From the figure shown, each of N people was asked to take 12 points ou

Question

GMATH  practice exercise (Quant Class 18)

https://i.postimg.cc/prJBksfr/01-Mar194z.gif

From the figure shown, each of N people will be asked to take 7 points out, so that the 3 points left form vertices of an equilateral triangle. If all of them proceed correctly, what is the minimum value of N to guarantee that at least two people will take out exactly the same 7 points?

(A) 13
(B) 14
(C) 15
(D) 16
(E) 17

fskilnik · Answer

https://i.postimg.cc/1XW4BFjw/01-Mar19-9b.gif

?\,\, = \,\,\left( {\# \,\,{\rm{possible}}\,\,\Delta \,\,{\rm{equilateral}}\,\,{\rm{configurations}}} \right)\,\,{\rm{ + }}\,\,{\rm{1}}

? = 15 + 1 = 16

The correct answer is (D).

We follow the notations and rationale taught in the  GMATH  method.

Regards, 
Fabio.

OhsostudiousMJ · Answer

Hi, please explain why 1 is added?

fskilnik · Answer

Sure, OhsostudiousMJ ! This is a trivial application of the Pigeonhole Principle. There are 15 different possibilities, therefore if (at most) 15 people must choose one configuration each, you will still have the possibility of all 15 people choosing different configurations. When 15+1 people are considered, at least one configuration must be chosen twice. Regards, Fabio. P.S.: https://en.wikipedia.org/wiki/Pigeonhole_principle

Debashis Roy · Answer

fskilnik   Bunuel   chetan2u  
Hi could you please explain how are the 2 combinations (4th and 5th from left in the 2nd row of the seven diagrams) equilateral triangles?
Thanks in advance

fskilnik · Answer

Hi  Debashis_Roy  !

Thank you for your interest in my problem and also in my solution.

https://i.postimg.cc/jqNwNBSx/04-Mar19-2w.gif

In the figure above I present ONE of the diagrams you have asked about (for the other, the reasoning is the same).

Please note that I must explain to you why the purple segments (here unfortunately they appear as light grey) have the same lengths.

Reason: we have three congruent triangles (shown), by the SAS case (same two sides and intermediate angle) and the purple segments are (in all cases) opposite to the 60-degrees angle.

(We know it is a 60-degrees angle because of the one-one-one corresponding triangles... see the FIRST equilateral triangle presented in my previous solution, also the first and the last ones in the last row!)

If something is still unclear, please let me (us) know!

Regards and success in your studies,
Fabio.

Debashis Roy · Answer

Thanks I got it!..Dint think of it though...

fskilnik · Answer

This explanation is (as you have realized) very GMAT-related, therefore it is important to be understood throughly... I am glad you did!

chetan2u · Answer

Hi,
 it might seem to you but since all triangles or dots make equilateral triangles, even 4th and 5th are equilateral triangle.
Calculate it you will get it
..draw a triangle in 4th case by joining the points and check the attachment

bumpbot · Answer

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From the figure shown, each of N people was asked to take 12 points ou

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