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 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:**

- #15. Fun & Brainy Haikus. Yours?

Caroline says

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

The Science Pundit says

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

Caroline says

Glad you liked it, and creative solutions are

alwaysencouraged!pjsk8 says

At least ONE person can speak all three languages.

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)

k says

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

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.

BJ says

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

Eric says

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

Mandarin and Spanish, and 10 speak Mandarin and Italian.

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

ddd69 says

10 cant speak spanish

20 cant speak italian

25 cant speak mandarin

so

100 — (10+20+25)= 45

can speak 3 languages

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.

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

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..

Brad Donner says

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

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.

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.

Javier says

Perhaps 54 is the maximum value and 45 the minimum value.

Javier says

xd obviously not cause maximum value is 75 of people that speak three languages.

My head will explode

Satyaraja Dasara says

The main thing here is to find the trilinguals.

Those who can’t speak a language can speak atleast one of other two languages.

But we strictly need trilinguals.

So those who don’t speak the language are summed up as non trilinguals.

So Total — Sum of non trilinguals = Possible Trilinguals

Btw I think the question should be atmost trilinguals not atleast.