Big Ben Maths

Big Ben is the nickname for the Great Bell of the clock at the north end of the Palace of Westminster in London, United Kingdom.  

A pile of old coins helps to keep the clock mechanism accurate.  The pennies are stacked on the pendulum of the clock and have acted as weights. Adding or taking away coins effects the pendulum’s centre of mass and the rate at which it swings. A single penny will change the clock’s speed by two fifths of one second per day.

How does the weekly time change if we add 5 pennies to the pendulum?

Answer

At twelve o’clock, twelve chimes  ring from the great bell in 44 seconds. ## How long does it takes to ring the hours at 15:00 ?

Answer

Singapore Math

Singapore math  is a teaching method based on the national mathematics curriculum used for kindergarten through sixth grade in Singapore. Singapore math, also known as the “mastery approach”: students learn a specific concept  in a linear progression before moving to the next more complex subject. Students learn and master fewer mathematical concepts at greater detail using a three-step learning process: concrete, pictorial, and abstract. These are techniques of the teaching methods: to cover fewer topics in greater depth, handling objects such as pencils, dice, or paper clips, pictorial visualisation by drawing diagrams, bar modelling (a pictorial method used to solve arithmetic problems). Bar modelling is far more efficient than the “guess-and-check” approach. A 2015 study of 140 schools in the UK by found that the mastery approach improved the speed at which students learned math skills.

This is how Jeff Bezos, the richest man on the planet teaches maths to his children.

 

 

A Self-educated Mathematician Established the Boolean Algebra

George Boole (1815–1864)

The term “Boolean algebra” honors George Boole (1815–1864), a self-educated English mathematician. Boolean algebra captures essential properties of both set operations and logic operations. Boolean logic is credited with laying the foundations for the information age (computers and cell phones).

One of the puzzles on Boolean logic:

There are three balls X, Y and Z. They are colored red, white and blue, but not necessarily in this order. One, but only one, of the following statements is true:

X is red

Y is not red

Z is not blue

How many solutions does the puzzle have?

Think yourself before checking the solution.

Follow Your Passion in Math

Grow your passion by motivating practice like the 11-year-old Math Marvel

“Chess is a mathematical poem” – Aarushi Maheshwari 

The Great Explainer Richard Feynman

How to explain the most complex ideas in simple terms?

Four steps of the Feynman technique are

1. Write down everything new you learn about a subject in a notebook.
2. Pretend to teach the topic in a classroom. Try to explain in a simple way.
3. Go back to the books when you get stuck.
4. Simplify and use analogies that everybody knows and understands.

 

“There is no better way to learn than to teach.” – Benjamin Whichcote

Complex Problem Solving is Your Top Skill

Be ready for the Fourth Industrial Revolution!

According to the new World Economic Forum Report “The Future of Jobs” the most required skill is Complex Problem Solving together with Creativity and Critical Thinking.

Train yourself! Start with simple problems, for example, at Aplisclick, do it regularly and grow your most important skill.

Daan Roosegaarde uses technology and creative thinking to produce imaginative, earth-friendly designs and highlights the importance of the problem solving skills.

“There are no passengers on spaceship earth. We are all crew.” – Marshall McLuhan

 

Selfie Math

Jim is going on a tour around the world. He has 5 tops, 4 bottoms, and 3 pairs of footwear. He wants to post a selfie everyday to show different places and his different outfits to his girl-friend Mary, who stays at home. He does not want to wear the same outfit twice. ## For how many days does he have enough clothing?

Leslie Green gives the answer : 96 days.

Can you explain why it is 96? What is logic behind?

 

 

Guesstimate Methods

Guesstimate is an estimate based on a mixture of guesswork and calculation.

“Everything should be made as simple as possible, but not simpler.” – Albert Einstein

“The purpose of computing is insight, not numbers.” – Richard Hamming

Guesstimate Methods and Examples from Igor Kokcharov

Famous Math and Logic Paradoxes

Math and Logic are full of paradoxes.

1. Achilles and the Tortoise The Paradox of Achilles and the Tortoise was described by the Greek philosopher Zeno of Elea in the 5th century BC. The great hero Achilles challenges a tortoise to a footrace. He agrees to give the tortoise a head start of 100m. When the race begins, Achilles starts running, so that by the time he has reached the 100m mark, the tortoise has only walked 10m. But by the time Achilles has reached the 110m mark, the tortoise has walked another 1m. By the time he has reached the 111m mark, the tortoise has walked another 0.1m, then 0.01m, then 0.001m, and so on. The tortoise always moves forwards while Achilles always plays catch up.  Why is Achilles always behind the tortoise?

2. Bermuda Triangle Paradox. Why the sum of the interior angles of the Bermuda triangle is not 180 degrees?

3. Simpson Paradox. The average score for dance of boys and girls in class A are 16 and 21, respectively.  The average score of boys and girls in class B are 15 and 20, respectively.  Twenty percent of class A students are girls. Forty percent of class B students are girls.  Which class has a higher average score?

4.  Braess paradox. The diagram shows a road network. All cars drive in one direction from A to F. The numbers represent the maximum flow rate in vehicles per hour. Engineers want to construct a new road with a flow rate of 100 vehicles per hour. Drivers randomly choose the road at  crossroads. What new road decreases the capacity of the network (the number of vehicles at point F)?

5. Barber Paradox  In a city, the barber is the ‘one who shaves all those, and those only, who do not shave themselves.’
Who shaves the barber?

6. The Two Envelope Paradox. One envelope has twice as much money as the second one. Gerry does not know which envelope contains the larger  amount. He takes one of the envelopes, counts the money, and is offered the chance to switch the envelope. He thinks “If the amount of money in the chosen envelope is X dollars, then the other envelope contains either 2X of 0.5X dollars, with equal probability of 0.5. The expected value of switching is  0.5 (2X) + 0.5 (0.5X) = 1.25X. This is greater than the value in the initially chosen envelope.  It is better to switch.”  What is your advice?

7. Potato Paradox. I have 100kg of potatoes, which are 99 percent water. I dry them until they are 98 percent water.  How much do they weigh now?

8.  Leonard Euler’s Paradox.   Why the average of all  of the numbers is not a zero?

1, -1, 2, -2, 3, -3, . . .

9. Friendship Paradox.  Your friends have more friends than you. Why?

10. Uninteresting Number Paradox. How many uninteresting numbers exist?

11. Gabriel’ Horn Paradox. The shape obtained from rotating the equation about x-axis resembles a trumpet. If we need an infinite volume of paint to paint the infinite horn, how much paint does the horn can contain inside itself?

12. Pop Quiz Paradox. A teacher announces that there will be a quiz one day during the next week. The teacher gives the definition that they would not when they come in to the class that the quiz was going to be given that day. The brightest student says that the quiz cannot be on Friday because they will know the day. With the same technique, she eliminates Thursday, Wednesday, Tuesday, and Monday. “You cannot give us a pop quiz next week” she says. When does the teacher give the pop quiz?  I know the paradox from Charles Carter Wald. Probably, Martin Gardner described it for the first time in The Colossal Book of Mathematics.

 

Famous Math and Logic Paradoxes from Igor Kokcharov

Answers

1. Achilles and the Tortoise

2. Bermuda Triangle Paradox

3. Simpson Paradox

4.  Braess paradox

5. Barber Paradox

6. The Two Envelope Paradox

7. Potato Paradox

8.  Leonard Euler’s Paradox

9. Friendship Paradox

10. Uninteresting Number Paradox

11. Gabriel’s Horn Paradox

12. Pop Quiz Paradox

 

Everyday Brain Training

Millions people do everyday physical training / exercises. They want to keep their body in good shape.

Why a smaller number of people intentionally do everyday brain training / exercises?

• The absence of intelligence is not so visible as defaults of the physical shape.
• Everybody thinks that he / she is enough smart.
• Playing computer games is considered as a mental work.
• No time, there are other priorities.
• Lack of motivation / self-discipline.
• There are not interesting intellectual challenges to meet everyday.
•  . . .

Many types of work, games, discussions concentrate on a limited types of challenges. You can play many hours everyday, but the types of exercises are limited: just kill / catch somebody, and you do it again and again.

Aplusclick tries to respond to the challenges by creating a collection of different logic puzzles for all ages. There are several thousands different types of problems. Enough to do everyday training for many months.  Many of them demands exceptional intelligence.

Do your everyday 5-minute brain training at the free website Aplusclick and keep your brain sharp.