*Mary has a cat, and a rat, and a hat, and a mat.*

**If the cat is on the mat, and rat is in the hat, but the hat is on the mat, where is the rat?**

This is a simple puzzle asked by Leslie Green, who collects dozens challenging puzzles. The collection of Leslie Logic and Math Puzzles is presented at the website Aplusclick. It’s worth to try the challenging questions.

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Leslie Green gives the answer : 96 days.

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

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

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

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Source: OCDE

More than one in four students in Beijing-Shanghai-Jiangsu-Guangdong (China), Hong Kong (China), Singapore and Chinese Taipei are top-performing students in mathematics, a higher share than anywhere else.

Students in larger schools score higher in science and are more likely than students in smaller schools. Students in smaller schools reported a better disciplinary climate.

Gender gap: the share of top performers in science is larger among boys than among girls. Finland is the only country in which girls are more likely to be top performers than boys. Girls mostly seek positions in the health sector and boys in becoming ICT professionals, scientists or engineers.

Five sample PISA Math questions:

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**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.

10. Uninteresting Number Paradox

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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 intelectual 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.

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For example, cars. How much does the gas cost in a month? What speed to choose? When do I need to change the tires? How to be on time? How much do I pay for a car during its life? . . .

What are other intersting subjects? Finance? Love? Sport?

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The classic Fermi problem, generally attributed to Fermi, is “How many piano tuners are there in Chicago?” – Wikipedia

Several simple examples of Aplusclick Fermi problems:

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This an example of classical math puzzles: http://www.aplusclick.com/k/5423.htm :

(10^{2} + 11^{2} + 12^{2}) – (13^{2} + 14^{2}) = ?

The picture “Mental Count” of Russian painter Nikolay Bogdanov-Belsky contains the simple calculation. Image source : Wikipedia

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**Be smart. Train your brain!**

Here several problems similar to the original Bongard problems go:

*What is the main difference between the pictures on the left page and on the right page? *

A.

B.

C.

Try to solve the problem yourself before looking for the answers in the A+Click Brainteaser Problems.

Harry Foundalis collected hundreds of Bongard problems.

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