# 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:** 1
**Author:** ![Jinx](https://avatars.discourse-cdn.com/v4/letter/j/c6cbf5/32.png) [@Jinx](https://boards.straightdope.com/u/Jinx)\ **Post date:** [February 5, 2003, 3:37am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/1 "2003-02-05T03:37:27Z")
Just wondering, has anyone been successful at building the numberline by using only math functions and the number 4? Let’s say, from 1-30. The simplest example: 4+4-4+4 = 0, for starters. Are some numbers simply NOT possible to “create” in this fashion? (Yes, you can use factorials, exponents, etc… It’s been awhile since I’ve thought about this. IIRC, I think 17 or maybe 23, as WAGs, might be such numbers. Hmm…what about the number “7”? That might be a tricky one, too. Just wondering if other SDopers have played this game and what their findings are… - Jinx ---
**Author:** ![C\_K\_Dexter\_Haven](https://avatars.discourse-cdn.com/v4/letter/c/b2d939/32.png) [@C\_K\_Dexter\_Haven](https://boards.straightdope.com/u/C_K_Dexter_Haven)\ **Post date:** [February 5, 2003, 3:46am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/2 "2003-02-05T03:46:54Z")
7 = 4 + 4 - (4/4) 17 = 4 x 4 + (4/4) ---
**Author:** ![Jinx](https://avatars.discourse-cdn.com/v4/letter/j/c6cbf5/32.png) [@Jinx](https://boards.straightdope.com/u/Jinx)\ **Post date:** [February 5, 2003, 3:50am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/3 "2003-02-05T03:50:40Z")
> [@](#): > > \*Originally posted by C K Dexter Haven \* > \*\*7 = 4 + 4 - (4/4) > 17 = 4 x 4 + (4/4) \*\* OK, ok, ok …so, I am a little rusty! 😉 What about 23 or 27? - Jinx ---
**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, 3:52am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/4 "2003-02-05T03:52:49Z")
> [@](#): > > \*Originally posted by Jinx \* > > Are some numbers simply NOT possible to “create” in this fashion? (Yes, you can use factorials, exponents, etc… No. Any number ‘n’ can be expressed as 1+1+1+1 … n times. Simply represent that summation as (m+m+m+m …n times)/m. ---
**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, 3:55am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/5 "2003-02-05T03:55:27Z")
> [@](#): > > \*Originally posted by Gyan9 \* > \*\*No. > > Any number ‘n’ can be expressed as 1+1+1+1 … n times. > > \*\* I mean, **integer** of course, above, in place of number. Any fraction is p/q. Which can be written as pm/qm. Which is (m+m+m…p times)/(m+m+m…q times) ---
**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, 3:58am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/6 "2003-02-05T03:58:08Z")
23 = 4! - (4)^(4-4) 27 = 4! + 4 - 4/4 ---
**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, 3:58am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/7 "2003-02-05T03:58:43Z")
> [@](#): > > \*Originally posted by Jinx \* > \*\*The simplest example: 4+4-4+4 = 0, for starters. > \*\* Although I’ve already answered, your example confuses me. Either it’s wrong or I haven’t understood your query in the first place. You could simply write 4-4 = 0. In fact, your example resolves to 8, not 0. ---
**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, 3:58am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/8 "2003-02-05T03:58:52Z")
**Gyan9** , I think you misunderstood. You must use exactly 4 4’s. 23 = [symbol]Ö[/symbol](4! - 4/4)[sup][symbol]Ö/symbol[/sup] ---
**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, 3:59am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/9 "2003-02-05T03:59:08Z")
There are two functions by which many of the familiar functions can be constructed. They are Z(x) defined by Z(x) = 0, and N(x), given by N(x) = x + 1. Given these two functions, it’s quite easy to map 4 onto any natural number. To wit: 0 is Z(4), 1 is N(Z(4)), 2 is N(N(Z(4))), and so forth. ---
**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, 3:59am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/10 "2003-02-05T03:59:41Z")
I don’t think factorial is allowed, after all, it’s just a shorthand for muliplying a whole bunch of numbers. ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [February 5, 2003, 4:00am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/11 "2003-02-05T04:00:20Z")
Depending on which actions you decide to allow, you can easily go up into the triple digits. I wasted a lot of time on time many, many years ago. ---
**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, 4:00am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/12 "2003-02-05T04:00:58Z")
> [@](#): > > \*Originally posted by RM Mentock \* > \*\*23 = 4! - (4)^(4-4) > 27 = 4! + 4 - 4/4 \*\* How do you define 4! without writing 1,2,3 ? ---
**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, 4:06am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/13 "2003-02-05T04:06:27Z")
Youch…triple post! 1 = 4/4_4/4 2 = 4/4_4/[symbol]Ö[/symbol]4 Can we use recurring decimals? Taking .4’ to mean .4444444…, 3 = (4 + 4)/(4[symbol]Ö/symbol) ---
**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, 4:11am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/14 "2003-02-05T04:11:04Z")
1 = 4/4 2 = (4_4)/(4+4) 3 = (4+4+4)/4 4 = (4_4)-(4+4) 5 = ((4_4)+4)/4 6 = 4+4+4/(sqrt(4)) (ok, i cheated here :-P) 7 = 4+4-(4/4) 8 = (4_4-4)-4 9 = 4+4+(4/4) ---
**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, 4:13am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/15 "2003-02-05T04:13:45Z")
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. ---
**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, 4:17am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/16 "2003-02-05T04:17:17Z")
10 = 4/.4\*4/4 11 = 44/([symbol]Ö/symbol) 12 = 4! - 4 - 4 - 4 13 = 4! - 44/4 ---
**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, 4:19am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/17 "2003-02-05T04:19:28Z")
Ooops…forgot to go back and change my 1. ---
**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, 4:21am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/18 "2003-02-05T04:21:48Z")
14 = 4 + 4 + 4 + [symbol]Ö[/symbol]4 15 = 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, 4:25am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/19 "2003-02-05T04:25:16Z")
16= 4!- 4-4 17 = 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, 4:27am UTC](https://boards.straightdope.com/t/math-game-combining-4-fours/152870/20 "2003-02-05T04:27:04Z")
Blew it on 16. Should be: 16 = 4+4+4+4 [Next page](https://boards.straightdope.com/t/math-game-combining-4-fours/152870.md?page=2)