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Maths Problems and Puzzles: How to Solve Them?

From Robin, published on 27/06/2017 We Love Prof - UK > Academia > Maths > The Challenge of Mathematical Riddles

Since antiquity, mathematics has been an integral part of our everyday lives.

Our understanding of it, and the constant desire to enlarge the extent of our knowledge about the world has taken us far, so that today, the world as we know it would not be possible without maths.

Had you considered, though, that you can have fun while learning maths?

How can subjects such as algebra, fractions, and probability, be made more entertaining? With puzzles!

That’s right, it is possible to practice maths while enjoying yourself!

Alan Turing was a pioneer of modern computing The mathematical pioneer Alan Turing broke the toughest of codes (Source: Wikipedia.org – Jon Callas)


Thomas

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Man and Mystery

Man has always sought to understand the world in which he evolved.

He has researched untiringly, consulted countless tomes and debated with his contemporaries, to better understand the world of maths. Each answer has lead to more questions.

The desire to solve riddles is part of our genetic heritage: We are born to seek answers.

Why are we here, on Earth? Is there life after death? Who were the first humans? How did they live?

Since antiquity, some of the great mysteries rooted in maths and physics have eluded our understanding:

  • How were the Pyramids of Egypt constructed; for what purpose were they so arranged?
  • How can we explain the mathematical genius of great men like Leonardo da Vinci, Archimedes, Newton, Henri Poincaré and Stephen Hawking?
  • Other archaeological mysteries of mathematical importance which still confound our understanding include: The Sphinx in Egypt, the pillar of Delhi; an iron pillar more than 7 meters high and 1600 years old which has never rusted, and the Megalithic Spheres of Costa Rica; 300 spheres each 2 meters in diameter and weighing 16 tonnes, whose period of origin and purpose remain unknown.

The film depicted the cracking of the secret Enigma code Breaking the Enigma code: The Imitation Game (Source: Flickr.com – Bagogames)

Becoming absorbed in maths problems can be a very good way for someone to forget their troubles.

To do so, you may need to use:

  • geometry
  • mental arithmetic
  • applied mathematics
  • long division
  • mathematical theorems
  • trigonometry

Maths and puzzles

To Love Maths… or not

Should we all love maths and mathematical equations?

Why do they matter?

Why do some people turn off when it comes to this topic?

What are some Reasons to Love Maths?

  • Maths helps to ascertain whether something is true or false.
  • There is a certain elegance in mathematical theories. Because of their very conciseness and simplicity, you may find that you’re able to gain new understanding with only a small amount of study. The important thing is to always seek to understand, rather than learning dozens of formulas and theorems without grasping their ins and outs.
  • Maths can be very useful in poker when it comes to winning a bet!
  • Maths is a very powerful tool: It is possible to achieve exceptional results and applications that at first seem beyond our reach.
  • Thanks to maths, you will not only gain a deeper understanding of the world around you, but be better able to approach other disciplines such as physics, chemistry and economics.
  • Maths is like a game in that it is logical, formal and stimulates your brain as do games like chess, sudoku and even Candy Crush Saga!
  • Once you have grasped the main principles, maths becomes a kind of second nature that helps you understand and solve the problems around you.

Why do many find Maths so Difficult?

  • Mathematics is its own language, and to use it well, you need to master its specific grammar, vocabulary and spelling. There are rules to be learned and those to be applied without question.
  • As a subject, it requires a great deal of self-discipline. It’s not good enough to settle for ‘almost’ in maths: You have to be concise and methodical.

How to go about cracking a maths problem? An Enigma decryption machine, called a “bombe” (Source: media.defense.gov – U.S. Air Force)

  • Finally, maths is a demanding discipline that requires regular and consistent practice. Whether you are alone in front of a screen, or your textbook, whether you are taking lessons with a private maths tutor, you need to be hard-working and persevere, especially outside of traditional maths classes, if you are in such a program of study.

Five Challenging Maths Problems

In a quest to understand the world around him, man has employed maths in an effort to tease out tangible proofs. The history of mathematics is punctuated by great minds wrestling with the great enigmas of their time.

Mathematical puzzles combine reasoning with numbers, calculations and figures.

To solve such puzzles, one needn’t be among the best mathematical minds, but it is important to take a logical approach and apply the maths skills one has learnt in school, from simple multiplication and division, quadratic equations and calculus.

Here, then, are 5 challenging puzzles that may be encountered in maths classes:

The 100 Prisoners Problem

There are 100 prisoners, sentenced to death, in a prison. Out of the blue, the prison’s director proposes a challenge

He assigns each prisoner a number between 1 and 100, then installs in his office a cabinet with 100 drawers, each containing a random number between 1 and 100, corresponding to those assigned to the prisoners.

Each number appears only once.

He asks each prisoner to open 50 drawers and check the number in each.

Once each prisoner has entered the office, he is forbidden from communicating with his fellow prisoners, nor to change the sequence of draws or leave any clues.

No prisoner will know what numbers the other inmates have seen.

The prison director gives two possible outcomes:

  • All of the prisoners find their respective numbers and all are pardoned.
  • None find their numbers and they are all are executed.

What is the chance that each prisoner finds the drawer corresponding to his number?

According to the law of mathematical probability, the chance that all would be pardoned is (1/2)100, or 0.0000000000000000000000000000008.

There is a clever strategy that offers the prisoners the chance to increase these odds, and live. What is it?

Solve these maths questions! Matrices like those in the Raven’s test: A nonverbal group test used in educational settings
(Source: Frontiers in Psychology – Daniel Little et. al.)

The 3 Gods

Behind three characters called A, B and C hide 3 gods known as ‘true’, ‘false’ and ‘random’.

The ‘true’ god always responds with the truth, the ‘false’ god always lies and the ‘random’ one alternates unpredictably between the two.

The challenge is ‘simple’!: Discover the respective identities of A, B and C by asking only three questions to which the answer is either true or false.

Each question can only be asked of one god, but if you decide to question a single god more than once (a maximum of three times), the other gods will not be able to answer.

Your questions may be unrelated to each other.

The Triangular Cake

When preparing his class for a mathematics competition, a teacher decides to offer his students a cake in the form of a triangle with three unequal sides.

He places an order with a cake shop, giving the measurements of the cake’s three sides.

The baker orders a box for the cake, giving the same measurements. When the cake is done, however, he finds that while the measurements have been respected, the shape is symmetrical, rather than identical, to that of its cake.

He calls the maths teacher to ask how he should cut the cake such that it fits in the box.

The teacher replies that two cuts will suffice.

How should these be made?

The Cat and Mouse Problem

A cat and a mouse decide to play “heads or tails”.

To liven up the game, they decide to change the rules: Each player must choose a combination of 3 results (e.g. heads, tails, heads).

They toss the coin many times, and the first to see one of his combinations appear in three consecutive coin tosses wins the game.

the two players cannot choose the same combination.

The cat, feeling himself to be the stronger player, starts first. The mouse, the smarter of the two, decides to let him go ahead.

How can the chance of winning be increased, for each player?

The Duck and the Cat

There is a duck in the middle of a circular pond. At the edge of this pond is an impatient cat.

While the duck would like to taste the grass at the edge of the pond, the cat would very much like to taste the duck!

The cat doesn’t know how to swim, and is too afraid of water to enter the pond.

The duck, meanwhile, has wings which are too small to let him fly away.

Knowing that the cat can run four times faster than the duck can swim, is it possible for the duck to reach the edge of the pond without getting caught by the cat?

Bonus: Three Maths Riddles

If you like fun maths games, word problems, mah-jong or brain-teasers, you’ll love these puzzles:

#1

The two possible answers are 3 + 3 = 6 and 8 - 3 = 5 There are two possibilities: 3 + 3 = 6 and 8 – 3 = 5 (Source: Preplounge.com)

#2

Find more maths puzzles at 9gag.com Answer: 100 (Source: 9gag.com)

#3

Find more maths games at brainfans.com Hint: the second digit is not important for the result.
Result = first digit * first digit, third digit * third digit
398 = 964 (3*3 8*8)
118 = 164 (1*1 8*8)
356 = 936 (3*3 6*6)
423 = 169 (4*4 3*3)
(Source: brainfans.com)

By now, hopefully, you’ll have seen, that maths lessons get your neurones firing!

Equally as important, they provide us with a better understanding of our world.

Finally, when you know where to look, it’s easy to see the imprint of maths in our daily lives!

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