## Algebra Basics: What Are Polynomials?

Are you struggling with Polynomials? Check out this video below to learn what Polynomials really is and how you can solve these math problems with ease!

## The Three Prisoner Puzzle

Here is one of my all-time favourite puzzles. This can be solved with just logic – no math needed. Sometimes those who are not so good at math, get it quicker! Try it out for yourself.

## Three Logic Problems

Here are three challenging logic problems to try out. Good luck!

## The Riddle Bridge

Taking that internship in a remote mountain lab might not have been the best idea. Pulling that lever with the skull symbol just to see what it did probably wasn’t so smart either. But now is not the time for regrets because you need to get away from these mutant zombies…fast. Can you use math to get you and your friends over the bridge before the zombies arrive? Alex Gendler shows how.

## The Burning Island Puzzle

A man is stranded on an island covered in forest.

One day, when the wind is blowing from the west, lightning strikes the west end of the island and sets fire to the forest. The fire is very violent, burning everything in its path, and without intervention the fire will burn the whole island, killing the man in the process.

There are cliffs around the island, so he cannot jump off.

How can the man survive the fire? (There are no buckets or any other means to put out the fire).

## The Black And White Hats Problem

Cannibals ambush a safari in the jungle and capture three men. The cannibals give the men a single chance to escape uneaten.

The captives are lined up in order of height, and are tied to stakes. The man in the rear can see the backs of his two friends, the man in the middle can see the back of the man in front, and the man in front cannot see anyone. The cannibals show the men five hats. Three of the hats are black and two of the hats are white.

Blindfolds are then placed over each man’s eyes and a hat is placed on each man’s head. The two hats left over are hidden. The blindfolds are then removed and it is said to the men that if one of them can guess what color hat he is wearing they can all leave unharmed.

The man in the rear who can see both of his friends’ hats but not his own says, “I don’t know”. The middle man who can see the hat of the man in front, but not his own says, “I don’t know”. The front man who cannot see ANYBODY’S hat says, “I know!”

How did he know the color of his hat and what color was it?

## The 5 Pirates Puzzle

5 pirates of different ages have a treasure of 100 gold coins.

On their ship, they decide to split the coins using this scheme:

The oldest pirate proposes how to share the coins, and ALL pirates (including the oldest) vote for or against it.

If 50% or more of the pirates vote for it, then the coins will be shared that way. Otherwise, the pirate proposing the scheme will be thrown overboard, and the process is repeated with the pirates that remain.

As pirates tend to be a bloodthirsty bunch, if a pirate would get the same number of coins if he voted for or against a proposal, he will vote against so that the pirate who proposed the plan will be thrown overboard.

Assuming that all 5 pirates are intelligent, rational, greedy, and do not wish to die, (and are rather good at math for pirates) what will happen?

## The Parking Spaces Problem

At a local business, parking spaces are reserved for the top executives: CEO, president, vice president, secretary, and treasurer with the spaces lined up in that order. The parking lot guard can tell at a glance if the cars are parked correctly by looking at the color of the cars. The cars are white, orange, red, black, and blue, and the executives names are Andrew, Brett, Caitlyn, Danny, and Erin.

• The car in the first space is black.
• A blue car is parked between the black car and the orange car.
• The car in the last space is red.
• The secretary drives a white car.
• Andrew’’s car is parked next to Danny’s.
• Erin drives a orange car.
• Brett’s car is parked between Caitlyn’s and Erin’s.
• Danny’s car is parked in the last space.

Question 1: Who is the secretary?

1. Erin
2. Danny
3. Caitlyn
4. Brett
5. Andrew

Questions 2: Who is the CEO?

1. Andrew
2. Brett
3. Caitlyn
4. Danny
5. Erin

Question 3: What color is the vice president’s car?

1. Orange
2. White
3. Blue
4. Red
5. Black

1. E – Caitlyn cannot be the secretary, since she is the CEO, nor can Erin, because she drives a orange car (and the secretary drives a white car). Danny’s, the red car, is in the last space. Andrew is the secretary, because his car is parked next to Danny’s, which is where the secretary’s car is parked.

1. C – The CEO drives a black car and parks in the first space. Erin drives a orange car; Brett’s car is not in the first space; Danny’s is not in the first space, but the last. Andrew’s car is parked next to Danny’s, so Caitlyn is the CEO.

1. A – The vice president’s car cannot be black, because that is the CEO’s car, which is in the first space. Nor can it be red, because that is the treasurer’s car, which is in the last space, or white, because that is the secretary’s. The president’s car must be blue, because it is parked between a black car (in the first space) and a orange car, which must be the vice president’s.

## The Snake Problem

The Snake Problem

You are asked to open one (and only one) of two boxes, labeled “A” and “B,” and are given the following facts:

Fact 1:  Each box contains either a poisonous snake or a large sum of money.  If you open a box containing a snake, which is highly poisonous, it is likely to attack you and kill you instantly.  If you open a box containing money, you are entitled to keep the money.

Fact 2:  Each Box has a sign attached with a written statement on it.  Either both of these statements are true, or both are false.

Here are the statements:

Statement on Box A:  At least one of these boxes contains a million dollars.

Statement on Box B:  The other box has a poisonous snake in it.

Question:  Which box should you open?

To answer that question we need to do an analysis of the given information. The first thing is to list all possibilities.

Notes To Help Answer The Question

1.  This is a logic problem, and a logic problem may present difficulties of “self-reference.”

2.  Do not jump into a problem right away.  Be sure you first understand the problem, even if it takes a few moments.

3.  If a problem clearly give a small number of possibilities, one way that is sometimes good to solve the problem is to “step” through it, and see how each possibility works out.

4.  There are two weak spots in this analysis—and they are typical of weak spots in almost any problem on a test or in real life that requires logical analysis.  They shuld be carefully investigated—on a test by reviewing when you are all done, or in real What life by someone in your company or on your team, especially if there is high risk, such as expense or health involve.  What are they?

(A)  Is the logic valid?  Is there a flaw in the logic?

(B)  Even if the logic is valid, the conclusion may be incorrect if you do not have the facts.  Are the claims used in the analysis reliable and complete?

– Reliable means honest and accurate (can have one without other).

– Complete means you have all the relevant information needed.

To do so we must be sure we understand all of the given information correctly.  It should be noted Fact 1 may be interpreted in one of two ways.

Wrong way:  One box has a snake and the other box has the money—two boxes cannot contain the same thing.  This gives us two possibilities:
Possibility 1:  A has a snake, and B has money.
Possibility 2:  A has money, and B has a snake.

Correct Way:  Any one box can contain either a snake or money—these possibilities are independent, and there is no guarantee that a snake is in either box or that there is money in either box.  Thus, there are four possibilities, the two above plus these two:
Possibility 3:  A has a snake, and B has a snake.
Possibility 4:  A has money, and B has money.

Question:  How do you know that the second way (as opposed to the first) is the correct one?

Answer:  Two possible ways.  First, from the wording itself , “Each box…”  This phraseology does not mean there is exactly one box with snake and exactly one box with money.  It merely means each time there is either a snake or money—there is no relationship between the two.  The second way is from the statement on box A saying “At least” one box has money—this statement allows for the possibility both have money (or both have snakes.)

Summarizing the gives us the following chart:

Our second step involves an analysis of the information once we have listed all cases.  Let us ask which statements, A or B, will be true in each case.  We put a Blue T or F (for True or False) next to each possibility if, in that case, Statement A is true or false, and we put a Red T or F next to each possibility if, in that case, Statement B is true or false.  NOTE: Our goal is to co-ordinate the two statements (Fact 1 & 2) with the two statements (A & B).  Here is our result:

Thus, we see in the first case, and only in the first case, both Statements A & B are true. There is no other case where both are true or both are false, as required by Fact 1.  Therefore, it must be that Possibility 1 is the only viable possibility, so open Box B.

In real life, however, we should not open any box, because, as humans, we can never be sure that (1) we did not make any mistake in our analysis, and (2) the information given to us (namely Fact 1 and Fact 2) are really correct.  We should consider these to be claims that other humans (who are fallible) have made, and not accept them as absolute fact.