On Friday I posed this puzzle….

Yesterday I bought a strip of multi-coloured cloth.  One third and one quarter of it is black, and the other 8 metres are red.  How long is the strip?

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

If x is the length of the cloth, then

x/3 + x/4 + 8 = x


4x + 3x + 96= 12x

and thus x=19.2 m

Did you solve it?  Any other answers?

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for theKindle (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. The statement “One third and one quarter of it is black” can be interpreted as only 1/3 of the cloth is black (because if 1/3 of it is black, 1/4 is also black).
    That would make the cloth 12m long (x/3 +8 = 8 ->2x/3 =8 ->x=12)

    1. It could be interpreted that way, but only by ignoring vox populi. Ever bought, say, “a pound and a half” of some product?

  2. The second line of the formulas is clearly wrong

    4x + 3x = 96= 12x (which would give x = 8)

    x/3 + x/4 + 8 = x yields

    x – x/3 – x/4 = (12 – 4 – 3)x/12 = 5x/12 = 8

    therefore x = 96/5 = 19.2

    1. I’d call it an excusable typo. “= 96” should be “+ 96”.

      On a classroom blackboard, that’d be easily (and possibly quietly 😉 fixed with a smudge of a finger.

  3. Got it, did it this way:

    1/4 + 1/3 = 3/12 + 4/12 = 7/12 (seven twelfths of the cloth is black, five twelfths of the cloth is red)

    5/12 = 8/length

    length = 8/5*12 = 19.2

  4. Yep; no surprises.

    I went the mental route of (1/3 + 1/4) = 7/12, hence x × 5/12 = 8m.

    Let me take the opportunity to rail a bit against the loosey-goosey handling of units in Richard’s equations above. “8” should be written as “8 m”. Otherwise it’s easy to confuse dimensionless numbers (“3” and “4”) with physical quantities, canceling and losing or mangling units in the process.

    It is good practice to handle givens in the calculation of a problem as symbolic parameters all the way through, and substitute in the numeric values at the end. This makes for more reproducible steps and more robust handling of units, metric or otherwise.

    Calling the cloth remainder length r, we can write up the problem as:
    (1/3 + 1/4) x + r = x,
    r = (1 – 7/12) x, hence
    x = 12/5 r.

    And since today’s value of r = 8 m, we get the stated solution as (12/5 × 8) m, naturally in the same unit.

  5. I got the answer above, but in doing so, I also got the answer that the strip is 8m long. The colours could be changing along the length, rather than along the width, or it could be a checkered pattern with very large squares!

    1. No, no, no. You see, not only would that make not much of a puzzle but we are clearly always encouraged to think laterally here.

    2. Oh, I though I was the one thinking laterally 😉

      From the wording of the question either of us could be right, or even Stephen above who pictured a chequered pattern.

      I think the real solution is that we can’t deduce anything about the length of the piece of fabric because we don’t have enough information.

    3. Actually Anonymous (above) was me! Sorry.

      P.S. Typo “though” should read “thought”

  6. What about this…
    1/4 = 3/12
    1/3 = 4/12
    => 1/4 + 1/3 = 7/12
    => 5/12 of the length = 8m
    => (12 x 8) /5 = 19.2
    (Same answer, just neater explanation)

  7. Not being a mathmathician, I realized that 5/12 = 8m, so I just tried to figure out what 1m was, by dividing 8 by 5, then I added it all up.

    1. But that means there is a twist, oh no. My brain explodes with the paradoxical nature of it all.

  8. The problem I had was parsing this “One third and one quarter of it is black” Is that 1/3 + 1/3*1/4?
    You could say that x/3 + 8 = x is legitmate. If 1/3 of it is black, then already 1/4 of it is black as well. But getting x/3 + x/4 + 8 = x from that phrase of English?

  9. And what if the stripes were in the other direction? The cloth would be 8 meter long! That’s another valid solution.

  10. Nope. It says “…and the OTHER 8 metres are red,” clearly excluding the black portion. If he’d just said, “…and 8 metres are red,” you’d have a point.

    1. Good point, I didn’t spot “the OTHER 8 metres”. I think that makes my otherwise elegant 8m solution less likely to be right.

    2. Not necessarily. It could be interpreted as meaning the “remaining” 8 metres, or the “rest of” the 8 metres. Part of the difficulty is that the “one third”, and “one quarter” parts don’t specifically refer to lengths, widths, areas, or anything else.

    3. It does refer to length, because saying “the other 8 meters are red” refers to that the first black part must also be considered length.

  11. The mathematical solution as stated here is the obvious answer, but I prefer the 8 metr solution, wherein the multi-colors are stripes running the length of the fabric, (as is indeed more likely, since it constitutes more of a ‘pattern’ one would see in material). In this
    solution, one could cut away the 1/3 and 1/4 black widths of the cloth, leaving the remainig 5/12 red width portion at the original 8 meters. This is a legitimate solution: 8 meters are black, another 8 meters are black, ” and the other 8 metres are red.”

  12. Richard, would you like me to audit your Q&A’s because, unless you are making deliberate mistakes to get debate going, you have got rather sloppy of late.

  13. Be patient. This man has now occupied the throne of global puzzle psychology king and should be allowed to eat drink and be merry to celebrate along with a little inattention to detail, which might be viewed as part of the puzzle.

    1. “This man has now occupied the throne of global puzzle psychology king” – he can’t even get the answers right.
      In the great man’s words – DREAM ON

  14. You could also solve it by plain arithmetic. 1/4 or 3/12 is black and 1/3 or 4/12 is black, so 7/12 is black. This leaves 5/12 to be red. Since we are told that 8 meters are red that means that 8/(5/12) meters is the whole length. 8/(5/12)=96/5=19.2.

  15. Whilst the answer as given was obvious, the wording was very odd, and it was not obvious to me that was the intended meaning. Moreover, with that interpretation, it’s a really strange piece of cloth whose like I have never seen.

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