Brainstorming Easy to Hard Tricky Puzzles

Impossible Puzzle - 003

2007-04-03T11:59:00.000+05:30

The classical mathematical puzzle known as water, gas, and electricity, the (three) utilities problem, or sometimes the three cottage problem, can be stated as follows:

Suppose there are three cottages on a plane (or sphere) and each needs to be connected to the gas, water, and electric companies. Using a third dimension or going through a company or cottage are illegal. Is there a way to do so without any of the lines crossing each other?

Why IMPOSSIBLE ? ?

The problem is part of the mathematical field of topological graph theory which studies the embedding of graphs on surfaces. In more formal graph-theoretic terms, the problem asks whether the complete bipartite graph K3,3 is planar. Kazimierz Kuratowski proved in 1930 that K3,3 is nonplanar, and thus that the three cottage problem has no solution.

Thomsen graph, Utility graph,

K 3,3

n = 6

m = 9

But K3,3 is toroidal, that is it can be embedded on the torus. In terms of the three cottage problem this means the problem can be solved by punching a hole through the plane (or the sphere). This changes the topological properties of the surface and using the hole we can connect the three cottages without crossing lines.

It is not possible to redraw these without edge intersections. Repeated attempts should convince you that this is plausible. To prove this mathematically requires knowledge of topology. The three cottage problem is a simplified version of a problem that electronic circuit board designers face. When all of the connections on a circuit board are limited to one side of a board, all nonplanar electronic circuits are impossible. If you are allowed to use a third dimension by using wires or connecting to the second side of the board, all electronic circuit paths are possible.

Impossible Puzzle - 002

2007-04-03T11:46:00.000+05:30

Seven Bridges of Königsberg

Seven Bridges of Königsberg is a famous solved mathematics problem inspired by an actual place and situation. The city of Königsberg, Prussia (now Kaliningrad, Russia) is set on the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges. The question is whether it is possible to walk with a route that crosses each bridge exactly once. Circa 1750, the prosperous and educated townspeople allegedly walked about on Sundays trying to solve the problem, but this might be an urban legend.

Map of Königsberg in Euler's time showing the actual layout of the seven bridges, highlighting the river Pregolya and the bridges.

Why Impossible ?

In 1736, Leonhard Euler proved that it was not possible. In proving the result, Euler formulated the problem in terms of graph theory, by abstracting the case of Königsberg — first, by eliminating all features except the landmasses and the bridges connecting them; second, by replacing each landmass with a dot, called a vertex or node, and each bridge with a line, called an edge or link. The resulting mathematical structure is called a graph.

→ →

The shape of a graph may be distorted in any way without changing the graph itself, so long as the links between nodes are unchanged. It does not matter whether the links are straight or curved, or whether one node is to the left or right of another.

Euler realized that the problem could be solved in terms of the degrees of the nodes. The degree of a node is the number of edges touching it; in the Königsberg bridge graph, three nodes have degree 3 and one has degree 5. Euler proved that a circuit of the desired form is possible if and only if there are at most two nodes of odd degree. Such a walk is called an Eulerian path or Euler walk. Further, if there are two nodes of odd degree, those must be the starting and ending points of an Eulerian path. Since the graph corresponding to Königsberg has four nodes of odd degree, it cannot have an Eulerian path.

An alternative form of the problem asks for a path that traverses all bridges and also has the same starting and ending point. Such a walk is called an Eulerian circuit or an Euler tour. An Eulerian circuit exists if and only if there are no nodes of odd degree. It can be seen that all Eulerian circuits are also Eulerian paths.

Euler's work was presented to the St. Petersburg Academy on August 26, 1735, and published as Solutio problematis ad geometriam situs pertinentis (The solution of a problem relating to the geometry of position) in the journal Commentarii academiae scientiarum Petropolitanae in 1741.

PRESENT STATE of BRIDGES

Two of the seven original bridges were destroyed by British bombing of Königsberg during World War II. Two others were later demolished by the Russians and replaced by a modern highway. The three other bridges remain, although only two of them are from Euler's time (one was rebuilt by the Germans in 1935).

In terms of graph theory, two of the nodes now have degree 2, and the other two have degree 3. Therefore, an Eulerian path is now possible, but since it must begin on one island and end on the other, it is impractical for tourists.

Impossible Puzzle - 001

2007-04-03T11:24:00.000+05:30

This is a famous puzzle first seen in a Scientific American under Mathematical Games entitled "The Mutilated Chessboard" by Martin Gardner. The puzzle was stated as follows:

The props for this problem are a chessboard and 32 dominoes. Each domino is of such size that it exactly covers two adjacent squares on the board. The 32 dominoes therefore can cover all 64 of the chessboard squares. But now suppose we cut off two squares at diagonally opposite corners of the board and discard one of the dominoes. Is it possible to place the 31 dominoes on the board so that all the remaining 62 squares are covered? If so, show how it can be done. If not, prove it impossible.

why impossible ??

The puzzle is impossible. Any way you would place a domino would cover one white square and one black square. A group of 31 dominoes would cover 31 white and 31 black squares of a chessboard, leaving one white and one black square uncovered. The directions had you remove opposite corner squares, and such squares are always either both black or both white.

Brainteasers - 010

2007-04-03T11:10:00.000+05:30

Square Puzzle

The five pieces shown below must be put together to a square.

The Question: How should this be done?

A Hint: Print the picture with the pieces, and cut the pieces out. It's more difficult than it looks!...

Brainteasers - 009

2007-04-03T11:05:00.000+05:30

Coin Weighing

We have 12 coins and a balance. 11 coins are of the same weight, but one coin differs in weight (note that you do not know whether the coin with different weight is heavier or lighter!). You may perform three weighings to find out which coin has a different weight, and whether this coin is heavier or lighter.

The Question: How should you perform these three weighings to find out which coin has a different weight, and whether this coin is heavier or lighter?

Brainteasers - 008

2007-04-03T10:50:00.000+05:30

Stacking Coins

You have an unlimited number of coins with a diameter d and you stack them. The goal is to let the topmost coin stick out as far as possible.

The Question: What is the maximal distance between the center of the topmost coin and the center of the lowermost coin?

Brainteasers - 007

2007-04-03T10:46:00.000+05:30

3 Heads & 5 Hats

In a small village in the middle of nowhere, three innocent prisoners are sitting in a jail. One day, the cruel jailer takes them out and places them in a line on three chairs, in such a way that man C can see both man A and man B, man B can see only man A, and man A can see none of the other men. The jailer shows them 5 hats, 2 of which are black and 3 of which are white. After this, he blindfolds the men, places one hat on each of their heads, and removes the blindfolds again. The jailer tells his three prisoners that if one of them is able to determine the color of his hat within one minute, all of them are released. Otherwise, they will all be executed. None of the prisoners can see his own hat, and all are intelligent. After 59 seconds, man A shouts out the (correct) color of his hat!

The Question: What is the color of man A's hat, and how does he know?

Brainteasers - 006

2007-04-03T10:42:00.000+05:30

Zebra

There are 5 houses. Each house has a unique color, and each owner has a different nationality. Each owner keeps a different pet, drinks a different type of beverage, and smokes a different brand of cigarettes.

The Brit lives in the red house, the Sweed keeps a dog, and the Dane drinks tea.
The green house is on the immediate left of the white house.
In the green house they drink coffee.
The man who smokes Pall Mall has birds.
In the yellow house they smoke Dunhill. In the middle house they drink milk.
The Norwegian lives in the first house.
The man who smokes Blend lives in the house next to the house with the cats.
In the house next to the house with the horse, they smoke Dunhill.
The man who smokes Blue Master, drinks beer.
The German smokes Prince.
The Norwegian lives next to the blue house.
They drink water in the house that lies next to the house where they smoke Blend.

The Question: Who owns the zebra?

Brainteasers - 005

2007-04-03T10:11:00.000+05:30

Colorful Dwarfs

In a distant, dark forest, lives a population of 400 highly intelligent dwarfs. The dwarfs all look exactly alike, but only differ in the fact that they are wearing either a red or a blue hat. There are 250 dwarfs with a red hat and 150 dwarfs with a blue hat. Striking however, is that the dwarfs don't know these numbers themselves and that none of them knows what the colour of his hat is (there are for example no mirrors in this forest). But the dwarfs do know that there is at least one dwarf with a red hat.

During a certain period of their year, there is a big party in this village, to which initially all dwarfs will go. However, this party is only intended for dwarfs wearing a blue hat. Dwarfs with a red hat are supposed never to return to the party again, as soon as they know that they are wearing a red hat.

The Question: How many days does it take before there are no more dwarfs with a red hat left at the party?

Brainteasers - 004

2007-04-03T10:07:00.000+05:30

Pirate Treasure

A pirate ship captures a treasure of 1000 golden coins. The treasure has to be split among the 5 pirates: 1, 2, 3, 4, and 5 in order of rank. The pirates have the following important characteristics:

* Infinitely smart.
* Bloodthirsty.
* Greedy.

Starting with pirate 5 they can make a proposal how to split up the treasure. This proposal can either be accepted or the pirate is thrown overboard. A proposal is accepted if and only if a majority of the pirates agrees on it.

The Question: What proposal should pirate 5 make?

Brainteasers - 003

2007-04-03T09:56:00.000+05:30

Bizarre Boxes

Someone shows you two boxes and he tells you that one of these boxes contains two times as much as the other one, but he does not tell you which one this is. He lets you choose one of these boxes, and opens it. It turns out to be filled with $10. Now he gives you the opportunity to choose the other box instead of the current one (and skip the $10 of the first box), because the second box could contain twice as much (i.e. $20).

The Question: Should you choose the second box, or should you stick to your first choice to maximize the expected amount of money?

A Hint: If you have $10, and you could double this with a chance of 1/2, or half it with a chance of 1/2, one would expect an average of 1/2 * $20 + 1/2 * $5 = $12.5 (so you would expect to gain $2.5)!...

Brainteasers - 002

2007-04-03T09:47:00.000+05:30

The TRUEL

On an early morning, three rivals get together on an open spot in a dark wood to compose a quarrel by means of guns. A kind of duel, but with three persons: A, B and C. The rules of the game are:

* They draw lots who may fire first, second and third.
* Next, they will continue firing at each other in this order until only a single person is alive.
* Every person decides himself at which person he fires.
* Everyone knows that A hits (kills) in 100% of all shots, B hits (kills) in 80% of all shots and C hits (kills) in 50% of all shots.
* Each person chooses his ideal strategy.
* No one is killed by a stray bullet.

The Question: Who has the largest chance of surviving the truel, and how big is this chance?

Brainteasers - 001

2007-04-03T09:28:00.000+05:30

Numbers and Dots

This is a famous problem from 1882, to which a prize of $1000 was awarded for the best solution. The task is to arrange the seven numbers 4, 5, 6, 7, 8, 9, and 0, and eight dots in such a way that an addition approximates the number 82 as close as possible. Each of the numbers can be used only once. The dots can be used in two ways: as decimal point and as symbol for a recurring decimal. For example, the fraction ¹/₃ can be written as

.

The dot on top of the three denotes that this number is repeated infinitely. If a group of numbers needs to be repeated, two dots are used: one to denote the beginning of the recurring part and one to denote the end of it. For example, the fraction ¹/₇ can be written as
. .

.142857

Note that '0.5' is written as '.5'.

The Question: How close can you get to the number 82?

Mirroring Clock

2007-03-27T21:07:00.000+05:30

Mirroring Clock

A boy leaves home in the morning to go to school. At the moment he leaves the house he looks at the clock in the mirror. The clock has no number indication and for this reason the boy makes a mistake in interpreting the time (mirror-image). Just assuming the clock must be out of order, the boy cycles to school, where he arrives after twenty minutes. At that moment the clock at school shows a time that is two and a half hours later than the time that the boy saw on the clock at home.

The Question: At what time did he reach school?

Sailors, monkey and coconuts ( Difficult )

2007-03-01T20:13:00.000+05:30


5 Sailors and a monkey landed on an island in the southern Pacific Ocean. They found a big pile of coconuts. Since they were so tired and it was getting late, they all went to sleep instead. Allan woke up at night. He wanted to get his share right then. He divided the coconuts evenly into 5 piles, but one coconut was left. He hid one pile for himself and gave the extra one to the monkey. He went back to sleep. Brad woke up (without knowing what had happened) after Allan. He divided the rest of the coconuts evenly into 5 piles, but one coconut was left. He hid one pile for himself and gave the extra one to the monkey. He went back to sleep. Charlie woke up (without knowing what had happened) after Brad. He divided the rest of the coconuts evenly into 5 piles, but one coconut was left. He hid one pile for himself and gave the extra one to the monkey. He went back to sleep. David woke up (without knowing what had happened) after Charlie. He divided the rest of the coconuts evenly into 5 piles, but one coconut was left. He hid one pile for himself and gave the extra one to the monkey. He went back to sleep. Earl woke up (without knowing what had happened) after David. He divided the rest of the coconuts evenly into 5 piles, but one coconut was left. He hid one pile for himself and gave the extra one to the monkey. He went back to sleep. When they all awoke in the morning, they determined to divide the rest of the coconuts evenly into 5 piles. Each sailor got a pile. Again one coconut was left, and they gave it to the monkey. Everybody got one pile. How many coconuts did each person get? 1. ) Now , Readers are requested to comment for providing the solution . You may put any funny answer or actual answers you have. 2. ) If needed go for the Google :) 3.) If readers didn't find the it , the solution will be provided on 10th day after asking the question.

Life Gate or Death Gate

2007-02-24T18:15:00.000+05:30



There is a prisoner who is about to be executed. The king decides to give him one last chance to live. There are 2 doors, the life door and the death door. There is one guard standing by each door. Those two guards know which door is the life door and which is the death door. However, one of them always tells the truth and the other always tells a lie. There is no way you can identify which door is the life door or the death door. There is no way you can distinguish who is the one telling the truth. The prisoner can only ask one guard one question. Then he needs to choose a door to walk in. If he walks in the death door, then he will be executed. If he walks in the life door, he can have a new life. He did choose the life door and lived. What was the question he asked? How did he choose the door after he got the answer from one of the guards? 1. ) Now , Readers are requested to comment for providing the solution . You may put any funny answer or actual answers you have. 2. ) If needed go for the Google :) 3.) If readers didn't find the it , the solution will be provided on 10th day after asking the question.