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+32 votes

Suppose you want to move from $0$ to $100$ on the number line. In each step, you either move right by a unit distance or you take a shortcut. A shortcut is simply a pre-specified pair of integers $i,\:j \:\text{with}\: i <j$. Given a shortcut $(i,j)$, if you are at position $i$ on the number line, you may directly move to $j$. Suppose $T(k)$ denotes the smallest number of steps needed to move from $k$ to $100$. Suppose further that there is at most $1$ shortcut involving any number, and in particular, from $9$ there is a shortcut to $15$. Let $y$ and $z$ be such that $T(9) = 1 + \min(T(y),T(z))$. Then the value of the product $yz$ is _____.

in Algorithms by Veteran (98.5k points)
edited by | 2.4k views

3 Answers

+27 votes
Best answer
$T(k)$ is the smallest number of steps needed to move from $k$ to $100$.

Now, it is given that $y$ and $z$ are two numbers such that,

$T(9) = 1 + \min (T(y), T(z))$ , i.e.,

$T(9) = 1 + \min ($Steps from $y$ to $100$, Steps from $z$ to $100)$, where $y$ and $z$ are two possible values that can be reached from $9$.

One number that can be reached from $9$ is $10$, which is the number obtained if we simply move one position right on the number line. Another number is $15$, the shortcut path from $9$, as given in the question. So, we have two paths from $9$, one is $10$ and the other is $15$.

Therefore, the value of $y$ and $z$ is $10$ and $15$ (either variable may take either of the values).

Thus, $yz = 150$.
by Loyal (9.3k points)
edited by
what a nice approach @Regina phalange

""T(9)=1+min(T(9)=1+min(Steps from yy to 100100, Steps from zz to 100)100), where y and z are two possible values that can be reached from 9." Where  it is mentioned???


Read the question carefully.


I am unable to find out how it,"where y and z are two possible values that can be reached from 9", is linked. Can you please explain where it is ?



@rahulgargnov4  here the 1 in the equation is telling that. Equation says calculate minimum of T(y) and T(z) and then simply add 1 to it. What does that mean? 9 is at a distance of 1 from both y and z.

Analogy : height of the tree is : 1 + height(left) + height (right). Why one is here? because root is at distance one from both right and left. 

+33 votes
$T(9) =$ Distance from $9$ to $100$
$T(9)=1+ \min(T(y),T(z))=1+$min(Distance from $y$ to $100$ , Distance from $z$ to $100$)

There are only two such values where we can reach from $9$ , one is simple step to right on number line , i.e $10$ and another is $15$ (given shortcut)

Hence , $y=10$ , $z=15$
$yz=10 \times 15 = 150$
by Active (3.8k points)
edited by
Can you please explain this question once again , I am not getting it precisely .
I am not getting .. Please explain anyone

it is solved like this








Now, shortcut for T(9)=15

So, yz=10*15=150

but how could it be say , it goes from 0 to 100?

@ Srinath

Solution of the equation


Now, 9 to 15 there is a shortcut.

Now, if I take solution like that



that means , T(9) to T(15) there exists 1 step, i.e. why we are adding 1.
+1 vote

From $k$, we have two paths-

  1. Path 1- This is the path where we go from $k$ to $k_1$ where $k_1 = k+1$. Here, $k_1$ is the next number to the right of $k$.
  2. Path 2- This is the path where we go from $k$ to $k_2$ where $k_2 = k+i$. Here, $k_2$ is the the number to which $k$ has a shortcut.

Therefore, we have the following recurrence relation-

$T(k) = 1 + min(T(k_1), T(k_2))$

This basically means that the smallest no. of steps from $k$ to $100$ includes at least one step (either from $k$ to $k_1$ or from $k$ to $k_2$) and then subsequent smallest no. of steps from either $k_1$ to $100$ or from $k_2$ to $100$, whichever is minimum.

$\therefore T(9) = 1 + min(T(10), T(15))


$k_1 = 9+1 = 10 \Rightarrow y = 10$, 

$k_2 = 15$ ,($\because$ there is at most a single shortcut involving any number and it is given that the shortcut from $9$ is $15$) $\Rightarrow z = 15$

$\therefore y.z = 10.(15) = 150$

by Junior (515 points)
edited by

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