# maths puzzle

**URL:** <https://boards.straightdope.com/t/maths-puzzle/151893>\
**Category:** Factual Questions\
**Created:** [January 31, 2003, 4:45pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893 "2003-01-31T16:45:42Z")\
**Posts on this page:** 20\
**Page:** 1

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**Author:** ![the\_idler](https://avatars.discourse-cdn.com/v4/letter/t/e8c25b/32.png) [@the\_idler](https://boards.straightdope.com/u/the_idler)\
**Post date:** [January 31, 2003, 4:45pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893/1 "2003-01-31T16:45:42Z")

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Right, first things first apologies to all those who answered my last query on vinyl for not responding earlier. Thanks for the info, unfortunately we had a cull that week in the office on abuse of internet facilities so i’ve been keeping my head down of late.

Anyway, the boss is away now so…

Add 6 numbers up to equal 20. You cannot use the same number more than once, 0 is not a number, no decimal places, no negative numbers.

I’ve been told that there is a mathematical solution to this but I’m damned if I can figure it out. I actually think it’s a wind-up but any help gratefully received

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**Author:** ![The\_Hamster\_King](https://avatars.discourse-cdn.com/v4/letter/t/8edcca/32.png) [@The\_Hamster\_King](https://boards.straightdope.com/u/The_Hamster_King)\
**Post date:** [January 31, 2003, 4:49pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893/2 "2003-01-31T16:49:52Z")

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If you’re limited to positive integers, it can’t be done.

1+2+3+4+5+6 = 21

Replacing any of these numbers with a different positive integer yeilds a higher total.

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**Author:** ![KneadToKnow](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/kneadtoknow/32/3999_2.png) [@KneadToKnow](https://boards.straightdope.com/u/KneadToKnow)\
**Post date:** [January 31, 2003, 4:51pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893/3 "2003-01-31T16:51:59Z")

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Does it have to be in base 10?

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**Author:** ![the\_idler](https://avatars.discourse-cdn.com/v4/letter/t/e8c25b/32.png) [@the\_idler](https://boards.straightdope.com/u/the_idler)\
**Post date:** [January 31, 2003, 4:56pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893/4 "2003-01-31T16:56:28Z")

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base 10? sorry mate, don’t get you

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<div class="post-metadata">

**Author:** ![the\_idler](https://avatars.discourse-cdn.com/v4/letter/t/e8c25b/32.png) [@the\_idler](https://boards.straightdope.com/u/the_idler)\
**Post date:** [January 31, 2003, 4:58pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893/5 "2003-01-31T16:58:15Z")

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I figured it was impossible, just checking i’d not missed something really obvious. It’ll be some daft answer, some kind of play on words. I’m off for my weekend now, later guys

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**Author:** ![squid](https://avatars.discourse-cdn.com/v4/letter/s/bbce88/32.png) [@squid](https://boards.straightdope.com/u/squid)\
**Post date:** [January 31, 2003, 5:03pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893/6 "2003-01-31T17:03:34Z")

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it says no decimals…how about fractions?

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**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:** [January 31, 2003, 5:54pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893/7 "2003-01-31T17:54:14Z")

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Assuming it is a trick with words:

40/4 + 50/10 + 60/12 = 20

No decimal places, no numbers repeated, no negative numbers.

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**Author:** ![Skammer](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/skammer/32/143_2.png) [@Skammer](https://boards.straightdope.com/u/Skammer)\
**Post date:** [January 31, 2003, 6:01pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893/8 "2003-01-31T18:01:30Z")

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If you allow fractions, it can be even simpler than that:  
2 + 3 + 4 + 5 + 6/1

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**Author:** ![Nametag](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/nametag/32/406_2.png) [@Nametag](https://boards.straightdope.com/u/Nametag)\
**Post date:** [January 31, 2003, 6:05pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893/9 "2003-01-31T18:05:52Z")

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Try it in hexadecimal: 2 + 4 + 5 + 6 + 7 + 8 = 20 (32 in base 10)

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**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:** [January 31, 2003, 6:12pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893/10 "2003-01-31T18:12:13Z")

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> [@](#):
>
> \*Originally posted by Exapno Mapcase \*  
> \*\*Assuming it is a trick with words:
> 
> 40/4 + 50/10 + 60/12 = 20
> 
> No decimal places, no numbers repeated, no negative numbers. \*\*

But you aren’t adding 6 numbers.

10 + 2/1 + 6/3 + 8/4 + 14/7 + 18/9 == 20

Although, I think the answer is some stupid play with words.

While we’re at it, how about a proper math puzzle ?

Six digit number : abcdef

bcdefa = 3 \* abcdef  
cdefab = 2 \* abcdef  
fabcde = 5 \* abcdef  
efabcd = 4 \* abcdef

Should be self-explanatory.

it’s pretty easy if you think a touch laterally.

In any case, those who know of this beforehand, don’t answer.

Those who don’t, post your solutions and explain…

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<div class="post-metadata">

**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:** [January 31, 2003, 6:12pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893/11 "2003-01-31T18:12:13Z")

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> [@](#):
>
> \*Originally posted by Exapno Mapcase \*  
> \*\*Assuming it is a trick with words:
> 
> 40/4 + 50/10 + 60/12 = 20
> 
> No decimal places, no numbers repeated, no negative numbers. \*\*

But you aren’t adding 6 numbers.

10 + 2/1 + 6/3 + 8/4 + 14/7 + 18/9 == 20

Although, I think the answer is some stupid play with words.

While we’re at it, how about a proper math puzzle ?

Six digit number : abcdef

bcdefa = 3 \* abcdef  
cdefab = 2 \* abcdef  
fabcde = 5 \* abcdef  
efabcd = 4 \* abcdef

Should be self-explanatory.

it’s pretty easy if you think a touch laterally.

In any case, those who know of this beforehand, don’t answer.

Those who don’t, post your solutions and explain…

---

<div class="post-metadata">

**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:** [January 31, 2003, 6:19pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893/12 "2003-01-31T18:19:19Z")

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> [@](#):
>
> \*Originally posted by Nametag \*  
> \*\*Try it in hexadecimal: 2 + 4 + 5 + 6 + 7 + 8 = 20 (32 in base 10) \*\*

In general, 20 can be written as 2\*base^1 + 0

= 2\*base (an even number)

Since, sum of first 6 numbers is 21 (odd), replace 6 with 7 in that sequence and you get 1+2+3+4+5+7 = 20 (in base 11).

Now it’s trivial. Keep on incrementing two digits at once. Whatever sum ‘S’ you get, if you use base ‘S/2’, the answer will be 20.

Although, I think the “actual answer” is more inane than this.

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**Author:** ![Munch](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/munch/32/5281_2.png) [@Munch](https://boards.straightdope.com/u/Munch)\
**Post date:** [January 31, 2003, 6:25pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893/13 "2003-01-31T18:25:05Z")

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Does the puzzle say “adding 6 numbers” or “using 6 numbers”?

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**Author:** ![Shade](https://avatars.discourse-cdn.com/v4/letter/s/2bfe46/32.png) [@Shade](https://boards.straightdope.com/u/Shade)\
**Post date:** [January 31, 2003, 7:20pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893/14 "2003-01-31T19:20:47Z")

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You could use modular (clock) arithmetic.

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**Author:** ![universe.zip](https://avatars.discourse-cdn.com/v4/letter/u/a9adbd/32.png) [@universe.zip](https://boards.straightdope.com/u/universe.zip)\
**Post date:** [January 31, 2003, 7:35pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893/15 "2003-01-31T19:35:15Z")

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Okay, **Gyan9** , I got it. Took me some puzzling and now I´m late for a party… 😃 - was fun though, being the geek that I am.  
I´ll post it in a spoiler, though, so nobody can blame me…  
Go on puzzling!!!

It´s 142857. How did I get it? Well, obviously it can´t include 0. Plus, it must be \<200,000, otherwise the multiplication with five would make it a 7 digit number. so a=1.  
so, let´s see - e_f yields 1 as the last digit, so f=7. 2_7=14, so b=4. 5_7=35, so e=5. 4_7=28, so d=8. the figures always run in the same sequence, so we take 14c857 and multiplicate it with 2. the last three digits give you 714 and a 1 to be added to c, so in order for it to yield 5, c must be 2. voila!  
Oh, this sounds terribly fuzzy, I´m not good at putting my thought processes into words, and maybe there´s an easier way - but it works 🙂

well, okay, if you´re trying to figure it out and don´t want to read the whole solution above - here´s a hint…  
only read this if you feel you must!

it´s a six digit number. it remains a six digit number. and there´s some order to it.

here´s another one, and it requires extremely lateral thinking (wrecked my brains on this one):

12  
1112  
3112  
211213  
312213  
212223

what´s the next one?

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<div class="post-metadata">

**Author:** ![universe.zip](https://avatars.discourse-cdn.com/v4/letter/u/a9adbd/32.png) [@universe.zip](https://boards.straightdope.com/u/universe.zip)\
**Post date:** [January 31, 2003, 7:40pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893/16 "2003-01-31T19:40:35Z")

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errrm… _wrecked_ my brains??? certainly looks like it… 😃  
okay, so I wrecked my brains racking my brains trying to figure out the solution. and it ate away my brain´s language centre, methinks.

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**Author:** ![Alescus](https://avatars.discourse-cdn.com/v4/letter/a/3d9bf3/32.png) [@Alescus](https://boards.straightdope.com/u/Alescus)\
**Post date:** [January 31, 2003, 8:05pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893/17 "2003-01-31T20:05:19Z")

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> [@](#):
>
> \*Originally posted by Gyan9 \*  
> \*\*While we’re at it, how about a proper math puzzle ?
> 
> Six digit number : abcdef
> 
> bcdefa = 3 \* abcdef  
> cdefab = 2 \* abcdef  
> fabcde = 5 \* abcdef  
> efabcd = 4 \* abcdef  
> \*\*

Nice little puzzle that, once I’d convinced myself that the answer was not 000000. Apart from the first step, my method was different from **universe.zip** s, but involved the same kind of logical steps.

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**Author:** ![merge](https://avatars.discourse-cdn.com/v4/letter/m/9f8e36/32.png) [@merge](https://boards.straightdope.com/u/merge)\
**Post date:** [January 31, 2003, 8:22pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893/18 "2003-01-31T20:22:37Z")

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> [@](#):
>
> \*Originally posted by universe.zip \*
> 
> what´s the next one? \*\*

I would go with 112223  
I would also say I am wrong…

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<div class="post-metadata">

**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:** [January 31, 2003, 8:54pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893/19 "2003-01-31T20:54:35Z")

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> [@](#):
>
> \*Originally posted by universe.zip \*  
> \*\*
> 
> 12  
> 1112  
> 3112  
> 211213  
> 312213  
> 212223
> 
> what´s the next one? \*\*

114213

Lateral indeed. 😃

**merge**

> [@](#):
>
> I would also say I am wrong…

You’re right.

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<div class="post-metadata">

**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\
**Post date:** [January 31, 2003, 8:59pm UTC](https://boards.straightdope.com/t/maths-puzzle/151893/20 "2003-01-31T20:59:55Z")

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> [@](#):
>
> \*Originally posted by universe.zip \*  
> \*\*here´s another one, and it requires extremely lateral thinking (wrecked my brains on this one):
> 
> 12  
> 1112  
> 3112  
> 211213  
> 312213  
> 212223
> 
> what´s the next one? \*\*

12113213

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