cotpi
http://cotpi.com/p/
cotpi puzzles.enBlack and white plane
http://cotpi.com/p/69/
<p>
How can you construct a plane where every point is coloured either black
or white such that two points of the same colour are never a unit
distance apart?
</p> Sun, 29 Sep 2013 00:00:00 +0000Susam Palhttp://cotpi.com/p/69/Boolean financial advisors
http://cotpi.com/p/68/
<p>
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.
</p><p>
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.
</p><p>
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?
</p> Sun, 31 Mar 2013 00:00:00 +0000Susam Palhttp://cotpi.com/p/68/Composite factorial plus one
http://cotpi.com/p/67/
<p>
How many positive integers \(n\) are there such that \(n! + 1\) is
composite?
</p> Sun, 03 Mar 2013 00:00:00 +0000Susam Palhttp://cotpi.com/p/67/Average salary
http://cotpi.com/p/66/
<p>
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?
</p> Sun, 20 Jan 2013 00:00:00 +0000Susam Palhttp://cotpi.com/p/66/Red and blue hats
http://cotpi.com/p/65/
<p>
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.
</p><p>
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?
</p> Sun, 09 Dec 2012 00:00:00 +0000Susam Palhttp://cotpi.com/p/65/All horses are the same colour
http://cotpi.com/p/64/
<p>
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?
</p> Sun, 04 Nov 2012 00:00:00 +0000Susam Palhttp://cotpi.com/p/64/Same leading digit
http://cotpi.com/p/63/
<p>
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?
</p> Sun, 07 Oct 2012 00:00:00 +0000Susam Palhttp://cotpi.com/p/63/Efficient gossiping
http://cotpi.com/p/62/
<p>
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?
</p> Sun, 16 Sep 2012 00:00:00 +0000Susam Palhttp://cotpi.com/p/62/Separating each integer from its double
http://cotpi.com/p/61/
<p>
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?
</p> Sun, 19 Aug 2012 00:00:00 +0000Susam Palhttp://cotpi.com/p/61/Two-move chess
http://cotpi.com/p/60/
<p>
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?
</p> Sun, 05 Aug 2012 00:00:00 +0000Susam Palhttp://cotpi.com/p/60/Factorial and power
http://cotpi.com/p/59/
<p>
What are the possible positive integer values of \(n\) such that
\((n - 1)! + 1\) is a power of \(n\)?
</p> Sun, 22 Jul 2012 00:00:00 +0000Susam Palhttp://cotpi.com/p/59/Paths in an n-dimensional grid
http://cotpi.com/p/58/
<p>
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
\((a_1, a_2, \dots, a_n\))?
</p> Sun, 15 Jul 2012 00:00:00 +0000Susam Palhttp://cotpi.com/p/58/Divisibility of the largest n-bit integer
http://cotpi.com/p/57/
<p>
What are the possible positive integer values of \(n\) such that the
largest \(n\)-bit integer is a multiple of \(n\)?
</p> Sun, 08 Jul 2012 00:00:00 +0000Susam Palhttp://cotpi.com/p/57/Position in a random permutation
http://cotpi.com/p/56/
<p>
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 \(1 \le k \le n\)?
</p><p>
</p> Sun, 01 Jul 2012 00:00:00 +0000Susam Palhttp://cotpi.com/p/56/Partitioning an integer and its double
http://cotpi.com/p/55/
<p>
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.
</p><p>
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?
</p> Sun, 24 Jun 2012 00:00:00 +0000Susam Palhttp://cotpi.com/p/55/Gambling with a die
http://cotpi.com/p/54/
<p>
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.
</p><p>
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?
</p> Sun, 17 Jun 2012 00:00:00 +0000Susam Palhttp://cotpi.com/p/54/Common divisors of sum and difference
http://cotpi.com/p/53/
<p>
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?
</p> Sun, 03 Jun 2012 00:00:00 +0000Susam Palhttp://cotpi.com/p/53/100 pirates and 1000 gold coins
http://cotpi.com/p/52/
<p>
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.
</p><p>
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.
</p><p>
Which pirate gets the greatest number of gold coins? How many coins does
he get?
</p> Sun, 27 May 2012 00:00:00 +0000Susam Palhttp://cotpi.com/p/52/Behind a man there is a woman
http://cotpi.com/p/51/
<p>
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?
</p> Sun, 20 May 2012 00:00:00 +0000Susam Palhttp://cotpi.com/p/51/Two sequences of natural numbers
http://cotpi.com/p/50/
<p>
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.
</p><p>
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?
</p> Sun, 06 May 2012 00:00:00 +0000Susam Palhttp://cotpi.com/p/50/