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+7 votes

An infinite two-dimensional pattern is indicated below.

The smallest closed figure made by the lines is called a unit triangle. Within every unit triangle, there is a mouse.

At every vertex there is a laddoo. What is the average number of laddoos per mouse?

  1. $\quad 3$
  2. $\quad 2$
  3. $\quad 1$
  4. $\left(\dfrac{1}{2}\right)$
  5. $\left(\dfrac{1}{3}\right)$
in Numerical Ability by Boss (30.8k points)
edited by | 465 views

3 Answers

+3 votes
Best answer

Let the number of lines per direction be $1$ as shown bellow:

Here $x,y,z$ depict the directions of the line. 

$\eta_{laddoo} = 1$

$\eta_{mouse}= 0$

Add one more parallel line to each dimension $x,y,z$ as shown bellow:

Encircled points represent laddoo  $\Rightarrow \eta_{laddoo} = 3$

and triangle enclosed by them represent mouse $\implies \eta_{mouse} = 1$

Similarly for $3$ lines in each direction

$\eta_{laddoo} = 6\quad (1+2+3)$

$\eta_{mouse} = 4 \quad(2^{2})$

As we continue we get a series which depends upon the no. of lines per direction $($let say $l)$

So, $\eta_{laddoo} =  \frac{l\left ( l+1 \right )}{2}$

$\eta_{mouse} =\left ( l-1 \right )^{2}$

$\lim _{l\to \infty} \frac{ \eta_{laddoo}}{\eta_{mouse}} =1/2.$         

So, $D:$ $1/2$ is the correct answer.

by (235 points)
edited by
this should be chosen as best answer!! (y)

Consider the middle laddoo marked with blue color. This blue laddoo will be equally shared by 6 mice surrounding it. So each mouse will be having 1/6th share of this laddoo.

And each mouse can have access to 3 laddoo-s on 3 vertices. 

Share on one laddoo -> 1/6

Share on 3 laddoo-s -> 3*1/6 = 1/2


In 3rd figure, there are Only 2 lines in z-direction

Share on 3 laddoo-s -> 3*1/6 = 1/2

@MiNiPanda Can you please explain this?

On what basis you said this. Didn't able to get it.




Can you please check this commented answer, if it's mathematically valid or not?

+5 votes

Ans will be 1/2.Every Laddoo shared by 2 mouse

by Veteran (119k points)
can you please clarify, I am getting 1
every laddoo is shared by 6 mouse ....there are 6 triangles surrounding a laddoo
0 votes
This question can also answered by using little bit of graph theory.

Consider graph $G=(V,E)$ where $V$ is infinite set of all laddoos and all mouses. $E$ is edge set where it contains an edge between mouse $M_i \in V$ to laddoo $L_j \in V$ if and only if $M_i$ can legally eat laddoo $L_j$.

Now, total number of edges be $E$ = $6L$ = $3M$ (Because every edge is only between some mouse and some laddoo. No other case is possible.)

$\therefore \frac{L}{M} = \frac{3}{6} = 1/2$

And that remains correct even if $L, M \rightarrow \infty.$
by (373 points)

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