Log In
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 Quantitative Aptitude
edited by

3 Answers

4 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.

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

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 $L, M \rightarrow \infty.$

Related questions

7 votes
5 answers
Consider a well functioning clock where the hour, minute and the seconds needles are exactly at zero. How much time later will the minutes needle be exactly one minute ahead ($1/60$ th of the circumference) of the hours needle and the seconds needle again exactly at zero? ... an integer multiple of $1/60$ th of the circumference. $144$ minutes $66$ minutes $96$ minutes $72$ minutes $132$ minutes
asked Nov 5, 2015 in Quantitative Aptitude makhdoom ghaya 714 views
5 votes
2 answers
Consider a sequence of numbers $\large (\epsilon _{n}: n= 1, 2,...)$, such that $\epsilon _{1}=10$ and $\large \epsilon _{n+1}=\dfrac{20\epsilon _{n}}{20+\epsilon _{n}}$ for $n\geq 1$. Which of the following statements is true? Hint: Consider the sequence of ... $\large (\epsilon _{n}: n= 1, 2,...)$ is decreasing and then increasing. Finally it converges to $1.$ None of the above.
asked Nov 5, 2015 in Quantitative Aptitude makhdoom ghaya 416 views
3 votes
2 answers
Let $\DeclareMathOperator{S}{sgn} \S (x)= \begin{cases} +1 & \text{if } x \geq 0 \\ -1 & \text{if } x < 0 \end{cases}$ What is the value of the following summation? $\sum_{i=0}^{50} \S \left ( (2i - 1) (2i - 3) \dots (2i - 99) \right)$ $0$ $-1$ $+1$ $25$ $50$
asked Nov 4, 2015 in Quantitative Aptitude makhdoom ghaya 305 views
6 votes
1 answer
Among numbers $1$ to $1000$ how many are divisible by $3$ or $7$? $333$ $142$ $475$ $428$ None of the above.
asked Nov 4, 2015 in Quantitative Aptitude makhdoom ghaya 376 views