -\> 4 😃 ---
**Author:** ![Sunspace](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/sunspace/32/1250_2.png) [@Sunspace](https://boards.straightdope.com/u/Sunspace)\ **Post date:** [February 5, 2003, 6:24am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/40 "2003-02-05T06:24:57Z")
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? [Previous page](https://boards.straightdope.com/t/math-game-combining-4-fours/152870.md?page=1) [Next page](https://boards.straightdope.com/t/math-game-combining-4-fours/152870.md?page=3)