coverOn Friday I posted this puzzle….

A boy and a girl are chatting.

“I am a boy”, said the child with black hair.

“I am a girl”, said the child with white hair.

At least one of them lied. What colour hair does the boy have?

 

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

Both the boy and the girl must have lied, and therefore the boy has white hair.

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.

42 comments

  1. How is this worked out without evidence or clues to work it out from? It seems the answer could only have been guessed at in which case any answer would be permissible.

    1. The two statements are complementary so either they are both telling the truth or both lying. As “At least one of them lied”, they must have both lied. This means that the child claiming to be the girl is, in fact, the boy and has white hair.

    2. There are only two scenarios to consider:
      1) Boy with black hair – says “I am a boy”
      Girl with white hair – says “I am a girl”
      In this case both are telling the truth
      2) Boy with white hair – says “I am a girl”
      Girl with black hair – says “I am a boy”
      In this case both are telling a lie

      Only true statements are that the child with black hair says “I am a boy” and the child with white hair says “I am a girl”. If at least one of them is telling a lie it must be the second scenario.
      Thus the boy has white hair.

    3. We are given the information that at least one of the children has lied. The child with black claims to be a boy. The child with white hair claims to be a girl. At least one of then is lying. Assume only one is lying and is in fact a boy. The other is a boy. At the start, Richard started that there is both a boy and a girl. The only way this could remain the case, with the statements given is that they both lied. If they both lied, the one with white hair claiming to be a girl must be the boy.

    4. Would someone be kind enough to restate the question and the answer?
      This has only been done about four times in the thread above.
      I’ll need to have it shoved under my nose at least 23 times before I get it.

  2. the boy’s hair is brown. or blue. or pink. we can’t tell from the problem, which allows the possibility that the girl has both black and white hair (or the other way around).

    1. RTQ, you are assuming that both children said something, but the question allows for one child to make two statements.

    2. You do realize that only a white headed kid and a brown headed kid are talking? Not pink or whatever color. Also, it is pretty rare to have a white headed kid.

  3. Blimey, I made a meal out of this one. I tangled myself up in all sorts of unnecessary knots and it was so basic in the end. Good puzzle.

  4. I have thought about this for some days and I am not convinced Richard’s answer is correct.

    It is an axiom of Boolean Logic that any value can arise from a falsehood. Thus IF A THEN B is TRUE when a is FALSE,

    If I AM A BOY is FALSE we cannot conclude anything about their hair. Rather we can conclude ANYTHING about their hair, Their hair may be blue or green or made up of cornflakes or any one of many other things all with an equal TRUTH value.

    This puzzle therefore is not solvable in any meaningful sense. There may not even be two children with distinct heads. Conjoined siamese is still possible from the given description.

    In future it will help if the puzzles are set more clearly and with given solutions. Thank you.

    1. I thought this when I first read through it, but I realised that the child existing and speaking is provided as fact. The child with the white hair DID speak. We also know that this child said “I am a girl”. Finally, we know that there are a boy and a girl present. The only statements made by the children are pertinent solely to their status as a boy of a girl.

    2. Good try BG, but neither child was making any statement about their hair colour. That information is conveyed by the poser of the question, and if he/she lied then anything goes.

      So you do have to place some trust in the question setter, but if you don’t, then what’s the point anyway?

    3. The children didn’t need to make a statement about their hair colour, that was already stated (‘said the child with black hair’). There were two children chatting. One had white hair and one had black hair. One was a boy and one was a girl. They were both lying. This is a puzzle which has, quite literally, a black and white solution.
      Now go and save the world instead of being a failure at pedantry.

    4. Thank you all for the support. It is clear that there was “something not quite right” with this puzzle as you cannot reach Truth through multiple Falsehoods.

      Now I know a modern approach is to agree with the doctor Giovanni House (as played by Huge Lorry) that “everyone lies”.. But House does not reach his truth by listening to the lies. He does so by looking further afield.

      I think we have all agreed now that this puzzle was not a suitable one for the website. Let us look forward to better ones in the Future.

    5. skatzsinger, the problem allows for child with black and white hair to speak twice. if so, then one of the child’s two statements must have been false. if the boy made the two statements, then his hair is black and white. but if the girl made the two statements, the color of the boy’s hair remains unknown.

      not quite a “black and white solution.”

    6. ctj,
      Even using your own, somewhat idiosyncratic, logic, your statement
      “if the boy made the two statements, then his hair is black and white.”
      may prove incomplete.

  5. I cannot believe some comments. It’s a simple puzzle. I am really bad at them usually.This one was easy. Some comments below are ridiculously pompous. Remove your head from your arse and just enjoy.
    PS Love your Friday
    Puzzles Richard

  6. When I saw this on Friday, I thought that I had surely missed something as this seemed so simple. I’m not even sure I’d call it a puzzle, it is so obvious.

    From the number of comments proudly claiming to have worked it out and to suggesting some kind of trick to figuring it out , I was convinced that there must be some trick to the wording that had eluded me. But it seems it was as glaringly obvious as I thought it was – so what were all the commenters on Friday going on about? Were they seriously taking pride in working this out? had the spotted an alternative answer? or did they just spectacularly overthink it?

  7. Says If it lied
    CBH: “I am a boy” ; “I am a girl”
    CWH: “I am a girl” ; “I am a boy”
    Only one lying is impossible, hence CBH = girl and CWH = boy.

    1. ………Says ………If it lied
      CBH: “I am a boy” ; then a girl
      CWH: “I am a girl” ; then a boy
      Only one lying is impossible, hence CBH = girl and CWH = boy.

  8. I also thought this one was a nice, straightforward puzzle. Objections to the wording, etc. are getting boring. From the wording of the puzzle, it’s reasonable to assume that (1) there are 2 children, (2) that one clams to be a boy and the other claims to be a girl, and (3) one has white hair and one has black hair. If you want to take exception with reasonable assumptions, I’m sure you can come up with any answer or denial thereof that you like. But I find it tiresome to read comments with objections like this every week.

    I know that sometimes the wording of a puzzle is deliberately tricky, and following reasonable assumptions leads you astray. E.g. “Two US coins add up to 30 cents. One of them isn’t a nickel. What are they?” The reasonable assumption in this case is that neither coin is a nickel, but the trick is–one isn’t a nickel, but the other one is. So the answer is a quarter and a nickel. But this is a trick, if not humorous, puzzle.

    So, if an answer exists without questioning reasonable assumptions, then that’s it.

  9. I got the correct answer even though i missed the first bit which tells us that there is 1 boy and 1 girl. I got as far as a truth table with three possibilities, including the 2 possibilities that they are both boys or both girls. Then i thought I was being clever in noticing the final sentence which refers to “the boy”, which confirmed that there was only 1 boy rather than 2 or 0 boys.

  10. My solution was that children with white hair are vanishingly rare and so the child with white hair must be lying and the boy has black hair.
    I realise now this is totally wrong but thought you might find the logic interesting.

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  12. Suppose the child with white hair is not a girl.
    Then the child with white hair is a boy.
    If the child with white hair is a boy, then the child with black hair is a girl.

    Suppose the child with black hair is not a boy.
    Then the child with black hair is a girl.
    If the child with black hair is a girl, then the child with white hair is a boy.

    Simply put, if one of them lying, then his/her hair is just the other color.

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