Author Archives: shivang1729
The Tower of Hanoi (also called the Tower of Brahma or Lucas’ Tower, and sometimes pluralised) is a mathematical game or puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape.
The objective of the puzzle is to move the entire stack to another rod, obeying the following rules:
- Only one disk may be moved at a time.
- Each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod.
- No disk may be placed on top of a smaller disk.
With three disks, the puzzle can be solved in seven moves.
for disks and towers, the number of steps required to transfer the disks from tower to another is .
Proof of our lemma:
Now we transfer the th disc to peg 2 . One move will be required for this.
Finally we will transfer discs which are on peg to peg . This will require another moves. Thus the transfer will be done in moves which will be equal to movesTherefore we have formed a recurrence relation that
Now we will obtain a general formula for
We will multiply both sides of the above relations by
Solving this gives us
Since ( one move will be required to move )
THIS FINALLY GIVES
1. wikepedia ( for the introduction) : http://en.wikipedia.org/wiki/Tower_of_Hanoi
The AMTI is a pioneer organisation in promoting and conducting Maths Talent Tests in India. Last year (43rd TC Data) (in the 43rd National level tests) 54058 students from 332 institutions spread all over India, participated at the screening level; 10% of them insitutionwise were selected for the final test. For the benefit of final level contestants and the chosen few for INMO, special orientation camps were conducted. Merit certificates and prizes were awarded to the deserving students.
DOWNLOAD THE 2012 PAPER :
NMTC Question Paper
Find the number of integers in the set such that is a multiple of
Let be a sequence of nondecreasing positive integers such that for some positive integers and . Prove that there exists a positive integer such that
There are cities, airline companies in a country. Between any two cities, there is exactly one -way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of
As shown in the figure below, the in-circle of is tangent to sides and at and respectively, and is the circumcenter of . Prove that .
There is a stone at each vertex of a given regular -gon, and the color of each stone is black or white. Prove that we may exchange the position of two stones such that the coloring of these stones are symmetric with respect to some symmetric axis of the -gon
Find all pairs of integers satisfying: there exists an integer such that is divisible by for all positive integers
Circles and are tangent to each other externally at . Points and are on , lines and are tangent to at and , respectively, lines and meet at point
Let be non-negative real numbers. Prove that