FRUITful Problem

There are three boxes. One has apples, one has oranges and the other has apples and oranges. The boxes are labeled wrong so that no label is correct. Sue opens just one box, and without looking in the box, takes out one piece of fruit. She looks at the fruit and immediately labels all the boxes correctly. Which box did she open and how did she know?

Communicated by G. Mejia.
apples_oranges

119 Comments

  1. Steven Miller on February 16, 2017 at 2:41 pm

    It’s some comments by myself / my students on how to attack these and related problems, and how to use in classrooms. Once you send a correct soln to a problem I’ll send the password -s- ([email protected])



  2. Terry Henderson on February 16, 2017 at 2:17 pm

    Steve,
    Like many others above, I teach Math and AP Computer Science at the high school level and love using riddles like yours to teach problem solving techniques. For my classes, every Thursday the warm-up is some kind of brain-teaser/puzzle. I would love to know more about the teacher/student corner.

    Mr. H.



  3. Steven Miller on September 5, 2016 at 1:18 am

    email [email protected] for a hint –s



  4. isuhrdgiourhg on September 4, 2016 at 8:46 pm

    what is the answer could you email it to me



  5. Steven Miller on August 20, 2015 at 2:31 am

    I prefer to send hints first — I think it’s very useful to go through the same process as the students. after you solve one I can share the student /
    teacher corner password



  6. marie on August 19, 2015 at 7:13 pm

    Can you send the answer to me I would like to show my students this



  7. Steven Miller on August 1, 2014 at 2:26 am

    Sure: contact me at sjm1 AT williams.edu



  8. Samantha on June 4, 2014 at 3:30 pm

    Would it be possible to have solutions to all of the medium difficulty riddles emailed to me? I will be using these in my Math Games summer school classes and want to double check my own solutions.



  9. Steven Miller on April 4, 2014 at 2:21 pm

    I believe I emailed you, but if not please do contact me at sjm1 AT williams.edu; I have a student / teacher corner with supplemental material to use these in classes, and happy to share the password. /s



  10. Anna on March 18, 2014 at 12:54 pm

    I LOVE these riddles. I use them a lot as warm-ups for my class. If you still need classrooms for you student/teacher corner, please let me know.



  11. Steven Miller on March 9, 2014 at 4:03 am

    Glad you’re enjoying.



  12. James on March 8, 2014 at 8:11 pm

    It’s a shame you don’t have a donate button! I’d most certainly donate to this superb blog!
    I suppose for now i’ll settle for book-marking and adding your RSS
    feed to my Google account. I look forward to brand
    new updates and will share this blog with my Facebook group.
    Talk soon!



  13. Steven Miller on January 21, 2014 at 6:19 pm

    Your email didn’t work. let me try a hint first. open the MIXED box //s (sjm1 AT williams.edu)



  14. Jen Dennehy on January 21, 2014 at 2:52 pm

    Could you please give me the answer for “FRUITful Problem”! This is making me crazy.
    Thank you Jen



  15. Steven Miller on November 15, 2013 at 1:48 pm

    email to you bounced; glad you’re enjoying it ..s



  16. Imogene on November 15, 2013 at 12:22 pm

    There is definately a great deal to know about this subject.
    I really like all the points you’ve made.



  17. Steven Miller on October 6, 2013 at 4:07 pm

    email me (sjm1 AT williams.edu) for a hint ..s



  18. LUVPUPPYS on October 6, 2013 at 4:03 pm

    Does anyone know the answer? If so, would you share it?! PULEEEEEAAAASE!!!!!!



  19. Steven Miller on May 21, 2013 at 3:15 pm

    sent hint – let me know if that helps



  20. Valerie on May 21, 2013 at 2:45 pm

    Email solution please – I’d like to share with my students.



  21. Indra on December 25, 2012 at 4:35 pm

    Can u email me the answer please? I am gonna use it for my team building activities.



  22. Steven Miller on December 5, 2012 at 1:44 am

    Meeeeep: good! sjm1 AT williams.edu



  23. Steven Miller on October 22, 2012 at 6:25 am

    email me — sjm1 AT williams.edu



  24. Anonymous on October 22, 2012 at 6:15 am

    ??



  25. Steven Miller on October 11, 2012 at 7:03 pm

    Correct anonymous



  26. Steven Miller on October 7, 2012 at 12:41 am

    I am — help is always appreciate (sjm1 AT williams.edu)



  27. Tina on October 6, 2012 at 7:23 pm

    I use these riddles with my students if you are still looking for student/teacher corner.



  28. Steven Miller on September 19, 2012 at 5:56 am

    I’ll send (email me at sjm1 AT williams.edu) — we’re also looking for students and teachers to help us with a ‘student / teacher’ corner to help use these riddles in classes.



  29. Hazel on September 18, 2012 at 10:21 pm

    I would like to share some of the riddles with my students, how can i get the solutions ?



  30. Steven Miller on September 11, 2012 at 2:01 pm

    anonymous: correct (email sjm1 AT williams.edu)



  31. Steven Miller on September 6, 2012 at 12:55 pm

    sure — sjm1 AT williams.edu



  32. Marlene Armstrong on September 6, 2012 at 4:15 am

    Can you email the solution? I am utilizing your riddles in my engineering class for bell work.



  33. Steven Miller on August 2, 2012 at 3:26 pm

    if you can’t explain it you don’t ‘know’ it. knowing something does entail being able to explain it to someone — you have to say what you
    did and why.

    so, start with what box you opened, what you saw, and what you then figured out because of what you saw. (sjm1 AT williams.edu)



  34. Steven Miller on August 2, 2012 at 3:25 pm

    hint sent (sjm1 ATwilliams.edu)



  35. cfasce on August 2, 2012 at 1:33 pm

    i know the answer but i dont know how to explain it, could you explain me?



  36. ammy on August 2, 2012 at 12:08 pm

    help!!!!!!!



  37. Steven Miller on June 24, 2012 at 3:15 am

    I don’t think this works. will email you //s



  38. isheanesu on June 23, 2012 at 10:50 pm

    She can pick an apple in a box labelled apple then this wud mean that the box is assorted and the box labelled orange can onli b apples and the one labelled assorted can onli b oranges! Or pick up an orange in a box labelled orange this wud mean the box is assorted and the box labelled apples they r oranges and the one labelled assorted are apples



  39. Steven Miller on June 22, 2012 at 8:58 pm

    sent a hint



  40. sailaxmi on June 22, 2012 at 4:10 pm

    answer



  41. Steven Miller on June 12, 2012 at 12:43 am

    I do not believe this can work — email me (sjm1 AT williams.edu)



  42. Steven Miller on June 6, 2012 at 4:24 am

    Double zero: having trouble following what you wrote. email me (sjm1 AT williams.edu) which box you pick first and what you do. //s



  43. Steven Miller on May 17, 2012 at 10:34 am

    robert: correct



  44. Steven Miller on May 17, 2012 at 3:01 am

    email me with more details: what box is opened, what fruit is seen (sjm1 AT williams.edu)



  45. Tyler on May 17, 2012 at 12:21 am

    If all boxes are mislabeled and she finds an orange that means she switches the label from the box that says oranges. Then puts the one from the first box on the 3rd and puts the 3rd label on the 2nd box



  46. Steven Miller on April 12, 2012 at 6:24 pm

    Mike L: correct



  47. Steven Miller on April 3, 2012 at 1:40 pm

    ah, but what if she opens the apple box and gets an orange? then we don’t know what it is….



  48. Banani on April 3, 2012 at 11:45 am

    i ddnt read the question properly.

    Solution: She opens the apple box and finds an apple (or the orange and finds an orange) so that means that this one was mixed , which means that the orange labeled box is actually apples and the mixed labeled box is actually oranges



  49. Steven Miller on April 2, 2012 at 1:39 pm

    Yes



  50. E on April 2, 2012 at 7:42 am

    So… she can see the labels and she knows they’re all wrong?



  51. Steven Miller on March 29, 2012 at 8:18 pm

    xiaomilk: correct, not posting as it’s the soln



  52. xiaomilk on March 29, 2012 at 2:56 pm

    I believe I have solved a similar question before. There are three boxes with all labels in the wrong places. Since the labels are on the wrong boxes, Sue should pick a fruit from the box labelled as apples and oranges. If she gets an orange, the apples would be in the box labelled oranges and the apples and oranges in the oranges box. If she gets an apple, the oranges would be in the box labelled apple and the apples and oranges in the box labelled oranges. So either way the apples and oranges would be in the box labelled oranges, and the apples or oranges, depending on the fruit Sue picked, in the box that is labelled orange ( apple ) or apple ( orange ).



  53. Steven Miller on March 27, 2012 at 6:40 pm

    if she opens apple and sees an orange, here are TWO possibities

    A – O AO
    AO – A O
    O – AO A

    note no box hsa the right label



  54. Morwenna on March 27, 2012 at 11:51 am

    She opened the apple box and found an orange. Therefore, it must actually be either the orange box or the apple and orange box. If the box she opened were the orange box, then she would be inclined to switch the labels on the apple and orange box. However, since all labels are incorrect, it means that the apple and orange box would have the same wrong label, so she would put the apple label on the apple and orange box, the apple and orange label on the orange box, and the orange box label on the apple box. Apple -> Apple and Orange -> Orange -> Apple



  55. Steven Miller on March 22, 2012 at 12:48 am

    ah, but what if she pulls out an apple?



  56. Andy Cai on March 21, 2012 at 11:05 pm

    Let’s say that she opens the orange box. ( It will work for the apple box too) If she pulls out an orange, she knows that is is the mixed box because everything is labeled incorrectly. The other two labels can be switched to make all the boxes labeled correctly.



  57. Steven Miller on March 21, 2012 at 3:17 am

    I can’t read t his ..s



  58. gbnfdb on March 21, 2012 at 2:59 am

    非佛殿发国仏



  59. Steven Miller on March 12, 2012 at 1:15 am

    Not sure — if you open Apple and see an Apple, you know it must be apple-orange. BUT, if you see orange, it’s either orange OR apple-orange.



  60. Chinmay on March 12, 2012 at 1:06 am

    Open the box labelled apple..
    CASE 1
    it contains orange then the box labelled orange has both and the box labelled both has apple
    CASE 2
    it contains both then the box labelled orange has apple and the box labelled both has orange..
    am i right?



  61. Steven Miller on March 9, 2012 at 2:14 am

    A little bit. Think which box is best to open….



  62. StarFox on March 9, 2012 at 2:11 am

    This is almost like the one riddle with two people, one tells the truth, the other lie, you say, take me to your town….but I can’t solve this….



  63. Steven Miller on February 26, 2012 at 6:02 am

    Stan: correct, well done



  64. Steven Miller on February 18, 2012 at 9:50 pm

    To anonymous; correct (not posting as it’s the soln)



  65. Steven Miller on February 9, 2012 at 7:10 pm

    Glad you’re enjoying, but not quite right. Say she opens A&O and sees O, so she labels it O. The other two boxes were initially labeled A and O; she can’t just switch the labels as she’s just used O for the box originally labeled A&O.



  66. Nafi on February 9, 2012 at 7:06 pm

    dear steve, really wonderful website.. im training my brain, so i come to ur website every day to solve one riddle. hope my solution is correct this time, the steps are:
    since none of the boxes is labeled correctly. then we have two scenarios.
    A- the orange box labeled :apple or mixed
    B- the apple box labeled :mixed or orange
    c- the mixed labeled as :orange or apple
    so she need to open the box labeled mixed, if she got orange then it’s the orange box(since it can’t be mixed since it’s labeled mixed and it’s wrong, and it can’t be apples goes she got orange), if she got apple then it’s the apple box ,the other two boxes just swap the labels



  67. Steven Miller on February 6, 2012 at 3:40 am

    to awesome: very close. you write she takes one from ** … then the one labeled X is … and the one labeled Y is …; I think you need to switch the order of the two. Email me at [email protected] if you want to discuss more.



  68. Steven Miller on February 1, 2012 at 9:11 pm

    sure — hope your kids enjoy ..s



  69. Cal on February 1, 2012 at 2:46 am

    This is great! Am thinking u have to go to both first, but cud u email the solution? Wud love to work w my kids on this…….thank you!



  70. Steven Miller on January 30, 2012 at 3:51 am

    but what if she had pulled out an orange?



  71. kevin K on January 29, 2012 at 11:59 pm

    she opened the box labled apples and pulled out an apple. since she knows that none of the lables are correct she knows that this box has apples and oranges in it and the box that has oranges in it is labled apples plus oranges since the box which contains apples having this lable would cause the box of orranges to be labled correctly. subsequently the box of apples must be labled oranges before being rearranged.



  72. Steven Miller on January 25, 2012 at 12:30 am

    Tomek K: correct, well done — not posting as it’s the soln



  73. Tomek K on January 24, 2012 at 4:36 pm

    [A+O] [A] [O]
    ^
    she grabs one fruit out of the box with the label “both”. since the label is wrong, it will contain only oranges or apples.

    She then attached the right label (O, or A) to the first box, lets say she grabbed an apple:
    [A] [ ] [O]

    The box she took the A-lable and the other box (with the O) have false labels as well, so the second must be the orange box and the third the box with both:
    [A] [O] [A+O]



  74. Steven Miller on January 18, 2012 at 4:29 am

    Kieron: correct, well done — not posting as it’s the soln



  75. Steven Miller on January 12, 2012 at 8:14 pm

    I don’t think this works. What is the box: oranges, or oranges and apples?



  76. Rob M on January 12, 2012 at 8:12 pm

    She opened the apple box and took out an orange



  77. Steven Miller on January 9, 2012 at 3:01 pm

    Sure. //s



  78. Bettina DeBell on January 9, 2012 at 2:57 pm

    Can you email me the solution? I’ve got students in my building who are working on it and on Friday I will post a solution and I’d like it to be worded clearly.



  79. Steven Miller on December 20, 2011 at 2:50 pm

    Glad you’re enjoying it. We’re also working on a student / teacher’s corner to use these riddles and give connections to higher mathematics — let me know if you want updates or if you want to help.



  80. arushi arora on December 20, 2011 at 1:28 pm

    very good site useful for children of every class.



  81. Steven Miller on December 20, 2011 at 1:46 am

    will email you. //s



  82. matt on December 19, 2011 at 11:21 pm

    whats the answer



  83. Steven Miller on December 16, 2011 at 4:40 am

    Chris: correct! not posting as it’s the soln



  84. chris on December 16, 2011 at 4:17 am

    Sue opened the mixed fruit box first. She knows because the fruit she pulled had a label on the box that matched the fruit that she pulled. If She knows the labels are incorrect then she would know that it was not the apple box if she pulled an apple. She would then know that the box that says mixed would be the orange box because the apple label was not available as it was on the box that she pulled the apple from. That would leave the orange label to remain on the apple box.



  85. Steven Miller on December 15, 2011 at 9:57 pm

    were you able to get it? email me at [email protected] to check a soln



  86. rebeca on December 15, 2011 at 9:50 pm

    wow so ard i had to have help



  87. Steven Miller on December 15, 2011 at 3:02 am

    Lance: correct — not posting as it’s the soln



  88. Steven Miller on December 6, 2011 at 5:03 pm

    email me at [email protected] if you want to discuss.



  89. Kiana and Lane on December 6, 2011 at 3:34 pm

    Ugh….wait….I just saw my flaw. If she pulls and ORANGE, then she could be in the both OR the orange container….grrr…hint please??



  90. Kiana and Lane on December 6, 2011 at 3:31 pm

    This is actually a really easy one. She goes into the container with APPLES.
    If she pulls an APPLE: she knows that she must be in the container with both because all of the labels are wrong (as previously stated). The orange-labeled container must have Apples in this case, and the box labeled with both produce must have oranges.

    If she pulls an ORANGE, it’s the same concept.



  91. Steven Miller on November 30, 2011 at 7:08 pm

    Singh: correct, well done — not posting as it’s the soln



  92. Steven Miller on November 26, 2011 at 1:39 am

    correct, well done; not posting as it’s the soln



  93. Steven Miller on November 26, 2011 at 1:38 am

    twig: correct, well done; not posting as it’s the soln



  94. Steven Miller on November 2, 2011 at 5:59 am

    but what if she picks an apple?



  95. Daniel on November 2, 2011 at 5:05 am

    Box 1 – Label: Apples
    Box 2 – Label: Oranges
    Box 3 – Label: Both

    She picks an Orange from Box 2 – Label Oranges. This must be the box with Both. Apples cannot be in Box 1, so that is the Orange Box. Thus Box 3 is the Apple box.

    Labels rearranged
    Box 1 – Label: Oranges
    Box 2 – Label: Both
    Box 3 – Label: Apples



  96. mukuro on October 29, 2011 at 4:19 am

    oh, thanks.



  97. Steven Miller on October 27, 2011 at 2:45 am

    sure — if you pick something from the box labelled apples, what happens if you get an orange? in that case, you can’t determine what you have. so maybe we shouldn’t pick from the apple box (and thus by symmetry shouldn’t pick from the orange box….)



  98. Anonymous on October 27, 2011 at 1:31 am

    can u give me a hint



  99. Steven Miller on October 26, 2011 at 3:54 pm

    mukuro: correct, well done — not posting as it has the solution.



  100. Steven Miller on October 26, 2011 at 2:10 pm

    I don’t think it works starting with picking apple. how do you know if you have orange OR apple/orange?



  101. mukuro on October 26, 2011 at 6:41 am

    there are three box
    ~
    box no——-label(incorrect)
    –1————-apple
    –2————-orange
    –3————-assorted
    lets assume that im sue and i picked box1(apple) and the fruit is orange so:
    ~
    box no———-label(incorrect)
    –1————-apple<——-fruit inside is orange
    –2————-orange<——it must be assorted
    –3————-assorted<—–it must be apple
    i think box no 2 is assorted because if it was like this…:
    ~
    box no———-label(incorrect)
    –1————-apple<——-fruit inside is orange
    –2————-orange<——it must be apple
    –3————-assorted<—–it must be assorted
    box3 label is assorted but it was not assorted so it must be apple!



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

    Noah: correct



  103. Steven Miller on October 24, 2011 at 3:05 am

    yes but need to explain logic — can email me at [email protected]



  104. Anonymous on October 23, 2011 at 3:43 pm

    i thinks its orange and apple is it correct



  105. Steven Miller on October 19, 2011 at 3:11 pm

    I don’t think that works. Say she opens apples. what happens if she sees an orange?



  106. William Fox on October 19, 2011 at 3:10 pm

    Lets see she opens one of the two just apple or orange box, correctes that label then since we know All of them are misslabeled she switches the two left?



  107. Steven Miller on October 11, 2011 at 1:53 am

    A bit more detail is needed — which bin is picked? Your second post was correct (I’m not posting that one as it’s the soln).



  108. Dan on October 10, 2011 at 7:27 pm

    if she picks an orange from a wrong labeled bin, and labels it oranges, she would only HAVE to switch the other two labels around on the other two bins.



  109. Steven Miller on October 2, 2011 at 2:43 pm

    Not quite. You need to tell me what box is picked, what is observed, what she deduces, ….



  110. Al on October 2, 2011 at 1:11 pm

    if we assume that the right distribution is
    apple orange appleandorange and all were misslabeled then the possible mix would be other than the origional correct labeling is
    apple appleandorange orange not possible since apple is correct
    orange appleandorange apple possibly
    appleandoragne apple orange possibly
    cancel reduadencies
    so orange vs appleandorange yields apple
    appleandorange vs apple yields orange
    apple vs orange yields apple and orange. which is the correct labeling



  111. Steven Miller on August 2, 2011 at 1:03 pm

    you’re close. if all boxes are mislabeled and sue opens the mixed box and gets an orange, what must the correct label be?



  112. Unknown on August 2, 2011 at 7:32 am

    how is this possible she has 3 ways and ill tell u how its impossible
    Way 1 = Sue pulls apple out she has no idea whatsoever if it belongs to the apple or mix
    Way 2 = Sue pulls orange out and she has no idea if its orange or mix
    Sue gets an apple and doesent know if its apple or mix
    Way 3 = Sue pulls from mix now this goes 2 ways
    Sue gets an orange and doesent know if its orange or mix



  113. Steven Miller on May 30, 2011 at 1:41 am

    ??



  114. abed on May 29, 2011 at 8:10 pm

    btza3ber 😛



  115. Steven Miller on May 4, 2011 at 1:34 am

    Not sure what you mean by do the boxes have the same order.



  116. Steven Miller on May 4, 2011 at 1:33 am

    So far so good!



  117. Sachin on May 3, 2011 at 8:32 pm

    do the boxes have the same order?



  118. Sachin on May 3, 2011 at 8:32 pm

    If she goes to the box which is labelled both and she picks up either an apple or an orange, then this implies that the box she just opened must contain ONLY apples or oranges (due to the labels being wrong) and hence she knows what’s inside that box.

    That’s as far as I have gotten, am i right?



  119. Steven Miller on April 6, 2011 at 1:32 pm

    Richard C: Your solution is correct.



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