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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.

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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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