8
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Move a single match to make the following expression true:

6 + 4 = 4

So far I've found 6 valid solutions, can you find all 6? (or more!)

Edit: Lots of great, creative answers! All 6 I found are represented among the answers, plus quite a few more!

A few clarifications:

  1. I changed "equation" to "expression" in the title and question, for the mathematically pedantic. This officially allows for inequalities.
  2. The expression still has to logically evaluate to true or false in a boolean sense. So "8 + 4 - 4" wouldn't count, even though it might be treated as TRUE by most programming languages.
  3. You're not limited to "perfectly-formed" LED-style numbers, although using a single vertical match as a "1" is kind of stretching it. But I'll allow it, if it gets us more good answers.
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3
  • 1
    $\begingroup$ do inequalities count? $\endgroup$
    – Matt
    Jan 19, 2017 at 21:31
  • $\begingroup$ Inequalities are fine, I've clarified the question. $\endgroup$
    – BradC
    Jan 19, 2017 at 23:00
  • $\begingroup$ A mathematical expression does not consist of an equals sign, larger than/smaller than, etc. So changing 'equation' to 'expression' is making the problem about a thousand times worse. $\endgroup$
    – Wen1now
    Jan 21, 2017 at 3:19

6 Answers 6

5
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Here are the compiled expressions from the current answers. These are from Glorfindel's answer, Matt's answer, humn's comment, as well as the various other comments on other answers. I have also added alphabetic labels to identify the unique relations and the different expressions.

Expressions A through G are either equations and strict inequalities with one operator:

[A] 0 + 4 = 4
[B] 5 + 4 = 9
[C] 8 - 4 = 4
[D] 6 + 4 > 4
[E] 6 ≠ 4 - 4
[F] 5 + 4 ≠ 4
[G] 6 - 4 ≠ 4

If expressions can have multiple inequality operators, then you also get expressions H and I:

[H] 5 ≠ ­4 ­= ­4
[I] 6 > 4 = 4

In total, that's 9 unique relations listed above.

For some of the inequalities that contain ≠, you can rewrite the expression with a different inequality operator. Because these new expressions have similar structure, I indicate them with a * suffix.

If the + was changed into ≠, then the + can also be changed into a negated strict inequality (either ≮ or ≯). Or, if the = was changed into ≠, then the = can also be changed into a non-strict inequality (either ≥ or ≤).

[E*] 6 ≮ 4 - 4
[F*] 5 + 4 ≥ 4
[G*] 6 - 4 ≤ 4
[H*] 5 ≮ ­4 ­= ­4

This raises the total to 13 different expressions.

If you allow a single vertical match to count as the number 1, then there are more expressions that you can form, using the same rules as above:

[J] 614 ≠ 4
[J*] 614 ≥ 4
[K] 6 + 4 ≠ 11
[K*] 6 + 4 ≤ 11
[L] 6 + 11 ≠ 4
[L*] 6 + 11 ≥ 4

These additional expressions raise the total to 19.

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5
  • $\begingroup$ This answer would be well referenced by other matchstick puzzles $\endgroup$
    – rotta
    Jan 20, 2017 at 6:20
  • $\begingroup$ @humn Could you elaborate what you mean? $\endgroup$
    – MikeQ
    Jan 20, 2017 at 9:01
  • $\begingroup$ This answer shows and explains so many different kinds of possibilities that other matchstick puzzles, perhaps the tag blurb too, would be well off to mention it rather than having to go through the same kinds of comment questions again and again. This answer also provides ideas for new puzzles. A valuable resource all of a sudden, would even serve well as the basis for a good, new, self-answered question about making math-chstick puzzles. $\endgroup$
    – rotta
    Jan 20, 2017 at 9:04
  • 1
    $\begingroup$ Our/My methods of generating solutions for this specific puzzle can't necessarily be generalized to all other matchstick puzzles... But maybe there is a formula? I'll look into this. $\endgroup$
    – MikeQ
    Jan 20, 2017 at 9:11
  • 1
    $\begingroup$ Good summary, Mike. To me the two "purest" answers (ie, the most likely solution the original puzzle maker envisioned) are A) and C), since they both maintain the equality and keep full LED-style numbers, so to me they almost belong in their own group. The 6 solutions I found before posting were A, B, C, E, F, G. $\endgroup$
    – BradC
    Jan 20, 2017 at 14:50
10
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The first obvious solution is

8 - 4 = 4

Another one which might be valid depending on how you write the 9 was mentioned first by @Written in the comments and then by two other answerers:

5 + 4 = 9

Another one:

0 + 4 = 4

A couple of others, all from the same family:

6 - 4 $\neq$ 4
5 + 4 $\neq$ 4
614 $\neq$ 4
6 + 4 $\neq$ 4
6 + 11 $\neq$ 4

(the last one uses a 6 without the top segment, and some of the 1's use only one match)

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12
  • 1
    $\begingroup$ @Written Can't make a 3 without moving 2 sticks $\endgroup$ Jan 19, 2017 at 21:29
  • 2
    $\begingroup$ Some of your solutions hold for Planck's constant h, upside down $\endgroup$
    – rotta
    Jan 19, 2017 at 21:30
  • 1
    $\begingroup$ 6 + 4 = 11 (in base 9) $\endgroup$
    – MikeQ
    Jan 19, 2017 at 21:33
  • 1
    $\begingroup$ There's also 6 ≠ 4 - 4, and 6 + 4 ≠ 11. $\endgroup$
    – MikeQ
    Jan 19, 2017 at 21:56
  • 1
    $\begingroup$ Good answers, although I'm not sure that a single vertical match by itself should qualify as a "1". $\endgroup$
    – BradC
    Jan 19, 2017 at 22:30
4
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There's this one

5 + 4 = 9
take the bottom left from the 6, and add it to the top of the last 4.

You can also stretch to

6 + 4 > 4
If you like your angle brackets flat on the bottom.

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6
  • $\begingroup$ Misses the bottom of the 9 on a standard 7-segment display. :( $\endgroup$ Jan 19, 2017 at 21:27
  • $\begingroup$ still recognizable as a 9, though $\endgroup$
    – Matt
    Jan 19, 2017 at 21:28
  • $\begingroup$ Similar: 6 > 4 = 4 $\endgroup$
    – MikeQ
    Jan 19, 2017 at 21:45
  • 1
    $\begingroup$ as a programmer, I see that as the result of (6 > 4) is equal to 4, and it isn't, usually. that's not to say it doesn't make mathematical sense. $\endgroup$
    – Matt
    Jan 19, 2017 at 22:03
  • $\begingroup$ @Matt 6 > 4 is true also 4=4 is true, so it's mathematically correct, it's not like in programming where they would be compared differently so I see what you are saying. $\endgroup$
    – Melkor
    Jan 19, 2017 at 22:12
3
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Rather than using the "not equal" symbol, you can take a match from the + and move it by the = to make 6 - 4 ≤ 4 Note that the character can be written with the middle line being flat.

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1
  • $\begingroup$ Very nice. Using that same strategy, you could do 5 + 4 ≥ 4. $\endgroup$
    – BradC
    Jan 19, 2017 at 22:41
2
$\begingroup$

5+4=9

Another one I found, that works.

6+4∼4

If you can bend matches.

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5
  • $\begingroup$ beat ya by a whole 10 seconds, haha $\endgroup$
    – Matt
    Jan 19, 2017 at 21:24
  • $\begingroup$ Misses the bottom of the 9 on a standard 7-segment display. :( $\endgroup$ Jan 19, 2017 at 21:27
  • $\begingroup$ @Matt the ninja :) $\endgroup$
    – Melkor
    Jan 19, 2017 at 21:27
  • $\begingroup$ ~? B..but.... the equation.... $\endgroup$ Jan 19, 2017 at 21:32
  • $\begingroup$ @IanMacDonald: I don't know what you mean by "standard". There are two reasonably-common shapes for each of "1", "6", "7", and "9" [whether the "1" uses the left segments or the right segments is irrelevant here, but could matter in some contexts]. The versions with fewer segments used to be more common in the days of diode logic, which would use N diodes to define a shape with N lit segments (so adding crossbars to the 6 and 9 would require the use of two extra diodes). $\endgroup$
    – supercat
    Jan 19, 2017 at 22:27
2
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Move the middle horizontal stick of 6 to make it:
0 + 4 = 4

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1
  • $\begingroup$ Nice answer.... $\endgroup$
    – KSR
    Jan 20, 2017 at 3:59

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