# 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:** 4
**Author:** ![dylan\_73](https://avatars.discourse-cdn.com/v4/letter/d/ecccb3/32.png) [@dylan\_73](https://boards.straightdope.com/u/dylan_73)\ **Post date:** [February 5, 2003, 4:26pm UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/61 "2003-02-05T16:26:53Z")
25 = 4! + (4/4)[sup]4[/sup] 26 = 4! + (4+4)/4 27 = 4! + (4-(4/4) 28 = 4! + 4[sup]4-4[/sup] 29 = 4! + 4 + (4/4) Kind of pointless, but this is fun. ---
**Author:** ![Ethilrist](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ethilrist/32/4604_2.png) [@Ethilrist](https://boards.straightdope.com/u/Ethilrist)\ **Post date:** [February 5, 2003, 4:27pm UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/62 "2003-02-05T16:27:00Z")
> [@](#): > > \*Originally posted by Jinx \* > \*\*The simplest example: 4+4-4+4 = 0, for starters. \*\* umm… way to get off to a bad start, dude. Try 4+4-4-4. ---
**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, 4:34pm UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/63 "2003-02-05T16:34:33Z")
He probably meant (4+4)-(4+4) = 0. ---
**Author:** ![dylan\_73](https://avatars.discourse-cdn.com/v4/letter/d/ecccb3/32.png) [@dylan\_73](https://boards.straightdope.com/u/dylan_73)\ **Post date:** [February 5, 2003, 4:35pm UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/64 "2003-02-05T16:35:05Z")
Oops. Patch 28 to read 28 = 4! + 4[sup]4/4[/sup] ---
**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, 4:37pm UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/65 "2003-02-05T16:37:14Z")
Sorry about the radical signs. **Sunspace** , a -\> 2 -\> b is supposed to be a^(a^(a^…^a…)), where there are b a’s. We can call this a^^b. a -\> 3 -\> a is supposed to be a^^(a^^(a^^…^^a…)), where there are b a’s. We can call this a^^^b. Etc. That’s how you do it if there are only 2 arrows. If there are 3 or more, then you can simplify it by these rules: [ol][li]If the last number is 1, then you can remove it, along with the arrow.[/li][li]If the second to last number is 1, then you can remove the last two numbers and arrows.[/li][li]If they’re both greater than one, then you do the following:[/li]~Decrease the last number by 1. ~Replace the second to last number with the original expression itself, except with the second to last number decreased by 1. [/ol] The simplification rules are mutually exclusive; only one of them can ever be used on a given chained arrow expression at once. ---
**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, 5:18pm UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/66 "2003-02-05T17:18:54Z")
Those of you discussing chained arrow notation might be interested in [this thread](http://boards.straightdope.com/sdmb/showthread.php?s=&threadid=149305), where I used a similar recursive notation to construct some very fast-growing functions. ---
**Author:** ![Shade](https://avatars.discourse-cdn.com/v4/letter/s/2bfe46/32.png) [@Shade](https://boards.straightdope.com/u/Shade)\ **Post date:** [February 5, 2003, 5:45pm UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/67 "2003-02-05T17:45:48Z")
(4-4), [symbol]Ö[/symbol]4, 4, … Or how about 4444!!! ? (I know, I know, it’d take a lot more to beat -\>s, but you get the idea) ---
**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, 5:49pm UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/68 "2003-02-05T17:49:26Z")
Mind-boggling. And intensely interesting. Thanks, all. ---
**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, 6:09pm UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/69 "2003-02-05T18:09:02Z")
And I’m now addicted to Mathworld. ---
**Author:** ![II\_Gyan\_II](https://avatars.discourse-cdn.com/v4/letter/i/bbe5ce/32.png) [@II\_Gyan\_II](https://boards.straightdope.com/u/II_Gyan_II)\ **Post date:** [February 5, 2003, 8:00pm UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/70 "2003-02-05T20:00:50Z")
> [@](#): > > \*Originally posted by RM Mentock \* > \*\*Gyan9 grabbed me and dragged me in. Sorry we got mud on y’all. \*\* 4! = 1_2_3\*4 but the number 4 itself can just be an elementary definition. That is, you look at any group of 4 objects and call it one unit and refuse to acknowledge any smaller group as a single unit. But, with factorials, those smallers units have to come into play. ---
**Author:** ![cybersnark](https://avatars.discourse-cdn.com/v4/letter/c/b9e5f3/32.png) [@cybersnark](https://boards.straightdope.com/u/cybersnark)\ **Post date:** [February 5, 2003, 8:06pm UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/71 "2003-02-05T20:06:55Z")
Do we have 29 yet? 4! + 4 + (4/4) ---
**Author:** ![cybersnark](https://avatars.discourse-cdn.com/v4/letter/c/b9e5f3/32.png) [@cybersnark](https://boards.straightdope.com/u/cybersnark)\ **Post date:** [February 5, 2003, 8:09pm UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/72 "2003-02-05T20:09:09Z")
Oh. Oops. ---
**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, 8:13pm UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/73 "2003-02-05T20:13:09Z")
> [@](#): > > _Originally posted by Gyan9 \* > \*\*4! = 1_2_3_4 > > but the number 4 itself can just be an elementary definition. > > That is, you look at any group of 4 objects and call it one unit and refuse to acknowledge any smaller group as a single unit.\*\* Not if you allow 4/4, or sqrt(4), or even 4 - 4. ---
**Author:** ![Splanky](https://avatars.discourse-cdn.com/v4/letter/s/b2d939/32.png) [@Splanky](https://boards.straightdope.com/u/Splanky)\ **Post date:** [February 5, 2003, 8:31pm UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/74 "2003-02-05T20:31:02Z")
30= 4! + 4 + (sqrt(4)) ---
**Author:** ![Splanky](https://avatars.discourse-cdn.com/v4/letter/s/b2d939/32.png) [@Splanky](https://boards.straightdope.com/u/Splanky)\ **Post date:** [February 5, 2003, 8:35pm UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/75 "2003-02-05T20:35:22Z")
31= 4 x 4 x (sqrt(4)) - 40 32= 4 x 4 x (sqrt(4)) x 40 ---
**Author:** ![Splanky](https://avatars.discourse-cdn.com/v4/letter/s/b2d939/32.png) [@Splanky](https://boards.straightdope.com/u/Splanky)\ **Post date:** [February 5, 2003, 8:36pm UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/76 "2003-02-05T20:36:39Z")
Oops…those zeroes are supposed to be superscripted…I thought I knew how to but I guess I was wrong. Can anyone enlighten me on how to do it? ---
**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, 8:40pm UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/77 "2003-02-05T20:40:59Z")
Use [**sup] and [**/sup]. ---
**Author:** ![Splanky](https://avatars.discourse-cdn.com/v4/letter/s/b2d939/32.png) [@Splanky](https://boards.straightdope.com/u/Splanky)\ **Post date:** [February 5, 2003, 8:54pm UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/78 "2003-02-05T20:54:56Z")
33= 4! x (sqrt(4)) - 4[sup]2[/sup] + 4[sup]0[/sup] 34= 4 x 4 x (sqrt(4)) + (sqrt(4)) 35= 36= 4 x 4 x (sqrt(4)) + 4 ---
**Author:** ![Splanky](https://avatars.discourse-cdn.com/v4/letter/s/b2d939/32.png) [@Splanky](https://boards.straightdope.com/u/Splanky)\ **Post date:** [February 5, 2003, 9:19pm UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/79 "2003-02-05T21:19:15Z")
Eh. I used 2 roots and 2 and 0 powers so none of those work if you disallow them. Ah well. ---
**Author:** ![cmosdes](https://avatars.discourse-cdn.com/v4/letter/c/a587f6/32.png) [@cmosdes](https://boards.straightdope.com/u/cmosdes)\ **Post date:** [February 5, 2003, 10:23pm UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/80 "2003-02-05T22:23:54Z")
This is actually a very old puzzle from [rec.puzzles](http://groups.google.com/groups?hl=en&lr=&ie=UTF-8&oe=UTF-8&threadm=394E340E.4E13C6D2%40someplace.com&rnum=1&prev=/groups%3Fq%3D4%2Bsqrt%2Blog%2Bgroup:rec.puzzles%26hl%3Den%26lr%3D%26ie%3DUTF-8%26oe%3DUTF-8%26selm%3D394E340E.4E13C6D2%2540someplace.com%26rnum%3D1). > [@](#): > > -log[log[sqrt[sqrt[sqrt[…{33 time sqrt}…sqrt[4]]]]]]/log(4)]/log[sqrt(4)] > > This solution uses only 3 times four. If you want to use all the 4, replace > sqrt[4] by 4/sqrt[4]. [Previous page](https://boards.straightdope.com/t/math-game-combining-4-fours/152870.md?page=3)