Fun with Geometry

Consider a pentagram (see picture below). There are five disjoint triangles initially. Just count the five ‘caps’ above the pentagon, do not count the larger triangles formed by using three of the five vertices, for those ‘triangles’ have lines going through them. By adding just two lines, you can go from five to 10 disjoint triangles. How?

Submitted by Bruce Lin, from friends at MIT, who got it from Andrew Russell, a graduate student in EE at MIT, who either made it up or got it from…?

triangle

Posted in Geometry, Hard

33 Comments

  1. Steven Miller on February 11, 2019 at 9:20 am

    Email me at [email protected] for a hint — your email address did not work



  2. Garima Jindal on February 11, 2019 at 7:09 am

    Can you send me the solution?



  3. Steven Miller on November 30, 2018 at 7:46 pm

    your email address didn’t work — email [email protected] for a hint



  4. Cynthia on November 30, 2018 at 7:38 pm

    ?



  5. Steven Miller on November 19, 2017 at 3:06 pm

    There is a solution, it works. email sjm1 AT williams.edu



  6. N on November 18, 2017 at 8:13 pm

    It is wrong, the question doesn’t have an answer, I proved it.



  7. Steven Miller on November 21, 2016 at 6:09 pm

    sure — email me at [email protected]



  8. Courtney on November 21, 2016 at 3:40 pm

    My students love your problems!
    I have two different solutions to this problem and I was wondering if you would check them?



  9. Steven Miller on November 12, 2016 at 3:23 am

    email me (emails to you bounce) at sjm1 AT williams.edu



  10. Steven Miller on May 11, 2015 at 7:08 pm

    I prefer to send hints first and then, if still stumped, solutions. You can email me at sjm1 AT williams.edu



  11. Lesesne Phillips on May 11, 2015 at 4:45 pm

    Great puzzles and this one is actually stumping my very intelligent software developers. Can I have the solution in case one of them does not solve it?



  12. Penny Dalton on April 4, 2014 at 1:59 pm

    my students came up with various solutions and we are curious about a couple of thiings
    1- is there only ONE solution
    2- do you have to end up with only TEN triangles
    3- We think there are more than five triangles to start with



  13. aaron on September 13, 2012 at 1:58 pm

    i got 13 triangles lol is that right or wrong?



  14. Steven Miller on August 24, 2012 at 2:30 am

    sure, hint set (sjm1 AT williams.edu)



  15. Steven Miller on July 23, 2012 at 1:06 am

    Let me send a hint first //s



  16. Steven Miller on July 7, 2012 at 2:25 am

    Nope. Very original, but once you divide one of the initial triangles you can’t count it again. email me at sjm1 AT williams.edu to chat more.



  17. Steven Miller on May 2, 2012 at 4:36 pm

    ok, the instructions weren’t clear

    you can’t use one edge in 2 triangles

    each triangle has 3 edges, and none of those edges are used in another triangle

    this isn’t like the chessboard problem, where you are asked how many squares are there and you keep grouping larger and larger blocks together



  18. Steven Miller on May 1, 2012 at 2:48 pm

    hint sent //s



  19. Steven Miller on April 29, 2012 at 1:46 pm

    email sjm1 AT williams.edu for a hint ..s



  20. Roman on April 29, 2012 at 10:48 am

    Can you please send me the solution ?



  21. Steven Miller on April 10, 2012 at 2:03 pm

    Sure. //s



  22. Steven Miller on March 9, 2012 at 8:57 pm

    sure, sent ..s



  23. Steven Miller on February 22, 2012 at 7:53 pm

    Sure ..s



  24. Steven Miller on February 1, 2012 at 9:10 pm

    can’t see the image — can you describe it to me (or if you look at it you can probably tell if it’s right). //s



  25. Mason on February 1, 2012 at 8:26 pm

    <a href="http://tinypic.com?ref=1ovghd&quot;



  26. Steven Miller on January 6, 2012 at 5:16 am

    ah, but what did you find? email me at [email protected] if you want a hint



  27. Bren on January 5, 2012 at 11:03 pm

    copy into paint, play, find the solution!



  28. Steven Miller on December 5, 2011 at 4:47 am

    draw two more lines. in the end, you need to have 10 triangles. each triangle is created by three lines. the same line can create many different triangles.

    if you draw a line straight down the middle (from the top vertex to the vertex underneath it) you replace the triangle on the top with 2 triangles.



  29. Thilina on December 5, 2011 at 4:44 am

    sir, could you please explain this problem a bit more. it’s bit confusing for me to understand what you’re trying to say in the middle part of the problem.



  30. Steven Miller on November 14, 2011 at 3:12 am

    sure //s



  31. Jessica on November 13, 2011 at 2:21 am

    Can you send me the solution?



  32. Steven Miller on November 9, 2011 at 5:03 am

    yes



  33. Phuc Nguyen on November 9, 2011 at 4:57 am

    I just have a question. Can any two of those ten triangles share the same edge?
    Thank you very much



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