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Several matchstick digits are still digits when rotated by 180 degrees:

invertible matchstick digits

Note that 2 is rotated to 2.

2 to 2

Thus, we can make a matchstick arithmetic expression which is valid when rotated 180 degrees. For example, 1+1=2.

1+1=2

But, this is trivial because two expressions are the same.

The following matchstick expression produces two different valid equalities: 10+02=12 and 21=20+01.

10+02=12,21=20+01

But, it's not natural because of the leading 0.

Can you find a natural matchstick arithmetic expression which produces two different valid equalities when rotated 180 degrees?

Maybe, the smallest answer is 0+6=6 which is turned into 9=9+0. Or 1x6=6 which is turned into 9=9x1. But, let us exclude trivial equalities like "0+...", "0x...", or "1x..."

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  • $\begingroup$ Your equations don't work, 2 becomes 5, and 5 ~= 1 + 1 $\endgroup$
    – warspyking
    Commented Jan 25, 2015 at 10:59
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    $\begingroup$ @warspyking So I eagerly drew the picture. :-) 2 is rotated to become 2. 5 is rotated to become 5. The mirror image of 2 is 5 and vice versa. $\endgroup$
    – P.-S. Park
    Commented Jan 25, 2015 at 11:23
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    $\begingroup$ Shouldn't the question be "Create an equation" instead of "Create an expression"? $\endgroup$
    – Spikatrix
    Commented Jan 25, 2015 at 14:19
  • $\begingroup$ From your wording of the question it is currently not clear you want the smallest number(s) in your equation. Rephrase? $\endgroup$
    – BmyGuest
    Commented Jan 25, 2015 at 17:27
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    $\begingroup$ Many experts proposed beautiful answers. Why don't you find an answer using only one operation? A trivial example: 1+1+1+1+1+1+1+1=8. $\endgroup$
    – P.-S. Park
    Commented Jan 26, 2015 at 10:32

6 Answers 6

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This first answer is kind of boring, but I think multiplication answers will be hard to come by:

11 x 6 = 66
99 = 9 x 11
You can have as many 1s as you like in one term. Also you can add as many extra x1 terms on either side as you want, just as you can add +0

(Removed an incorrect solution that assumed 2 rotated to 5. Erk).

Here's a more interesting one:

65 + 2 x 6 + 99 - 66 = 58 + 52
25 + 85 = 99 - 66 + 9 x 2 + 59

Edit: And here's another extendable answer:

66 + 55 - 22 = 99
66 = 22 - 55 + 99
You can do this with any number of digits.

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  • $\begingroup$ Good! I believed that your answer of 1-digit numbers was the smallest. Meelo and supercat created the smaller ones, though. $\endgroup$
    – P.-S. Park
    Commented Jan 26, 2015 at 10:18
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I'll start the ball rolling...

enter image description here

and

enter image description here

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  • $\begingroup$ Great rolling! But the smaller number can be expressed. $\endgroup$
    – P.-S. Park
    Commented Jan 25, 2015 at 10:21
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What answer is "best" depends on how one defines "trivial", but if the goal is to have an answer which uses different arithmetical relationships when inverted, how about 

 21-9                                   5-11
------ = 8-6    which becomes    9-8 = ------
 11-5                                   6-12

The values before and after inversion are different, and the things that make one side different have no analog on the other side (whereas in e.g. something like 26+12 = 16+22, the left-side "6" becoming a nine is balanced by having the right-side six become a nine).

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  • $\begingroup$ "Best" answer would vary under various conditions. Maybe Callidus' answer would be the best of the those form using only +/-. I didn't thought of the fantastic answer as yours. $\endgroup$
    – P.-S. Park
    Commented Jan 26, 2015 at 10:27
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65+2-5 = 5-2+59

Technically, it is not a different equality. But I believe it is in the spirit of the puzzle.

If not, just write

65+2-5 = 55+5+2
2+5+55 = 5-2+59

Another solution:

25-19 = 6

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  • $\begingroup$ I like your answer! :-) The last simplest answer is the same as Callidus. $\endgroup$
    – P.-S. Park
    Commented Jan 26, 2015 at 11:51
  • $\begingroup$ Indeed, so I changed it. $\endgroup$
    – Florian F
    Commented Jan 26, 2015 at 17:09
  • $\begingroup$ I especially like 25-19 = 6 since none of the numbers appear in the rotated version. $\endgroup$
    – Joffan
    Commented Jan 26, 2015 at 18:13
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$$6+6-5=5+1+(1+1)-1$$ Rotates to $$1-(1+1)+1+5=5-9+9$$

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    $\begingroup$ It's better to change (1+1) to 2. I don't know how to depict parantheses by matchsticks. :-) Anyway, it's a great answer. $\endgroup$
    – P.-S. Park
    Commented Jan 26, 2015 at 10:21
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How about.

$$0=28^0-28^0$$
$$0^{82}-0^{82}=0$$

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  • $\begingroup$ To be fair, it's only marginally not-an-answer. The two equalities it yields are essentially 1-1=0 and 0-0=0, which are indeed different. But I think they fall into the same "trivial" category as ones of the form x+0=x, explicitly excluded in the question. $\endgroup$
    – Gareth McCaughan ♦
    Commented Sep 3, 2016 at 19:00
  • $\begingroup$ @Gareth Check out the revision history of this answer: originally it was of the 'trivial' type explicitly excluded by the question, which was why I flagged it as NaA, but now I think it's OK. $\endgroup$
    – Rand al'Thor
    Commented Sep 4, 2016 at 17:55
  • $\begingroup$ Ohhh, I see. I should have thought of that. $\endgroup$
    – Gareth McCaughan ♦
    Commented Sep 4, 2016 at 22:40

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