Yoak: Simple Arithmetic

I recently got back in touch with an old friend and puzzler and he reminded me of a puzzle that he once told me about that confounded me for weeks.  Faced with a restatement of it, again I couldn’t come up with an answer for the life of me.  The mechanism is painfully simple, but there is something about the particulars here that short my mind out.

Combine the four number 1,3,4,and 6 with operators of addition, subtraction, multiplication and division (and parenthesis to indicate order of operation) to yield an expression equal to 24.

I assure you that you can take this in the most straight-forward manner possible.  You aren’t mean to smoosh them together to get “13” out of 1 and 3.  You aren’t meant to use “1” as a problem number or something of that sort.  An answer will look something like this:


except that is equal to 1.5 .  Your expression must equal 24.

I’m interested to hear if this is as difficult for others as it was for me.


  1. Blaine said,

    October 5, 2009 at 4:37 pm

    There’s a trick to it and it isn’t immediately obvious.[spoiler]I won’t give it away but if you can correctly answer “What is 10 divided by 1/2?” you’ll know how to solve this.[/spoiler]

  2. Andy said,

    October 5, 2009 at 10:18 pm

    I found this to be fairly difficult. I didn’t get it until the hint from Blaine (although then I got it immediately).

  3. Bud said,

    October 7, 2009 at 12:18 pm

    I assume that it’s OK to use the operators more than once, but each number can appear only once?

  4. jyoak said,

    October 7, 2009 at 12:49 pm

    Bud, yes, that’s exactly what I had in mind.  You use a total of three operators plus any number of parenthesis you like and they may either be all different, two the same and one different or all different.  The numbers must be used exactly once each.

  5. jyoak said,

    October 7, 2009 at 12:50 pm

    Andy, I’m happy to have you call it fairly difficult.  I was worried that this was trivial to anyone except me.  :-)  Calling it a trick might be a stretch, but Blaine’s comment would have caused me to realize it instantly as well.

  6. jyoak said,

    October 10, 2009 at 12:17 pm

    Here’s the answer:

    [spoiler] 6/(1-(3/4)) [/spoiler]

  7. Stephen Morris said,

    October 21, 2009 at 7:04 pm

    There is a very long running and popular game show  in Britain which has exactly this kind of problem.  You can try it out for yourself  here.
    If you search for ‘countdown number game’ you will find many sites which solve them for you.  I couldn’t find any which solve this particular problem, they all make an assumption which isn’t valid here.

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