cotpi

Boolean financial advisors

Susam Pal — Sun, 31 Mar 2013 00:00:00 +0000

Alex and Bob work as financial advisors for the same company. They draw equal salaries from the company. They behave well at the office. Both work on similar assignments. Each assignment requires a yes-no decision. The company uses the decisions made by them to make profits.

After the recession hit the company very badly, one of them has to be fired. Both Alex and Bob have worked on almost the same number of assignments in the last ten years. Alex has been consistently taking about 80% decisions correctly every year. Bob, on the other hand, has been taking only about 5% correct decisions every year.

Assuming that the performances of Alex and Bob would remain the same in future, who should the company fire to maximize its profits in the years to come? Why?

Composite factorial plus one

Susam Pal — Sun, 03 Mar 2013 00:00:00 +0000

How many positive integers \(n\) are there such that \(n! + 1\) is composite?

Average salary

Susam Pal — Sun, 20 Jan 2013 00:00:00 +0000

A group of friends wants to know their average salary such that no individual salary can be deduced. There are at least three friends in the group. How can this problem be solved with a pen and paper?

Red and blue hats

Susam Pal — Sun, 09 Dec 2012 00:00:00 +0000

There are 100 prisoners in a prison. The warden will set them free if they win a game involving red and blue hats. All the prisoners will be made to stand in a straight line. The warden will blindfold all the prisoners, then put either a blue hat or a red hat on each prisoner's head, and finally remove all the blindfolds. Each prisoner can then see the hats of all the prisoners in front of him but he cannot see his own hat or the hats of those behind him. If at least 99 prisoners can correctly declare the colour of his hat, the warden will set them free.

Once the game begins, each prisoner is allowed to utter "red" or "blue" only once to declare the colour of his hat. They will not be allowed to communicate in any other manner. The warden will give them one day to decide a strategy to win this game. What should their strategy be?

All horses are the same colour

Susam Pal — Sun, 04 Nov 2012 00:00:00 +0000

All horses are the same colour; we can prove this by induction on the number of horses in a given set. Here's how: "If there's just one horse then it's the same colour as itself, so the basis is trivial. For the induction hypothesis, horses \(1\) through \(n - 1\) are the same colour, and similarly horses \(2\) through \(n\) are the same colour. But the middle horses, \(2\) through \(n - 1\), can't change colour when they're in different groups; these are horses, not chameleons. So horses \(1\) and \(n\) must be the same colour as well, by transitivity. Thus all \(n\) horses are the same colour; QED." What, if anything, is wrong with this reasoning?

Same leading digit

Susam Pal — Sun, 07 Oct 2012 00:00:00 +0000

The powers \(2^n\) and \(5^n\) start with the same digit where \(n\) is a positive integer. What are the possible values for this digit?

Efficient gossiping

Susam Pal — Sun, 16 Sep 2012 00:00:00 +0000

Each of \(n\) male citizens knows a different piece of gossip. They are allowed to exchange the gossip they know by phone. During a call, just one of the men speaks and tells the other all the gossip he knows. What is the minimum number of calls required to enable each man to know all the gossip?

Separating each integer from its double

Susam Pal — Sun, 19 Aug 2012 00:00:00 +0000

The positive integers are to be partitioned into several sets such that two times an integer and the integer itself do not belong to the same set, and every positive integer belongs to exactly one set. What is the minimum number of such sets required?

Two-move chess

Susam Pal — Sun, 05 Aug 2012 00:00:00 +0000

The two-move chess game has the same rules as the regular one, with only one exception. Each player has to make two consecutive moves at a time. Does the White player, who makes the first two moves, have a non-losing strategy?

Factorial and power

Susam Pal — Sun, 22 Jul 2012 00:00:00 +0000

What are the possible positive integer values of \(n\) such that \((n - 1)! + 1\) is a power of \(n\)?

Paths in an n-dimensional grid

Susam Pal — Sun, 15 Jul 2012 00:00:00 +0000

There is an \(n\)-dimensional grid in an \(n\)-dimensional Euclidean space made of all points with integer coordinates of the form \((x_1, x_2, \dots, x_n)\) that satisfy the inequality \(0 \leq x_i \leq a_i\) where \(i \in \{1, 2, \dots, n\}\). Every pair of points in the grid that are a unit distance apart are connected by an edge. Through these edges, how many possible shortest paths are there from the point at \((0, 0, \dots, 0)\) to the point at \((a_1, a_2, \dots, a_n\))?

Divisibility of the largest n-bit integer

Susam Pal — Sun, 08 Jul 2012 00:00:00 +0000

What are the possible positive integer values of \(n\) such that the largest \(n\)-bit integer is a multiple of \(n\)?

Position in a random permutation

Susam Pal — Sun, 01 Jul 2012 00:00:00 +0000

In a random permutation of the first \(n\) natural numbers, what is the probability that \(k\) appears before all the numbers greater than itself where \(1 \le k \le n\)?

Partitioning an integer and its double

Susam Pal — Sun, 24 Jun 2012 00:00:00 +0000

A partition of a positive integer \(n\) is a list of positive integers, ordered from largest to smallest, such that the sum of the integers in the sequence is \(n\). Each integer in the list is called a part.

What is the ratio of the number of possible partitions of a positive integer \(n\) to the number of possible partitions of \(2n\) into \(n\) parts?

Gambling with a die

Susam Pal — Sun, 17 Jun 2012 00:00:00 +0000

A gambling game involves a fair die with six faces. Each face is marked with a distinct number of dots from one and six. You pay a fee to play every round of the game. The fee for the first round is ₹0.01. The fee increases by ₹0.01 after each round is played. Thus, the fee to play the second round is ₹0.02, the fee to play the third round is ₹0.03, and so on. Once you have paid the fee, it won't be returned to you.

In each round, after you pay the fee, the die is rolled and you win ₹1.00 times the number of dots that appear on the face facing up. You may quit the game at any moment. How long should you play this game if you want to maximize your profit? What is the profit you expect to make?

Common divisors of sum and difference

Susam Pal — Sun, 03 Jun 2012 00:00:00 +0000

There are two numbers that are coprime to each other. What are the possible common positive divisors of their sum and the difference between them?

100 pirates and 1000 gold coins

Susam Pal — Sun, 27 May 2012 00:00:00 +0000

There are 100 pirates on a ship. They have distinct dates of birth. They find 1000 gold coins. They want to distribute the coins among themselves. The oldest pirate is supposed to propose a distribution of the coins, and then all the pirates, including himself, vote on whether the distribution is acceptable. If at least half of the pirates on the ship vote for it, the coins are distributed according to the proposed distribution. Otherwise, the oldest pirate who proposed the distribution is killed and thrown overboard, and the oldest among the remaining pirates makes a new proposal. This continues until a proposal is accepted.

Each pirate decides his vote after considering three factors. For each pirate, survival is the most important necessity. After ensuring survival, each pirate maximizes the number of coins he gets. He prefers to throw the oldest pirate overboard if it doesn't affect his survival and the maximum number of coins he gets.

Which pirate gets the greatest number of gold coins? How many coins does he get?

Behind a man there is a woman

Susam Pal — Sun, 20 May 2012 00:00:00 +0000

Imagine a queue consisting of men and women only. The first person in the queue is a man, and the last person is a woman. An autistic child refuses to believe that somewhere in the queue there is a woman directly behind a man. Fortunately, some teachers have succeeded in teaching him the principle of mathematical induction. Now, can you show the child, using mathematical induction only, that somewhere in the queue, directly behind a man there is a woman?

Two sequences of natural numbers

Susam Pal — Sun, 06 May 2012 00:00:00 +0000

There are two sequences of natural numbers. The first sequence consists of all natural numbers that do not contain a particular digit, in ascending order. The second sequence consists of all the remaining natural numbers in ascending order. Every natural number occurs exactly once in exactly one of the sequences.

How many terms are there in the first sequence such that each of those terms is greater than the corresponding term at the same position in the second sequence?

Powers of ten

Susam Pal — Sun, 29 Apr 2012 00:00:00 +0000

How many positive integers n are there such that \(10^{10^{10^{n}}} + 10^{10^{n}} + 10^{n} - 1\) is prime?