On Friday I posted this puzzle…..

The following equation is obviously wrong:


1) Can you add 4 lines to make it correct?


2) Can you add 3 lines to the equation and make it correct to six decimal places?

If you have not tried to solve it, have a go now.Β  For everyone else, the answer is after the break.


2) Did you solve it?Β  Any other solutions?

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.


    1. I arrived at (2) wondering how one gets a decimal point in there in the first place. Then I realized that some sort of constant would do, from which it was a short leap to the known rational approximations of π.

      The notation in (2) denotes long division as written in the UK, the US, and several other countries. As I commented on Friday, this is by far not universal. I was not familiar with this notation until coming to the US. It does feel rather unusual for me still.

      In any case, a straightforward real, even irrational and downright transcendental puzzle this time, rather than a creative lateral one πŸ˜‰

    1. Very clever to turn the 1 into a plus sign, and your solution allows for infinite decimals in 3 lines.

      I also thought of extending each 1 downwards by using two lines to make an exponent on each side. That way you only need two lines for an absolutely correct answer.

      So you end up with something like this:
      |1 = |13355
      | |
      | |

      which is

      1^1 = 1^13355
      1 = 1

      A lot of cool solutions for this one.

    2. Some formatting problems with the spaces. It’s supposed to look like this:

      |1 = |13355

    1. Only if you weren’t educated in one of a few countries πŸ˜‰

      See my comment above.

  1. First one, yes. For the second I had 11=11+3/355 (=11.008450). That looks like the pi symbol in the answer. No idea but I loved trying it!

  2. If you ignore the spacing in the first equation, as well as operator precedence, you can also have: 1+1=113-3/55

  3. I didn’t think of long division. Even though I am English, I haven’t seen that for years (decades?), the best I could find was:
    1-1 = 1/133551 (0=0.00000748…)

    1. This is how I did it too.

      Personally I think it’s superior to the long division / pi one. πŸ™‚

  4. I got the first one as -11 = 11 + 33 – 55

    I did not understand the second. And still I think this is a weird way of displaying 355 / 113 = pi. It is clever though.

  5. I got both, but it took me absolutely ages (and the help of a friend) to remember that you could represent division like that. I realised quickly that you could turn the left hand side into pi with only 1 line, so it seemed like an obvious candidate. Then I looked up approximations of pi, and found 355/113 fairly easily, it just then took a while to work out how to show that with only 2 lines!

    You can solve #1 with 1 line by using it to cross out ‘3355’ as well πŸ˜›

  6. One can make the equation exact by adding to lines:
    11′ = 113355′
    A constant derived gives 0, thus 0 = 0.

    1. Same thing I came up with. A mathematician says it’s not an equation, a programmer looks at it and says “It’s a correct mathematical statement, which was the challenge.”


    1. No – it’s not the same as a solidus “/”.

      If you ask me, it is simply a historical approach that stuck. I don’t see much logic in that notation either – perhaps that’s why people dread long division so much. In Germany a : b is used to denote a divided by b, and I never heard that complaint there. Regardless of the specific operator symbol, that notation is just more congruent with the semantics.

  7. Damn! I got the first one, was so close to getting the second. I had the idea to make pi, but I couldn’t work out how to get there. Never thought of long division!

    It has just occured to me that you can achieve the same solution if you raise 113 to the power of -1 although that would be a rather awkward way of writing it.

    Ο€ = 113 355

    Good puzzles.

    1. Hmm. That didn’t display how I expected. The minus one is supposed to be a power i.e. Ο€ = 113^(-1) 355.

  8. Part 1 did not say that the four lines had to be added on the right hand side of the equation. It just said to add four lines to make the equation correct. So this works as well:

    -11 = 11 + 33 – 55

    1. That’s the solution I came up with for the first part as well.

      However, I didn’t get the second part ( Ο€ = 113 |Β―355Β― ) though.

  9. I knew I was looking for a way of writing pi = 355/133 at an immediate glance, but couldn’t do it in 3 lines. I don’t consider the proposed solution as a valid notation for division. Who has ever seen this, in the times when they used it, on one side of the = sign? No of course not, you put the answer on top. So it does not actually act as a representation of the answer, and thus is not a notation. What it is, in fact, is a construction line to facilitate the arrangement necessary for carrying out the operation manually on paper. We can compare it with the arrangement and aid lines used for carrying out a long multiplication: that is not a notation for multiplication either (apart from the x sign you may place there).

  10. I got the second one as stated but had a different solution to the first one:
    +1 = 11 – 3 – 35/5
    = 8 – 7
    = 1

    i.e. added a horizontal line to the one on the left to make it a plus sign, added a minus between 11 and 3 and between 3 and 35 on the right side, and finally added a divide by sign between the 35 and the 5 on the right side

    1. Which is in most respects a much better solution, but which does require four lines rather than three. Well, you can’t expect a mathematician to be able to count that high.

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