# Puzzle 69 |The Number Game

• Difficulty Level : Hard
• Last Updated : 24 Aug, 2021

Ankur and Vijay are fond of playing Number games. In one of the games, Ankur has to choose a number X, such that X belongs to [1, 10000] . Vijay has to guess the chosen number as soon as possible. Ankur will let Vijay know whether his guess is smaller than, larger than or equal to the number. The warning is that Vijay loses the game if his guess is larger than Ankur’s chosen number two or more times.

Q1. How to make guesses, and how many guesses are necessary?

Q2. What if Ankur is allowed to pick a very large positive number without any given limits ? .

Solution:

Both the problems can be treated in the following ways:

Ans1:
When Ankur chooses a number between 1 and n, here n=10000, Vijay should start guessing √n, 2√n, 3√n, 4√n, and so on. The first time his guess exceeds Ankur’s number, the range of numbers has been narrowed down to √n numbers; he then starts guessing sequentially in that range.

Ans2: If Ankur guesses an arbitrarily large number n, then Vijay may use Binary Search which mean to choose mid of 1and 10000 ie 5000 and if Ankur says it is less then choose mid term of 1 and 5000 ie 2500 and so on and at the end you will be meeting up with exact number n. It takes the time of order O(n) to complete.