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Given: six 6s
Goal: Construct an expression with value $1,000$

Rules:

  • Use all six digits in each expression.
  • No other digits are allowed.
  • Concatenation of digits is allowed.
  • One horizontal fraction bar is required.
  • No grouping symbols are allowed.
  • At most two subtraction signs are allowed.
  • At most two decimal points are allowed. (They are for regular use of the decimal point here, and not for repeated decimals.)

Other than the fraction bar for division, any subtraction signs used, and the role of the decimal point(s), no other arithmetic or any other different operations/functions//symbols are allowed, except as previously mentioned.

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  • $\begingroup$ All of these answers fall into a pattern, so if you get one or more, it is possible that they may help you to get other answers in the set of answers. $\endgroup$
    – Olive Stemforn
    Commented Mar 23, 2016 at 15:17
  • $\begingroup$ How many answers are you expecting? $\endgroup$
    – question_asker
    Commented Mar 23, 2016 at 15:23
  • $\begingroup$ @question_asker - I am expecting six answers, not including any variations that result by multiplying the numerator and the denominator by -1. $\endgroup$
    – Olive Stemforn
    Commented Mar 23, 2016 at 15:33

4 Answers 4

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Answer #1 is

$\frac{666 - 66}{.6} = 1000$ $~~~~$and the variation$~~~~$ $\frac{66 - 666}{-.6} = 1000$

Answer #2 is:

$\frac{666 - 6}{.66} = 1000$ $~~~~$and the variation$~~~~$ $\frac{6 - 666}{-.66} = 1000$

Answer #3 is:

$\frac{66 - 6}{.66-.6} = 1000$ $~~~~$and the variation$~~~~$ $\frac{6 - 66}{.6-.66} = 1000$

Answer #4 (due to Narmer) is:

$\frac{ 6}{.666-.66} = 1000$ $~~~~$and the variation$~~~~$ $\frac{-6}{.66-.666} = 1000$

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    $\begingroup$ Helping out here $\cfrac{6}{.666-.66}$ = 1000 $\endgroup$
    – Narmer
    Commented Mar 23, 2016 at 15:47
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    $\begingroup$ @ Narmer - Why don't you post it as a separate solution to get credit for it with arrows up for it!? $\endgroup$
    – Olive Stemforn
    Commented Mar 23, 2016 at 15:51
  • $\begingroup$ @OliveStemforn Nah, just having a little bit of fun, upvotes are not needed for it ;) $\endgroup$
    – Narmer
    Commented Mar 23, 2016 at 15:53
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    $\begingroup$ What about $\cfrac{6}{.6}\cdot\cfrac{66}{.66}$ = 1000? $\endgroup$
    – Narmer
    Commented Mar 23, 2016 at 16:03
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    $\begingroup$ @Narmer And from that $\cfrac{666}{.666} = 1000$ $\endgroup$
    – Lacklub
    Commented Mar 23, 2016 at 17:11
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I'll have a go at it

$\cfrac{666 - 66}{.6}$

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  • $\begingroup$ Good. Yes, this is one of them so far. $\endgroup$
    – Olive Stemforn
    Commented Mar 23, 2016 at 15:20
  • $\begingroup$ * * * * At this moment, other than Narmer's solution, which was not posted as a public solution, as well as all variations (with their changes in signs), there is one more basic one I'm looking for. But I'm going on a break away from here for an hour, give or take. * * * * $\endgroup$
    – Olive Stemforn
    Commented Mar 23, 2016 at 15:57
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Here's another onetwo

$\frac{666-6}{.66}$ or $\frac{666}{.666}$

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  • $\begingroup$ The first one was covered by JoeMalpass (first). Your second one is new. (edit) $\endgroup$
    – Olive Stemforn
    Commented Mar 23, 2016 at 15:36
  • $\begingroup$ According to the number of minutes ago on my screen, JoeMalpass posted it 2 minutes before you did. $\endgroup$
    – Olive Stemforn
    Commented Mar 23, 2016 at 15:39
  • $\begingroup$ I see I am wrong. $\endgroup$
    – Olive Stemforn
    Commented Mar 23, 2016 at 15:44
  • $\begingroup$ @olive no problem! I'm not super concerned with being first, tbh $\endgroup$
    – question_asker
    Commented Mar 23, 2016 at 15:45
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    $\begingroup$ @question_asker & Lawrence, that was it - the cfrac Vs. frac. That's what I get for copy/paste/editing instead of learning the syntax from scratch, hahaha. $\endgroup$
    – CactusCake
    Commented Mar 23, 2016 at 17:46
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I've got:

$$\frac{666 - 66}{.6} $$

And I think a new one:

$$\frac{\frac{6}{.6}}{\frac{.6}{66-6}}$$

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  • $\begingroup$ Dennis posted this an hour ago. But congrats on getting a correct answer $\endgroup$
    – Bishop
    Commented Mar 23, 2016 at 16:51
  • $\begingroup$ @Bishop yeah, just saw that so added a new one $\endgroup$
    – Paul Evans
    Commented Mar 23, 2016 at 16:57
  • $\begingroup$ I think this is disallowed by the rules (which sucks, because I like it) $\endgroup$
    – question_asker
    Commented Mar 23, 2016 at 17:02
  • $\begingroup$ @ Paul Evans - It violates the rules ( it uses two horizontal fraction bars instead of the required one), but I like the creativity. $\endgroup$
    – Olive Stemforn
    Commented Mar 23, 2016 at 17:18
  • $\begingroup$ * * * All of the answers have been posted in some form or another. * * * $\endgroup$
    – Olive Stemforn
    Commented Mar 23, 2016 at 17:24

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