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#14 Brain Teaser: Party For Polyglots

We are delight­ed to intro­duce you to Wes Car­roll who has gra­cious­ly cre­at­ed a few new puz­zles to bend all those sharp brains out there! Wes Carroll

Wes is the head of Do The Math pri­vate tutor­ing ser­vices, Puz­zle Mas­ter for the Ask A Sci­en­tist lec­ture series, and an inter­na­tion­al­ly tour­ing per­former and teacher of music. With no fur­ther ado, the first puz­zle!

Par­ty For Poly­glots

Dif­fi­cul­ty: MEDIUM

Of the 100 peo­ple at a recent par­ty, 90 spoke Span­ish, 80 spoke Ital­ian, and 75 spoke Man­darin. At least how many spoke all three lan­guages?

Have you solved it yet? If you are work­ing the prob­lem, mak­ing hypothe­ses, test­ing your ideas, and com­ing up with a solu­tion, you are using your frontal lobes. This is great exer­cise because the frontal lobes fol­low the “last hired, first fired” adage. They are they last areas of your brain to devel­op and the first to suf­fer the rav­ages of time and stress. So, keep exer­cis­ing them!


10 could not speak Span­ish, 20 could not speak Ital­ian, and 25 could not speak Man­darin. So there could have been 10 peo­ple who spoke none of those lan­guages.

How­ev­er, that would max­i­mize the num­ber of peo­ple who could speak all three, and the prob­lem asks at least how many speak all three. There­fore, we must assume that these 10, 20, and 25 peo­ple are all sep­a­rate peo­ple. Hav­ing iden­ti­fied 55 each of whom is miss­ing one lan­guage, the remain­ing 45 speak all three.

Next brain teas­er in Sharp­Brains’ top 25 series:

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19 Responses

  1. Caroline says:

    If you liked this puz­zle, you can try a few more by oth­er puz­zlers at The Car­ni­val of Math­e­mat­ics. Enjoy!

  2. I like it. I actu­al­ly solved it from the oppo­site direc­tion: I start­ed with the speak­ers of one lan­guage and sub­tract­ed.

  3. Caroline says:

    Glad you liked it, and cre­ative solu­tions are always encour­aged!

  4. pjsk8 says:

    At least ONE per­son can speak all three lan­guages.

  5. Beck says:

    Used same method as The Sci­ence Pun­dit.

    Pop­u­la­tion = 100
    With­in the pop­u­la­tion,
    Span­ish Speak­ers = 90
    Ital­ian Speak­ers = 80
    Man­darin Speak­ers = 75

    So, for the min­i­mum num­ber of all three lan­guage speak­ers or total over­lap…

    x = 90 — (100 — 80) — (100 — 75)

  6. k says:

    9/10 x 8/10 x 3/4 = 216/400 => 54 Peo­ple

  7. scott says:

    90+80+75=245 language/people


    there were 45 more language/people than if 2 lan­guages were spo­ken by all 100 peo­ple.

  8. BJ says:

    I chal­lenge some­one to cre­ate a Venn dia­gram with 45 peo­ple for 3 lan­guages and have the var­i­ous totals add up cor­rect­ly

  9. Eric says:

    For the Venn dia­gram: 25 speak Ital­ian and Span­ish, 20 speak
    Man­darin and Span­ish, and 10 speak Man­darin and Ital­ian.

  10. allstar says:

    i remem­ber doing this exact same ques­tion in maths at school. final­ly got it, took me a while to remem­ber how 🙂
    10 don’t speak span­ish
    25 don’t speak man­darin
    20 don’t speak ital­ian
    there­fore the rest speak all lan­guages — 45 … if that makes sense

  11. ddd69 says:

    10 cant speak span­ish
    20 cant speak ital­ian
    25 cant speak man­darin
    100 — (10+20+25)= 45
    can speak 3 lan­guages

  12. michelle says:

    pjsk8 is right. I don’t think the writer intend­ed it, but the way the ques­tion was phrased, it is ask­ing for the min­i­mum of speak­ers that can speak all three lan­guages. That being 1.

    the work “atleast” is key.

  13. Abhishek says:

    10 peo­ples can­not speak span­ish, 20 peo­ple can­not speak Ital­ian and 25 can­not speak man­drin. Adding all of them comes 55. So there are 55 peo­ple who can­not speak alteast one lan­guage. Remain­ing 45 peo­ple can speak all the three lan­guages. It is log­i­cal­ly very good puz­zle!!! Thanks for such stuff

  14. abhishek says:

    prob­lem can be solved by two method…the log­i­cal way give us 45..but per­son who do lat­er­al think­ing kind of puz­zle must reply 1..

  15. Brad Donner says:

    A min­i­mum of ten could not speak all three lan­guages; and a max­i­mum of 75 could, all con­sid­ered with­out assump­tions.

  16. Brad Donner says:

    I stand cor­rect­ed. A min­i­mum of 25 could not speak all three, and a max­i­mum of 75 could; allow­ing that some only speak one or two lan­guages.

  17. Javier says:

    i have a ques­tion about this rid­dle.
    the rid­dle is pro­posed as if there are exact­ly 100 of peo­ple at the par­ty.
    So, if you think so, this prob­lem maybe be equal to a prob­a­bil­i­ty prob­lem to find the num­ber of pos­si­bil­i­ties to find a per­son between 100 who speaks Span­ish, Ital­ian and Chi­nese lan­guage. If you applies prob­a­bil­i­ty the­o­ry to find event P(S ? I ? M) (S: Span­ish) (I: Ital­ian) (M: Man­darin) of a per­son that speaks these lan­guages. So, if you mul­tip­plies the prob­a­bil­i­ties of speak Span­ish, Ital­ian and Man­darin the answer is 54.
    I imag­ine that there was a mis­take of my part but if some­one could help me, I will thank very much.

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