Legal 21, Officer

Young Saul, a budding mathematician and printer, is making himself a fake ID. He needs it to say he’s 21. The problem is he’s not using a computer, but rather he has some symbols he’s bought from the store, and that’s it. He has one 1, one 5, one 6, one 7, and an unlimited supply of + – * / (the operations addition, subtraction, multiplication and division). Using each number exactly once (but you can use any number of +, any number of -, …) how can he get 21 from 1, 5, 6, 7?

Note: you can’t do things like 15+6 = 21. You have to use the four operations as ‘binary’ operations: ((1+5)*6)+7.

Problem submitted by [email protected], phrasing by yours truly.

73 Comments

  1. Steven Miller on November 16, 2017 at 3:32 pm

    Email me at sjm1 AT williams.edu for hints



  2. Steven Miller on November 16, 2017 at 3:30 pm

    Correct. Can use to group, cannot use as binomial (though a clever idea!)



  3. Martin on November 16, 2017 at 2:50 pm

    I see that you use parentheses in your explanation of the problem. But the wording leads me to assume that the binary operation of “n choose k” is not one of those allowed. Correct? But parentheses, as many as needed, are allowed? Thanks.



  4. Judy on November 6, 2017 at 5:22 am

    I can’t figure it out and I want to give it to my students. can you please email me the answer!!!
    Thanks,



  5. Steven Miller on February 11, 2016 at 2:08 am

    All emails to you bounced — please email me at [email protected]



  6. Boone walker on February 11, 2016 at 12:23 am

    5!/(7+1) +6. I know you can’t use factorial but cmon, this is awesome



  7. printer hero on November 25, 2015 at 1:50 am

    i realy enjoyed read all article and posting in this blog , and i like this website design with light purple backgroud ,



  8. Steven Miller on May 12, 2015 at 1:59 am

    nay but very clever!



  9. Philip on May 12, 2015 at 1:51 am

    1+5+6+7 = 21 in base 9. Yea or nay?



  10. Steven Miller on May 28, 2014 at 3:12 pm

    very original, but you can solve without introducing a decimal point.



  11. Mels Duncan on May 26, 2014 at 8:42 pm

    Not sure, but:

    (5 x 6 x 7) x .1 = 210 x .1 = 21



  12. Justin on March 22, 2013 at 10:30 pm

    Does it involve turning a number over (like 5=>2)
    Then it could be 2*7+6+1



  13. Steven Miller on October 18, 2012 at 1:03 am

    email me: sjm1 AT williams.edu



  14. Steven Miller on October 18, 2012 at 1:03 am

    sure: email me: sjm1 AT williams.edu



  15. Patrick on October 15, 2012 at 4:54 pm

    hint please 🙂 Spent too much time on it..



  16. Yolanda Castillo on October 15, 2012 at 4:27 pm

    Hello I know the answer to this problem but I want it to give to my students but what kind of hint can I give them so they can help them to come faster with the answer. Thanks.



  17. Steven Miller on October 6, 2012 at 12:56 am

    no — there is a way using +-*/



  18. Steven Miller on October 6, 2012 at 12:52 am

    highly original, beautiful answer, but there is one using +-*/



  19. Minas Graikos on October 5, 2012 at 6:13 pm

    hello!!!very nice riddle steve!!!i was thinking about the solution and something came up but i dont know if its legal with the rules of the riddle..my thought was: “7 choose 5” + log6(1) where “7 choose 5” is the combinations of seven to 5 and log6(1) is the logarithm to the base of 6



  20. Garrett on October 3, 2012 at 9:30 pm

    Could u do 1*5*6*7=210 and just drop the zero?



  21. Steven Miller on September 22, 2012 at 3:42 pm

    yoav: well done — email me at sjm1 AT williams.edu for more on this problem



  22. Steven Miller on September 19, 2012 at 5:55 am

    sure



  23. bryan on September 18, 2012 at 10:38 pm

    Can i have a hint for this one, I’ve been trying For a few days now and can’t get it. I’m thinking it has to do with fractions am I on the right track?



  24. Steven Miller on September 14, 2012 at 3:03 am

    sorry — I don’t want to post the hints — can you get an email account? sjm1 AT williams.edu



  25. sucks@math on September 14, 2012 at 2:42 am

    i don’t havde an actual email so could u just post it??



  26. Steven Miller on August 17, 2012 at 8:00 pm

    let me send a hint first. /s



  27. Clarice on August 16, 2012 at 7:01 am

    I can’t solve this after thinking about it for hours! Can I have the solution please?



  28. Steven Miller on July 27, 2012 at 4:43 am

    sure, I sent a hint (you can also email me at sjm1 AT williams.edu)



  29. Jessica on July 26, 2012 at 11:41 am

    I am pulled this problem for my class and no one figured it out. Could you please send me the solution. It would be greatly appreciated.



  30. Steven Miller on July 16, 2012 at 3:51 am

    sure, sent



  31. naomi on July 15, 2012 at 5:55 pm

    I need a hint, I’m trying to give this riddle to my dad but I cannot figure out the answer my self. He gave me a difficult riddle about 15 minutes ago and I couldnt figure it out. So this riddle is my Revenge! so help please. 🙂



  32. Steven Miller on June 3, 2012 at 3:06 am

    will send a hint



  33. Chris on June 1, 2012 at 7:25 pm

    I’ve spent too much time thinking about this, please send me the solution. Thanks.



  34. Steven Miller on May 28, 2012 at 1:00 am

    sure, sending //s



  35. Kevin on May 27, 2012 at 5:19 am

    May I have a hint?



  36. Steven Miller on May 13, 2012 at 1:15 am

    sure — sent



  37. Milind on May 12, 2012 at 6:10 am

    pls give me a hint



  38. Steven Miller on March 28, 2012 at 11:47 am

    sure. I’m working on adding a student/teacher corner, and wondering if you and/or your students are interested in helping. Details follow:

    I have several goals for the web site. Primary, of course, is that riddles are fun, and a great way to keep you mentally active.
    Another hope, though, is that these will be of use in our schools. Many of these riddles are appropriate for high school and
    junior high school settings. These aren’t just for the best and brightest (though these can be something for them to look at
    while teachers have to review other material), but a chance to help everyone think more creatively. We get so bogged down in
    passing exams and being taught to exams that we lose the sense of exploration. We learn or are taught the mechanics; this helps
    us become technicians, but more is needed. We want to learn how to ask questions, to see where things can go….

    A lot of these riddles are related to great concepts in modern mathematics, though the connections are not always apparent.
    That’s one of the goals of the student / teacher corner, to talk about these connections and try to excite people. Math is not
    an old, dusty, dead subject — it’s alive, with issues and wonrous applications. For example, the ‘C is for Cookie’ problem is
    related to some of the hardest partitioning problems in number theory, while the chess problem is related to designed a baseball
    schedule or an airline schdeule. A big part of the student / teacher corner is to make these connections explicit, and provide a
    guide to further reading.

    Another goal is to describe the thought process behind attacking and solving these riddles. What is a good thing to try, and
    why. This includes not only giving the right answers, but also the wrong ones. What was good about that wrong answer?

    Anyway, I hope this gives some sense of what I hope we can acocmplish.



  39. Caitlin on March 28, 2012 at 11:11 am

    May i have the solution? I am planning on giving this problem to my high school for a “math month” riddle



  40. Steven Miller on March 14, 2012 at 1:07 am

    sorry, no square-rooting. you can do it without.



  41. Steve Kench on March 13, 2012 at 8:30 pm

    Is it (7*6)/square root of 5-1? or no square roots allowed either ?



  42. Steven Miller on March 11, 2012 at 7:42 am

    beautiful — great thinking outside the box, but there is a way to solve it using the 6 as a 6.



  43. Alejandro on March 11, 2012 at 3:02 am

    If we can flip the 6 to make it a 9 then (9-5-1)*7 or 19-5+7



  44. Steven Miller on February 29, 2012 at 10:15 pm

    I’m missing something: isn’t this just 30 – 7 – 1 = 22, not 21?



  45. kevin k on February 29, 2012 at 6:50 pm

    5*6-7-1



  46. Steven Miller on February 24, 2012 at 2:39 pm

    I like it — great ingenuity. sadly, we don’t have a decimal point….



  47. Anonymous on February 24, 2012 at 12:13 pm

    1 x 6 x 7 x .5



  48. Steven Miller on January 30, 2012 at 3:49 am

    there is a standard soln involving just +-*/ as the operations are intended.



  49. kevin K on January 29, 2012 at 11:34 pm

    he has an infinite number of / and – right so if you put those two shapes together (using >1 “-” sign if nessary” you can make a 7 shap repete the proscess and you get 7+7+7. and before you say you can’t do that… the question used symbols that arn’t allowed… like “(” and “)”



  50. Steven Miller on January 27, 2012 at 3:13 pm

    Nice originality, but you can do it without splitting up something like 13 — you’re supposed to just use the four numbers and the standard operations.



  51. king on January 27, 2012 at 2:58 pm

    i know the answer now. 7+6=13 i separates the 13. i got 1 and 3 right?
    i 1+3=4*5=20+1=21!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!



  52. Steven Miller on January 25, 2012 at 12:22 pm

    very nice try, but you can’t do this as you don’t have a 0



  53. king on January 25, 2012 at 8:57 am

    how about 70/5=14+6+1=21

    i can do that right



  54. Steven Miller on December 31, 2011 at 4:56 pm

    Nice try, but it can be done without a decimal (as that is introducing a new symbol).



  55. Lance on December 31, 2011 at 3:57 pm

    First instinct: 6 * 7 * 1 * .5 (= 42 * .5 = 21) Decimals are not permitted?



  56. Steven Miller on December 30, 2011 at 11:31 pm

    Kher ([email protected]): hadn’t thought of that soln, but nice!



  57. Steven Miller on December 1, 2011 at 2:05 am

    correct kareem — not posting as it’s the soln



  58. Steven Miller on December 1, 2011 at 2:04 am

    you can do it without that; however, if you can do it that way, I’d love to see (email me at [email protected])



  59. kareem on November 30, 2011 at 10:43 pm

    can we arrange to numbers to make a bigger number? for example, using 1 and 5 to make 15?



  60. Steven Miller on November 22, 2011 at 8:03 pm

    No, but a very original attempt!



  61. lizard on November 22, 2011 at 7:36 pm

    is the answer 2.1 (1+5)/(6+7) 6/13 2.1666666



  62. Steven Miller on November 7, 2011 at 1:49 am

    Not quite — the numbers are 1,5,6,7 not 1,3,6,7



  63. Tommy Le on November 6, 2011 at 6:36 pm

    Done. (6/3+1)*7



  64. Steven Miller on October 25, 2011 at 2:00 am

    Going to, or is? Email me at [email protected] for a hint.



  65. Noah Haibach on October 24, 2011 at 1:30 pm

    This is going to drive me crazy.



  66. Steven Miller on October 20, 2011 at 7:23 pm

    no, only +-*/



  67. William Fox on October 20, 2011 at 5:54 pm

    Can we use exponents?



  68. Steven Miller on October 20, 2011 at 5:55 am

    doesn’t give exactly 21 with all 4 numbers being used, and is thus not the soln



  69. Anonymous on October 20, 2011 at 5:51 am

    Could he just say he’s (7+5)/6 1? I’m not completely sure I understand the question. The value of (7+5)/6 is 2, so by stating you’re (7+5)/6 1, this could be read as 2 1, or twenty one.



  70. Steven Miller on October 20, 2011 at 1:00 am

    Aldo: correct



  71. Steven Miller on October 18, 2011 at 1:27 pm

    If this is 1/(5*6*7) or (1/5)*6*7, neither gets you to 21. look at all the different ways you can structurally combine 4 numbers….



  72. William Fox on October 18, 2011 at 6:12 am

    1/5*6*7? Still thinking on the 1,2,3 Q, hard.



  73. Steven Miller on August 1, 2011 at 12:59 pm

    To: Yair.K: correct solution, glad you’re enjoying the riddles. I’m not posting your reply as it gives the answer.



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