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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
Type: LOGIC

QUESTION:
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!

ANSWER:
45

EXPLANATION:
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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17 Responses

  1. […] Many of this weeks submissions presented problems, of varying difficulty. As a warm-up, Sharp Brains‘ new puzzle master offers a brainteaser entitled ‘Party For Polyglots’. More demanding are two entries from MathNotations, aimed at calculus students: the first concerning properties of the ellipse, the second, more advanced post is on exploring infinite series. We close with the most amibitious material, the Unapolagetic Mathematician’s examination of the knot colouring problem: for the background, see here; this week’s submission explains what’s going on. I feel that I had an unfair advantage since my flatmate is a knot-theorist; I’m often intrigued by the techniques applied to this field. Plus I rarely get to draw pictures during my own research! […]

  2. Caroline says:

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

  3. […] Party For Polyglots A brain teaser puzzle, not too difficult. […]

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

  5. Caroline says:

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

  6. pjsk8 says:

    At least ONE person can speak all three languages.

  7. 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)

  8. k says:

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

  9. scott says:

    90+80+75=245 language/people

    245-200=45

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

  10. BJ says:

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

  11. Eric says:

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

  12. 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

  13. ddd69 says:

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

  14. 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.

  15. 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

  16. 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..

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