Logical Reasoning in Pattern Recognition

Bongard problem is a kind of puzzle invented by the Soviet computer scientist Michael Bongard (1924–1971) in the mid-1960s. He died in 1971 during a hiking expedition in the Pamir Mountains. The tests played an important role in the disciplines of cognitive psychology and cognitive science. Human logical reasoning has a great advantage over computer intelligence.

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. z5461

B. z5462

C. z5468

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



Harry Foundalis collected hundreds of Bongard problems.



Project Starters

Challenging puzzlers to create a positive attitude to real projects.

How to Solve a Problem – Read the Question

Try to solve the following problem suggested by Leslie Green:


What is the difference between the number of letters in UPPER CASE and lower case in the text?

Most of the people and me too answered the question fast: None.

If you give the same answer, read the question again.

The trivial advice is often useful: “Try to answer exactly to the question.”

Do you get another answer? What is it?

1932Another example:  A shopkeeper of a Dairy stands six feet tall and wears size 13 sneakers. What does he weigh?



Pancake Math

panecake2“Archaeological evidence suggests that pancakes are probably the earliest and most widespread cereal food eaten in prehistoric societies.” – Wikipedia


What is the maximum number of sections into which a pancake may be divided into by four straight cuts through it?

Here more difficult questions go:

1) Is it possible to divided it so that all sections have equal area?

2) How many pancakes are needed to reach your height if they are squeezed by the weight of the upper pancakes?

3) If I spent 30 grams of batter for a pancake (French-style crêpe) of 30 cm in diameter how much batter do I need for the square pancake of the same thickness and the side length of 30 cm?  


These and many other practical “pancake” questions are presented in the Applied Math section of A+Click series, which already includes more than 4500 questions.


Mensa Tough Puzzle

Tough question from A+Click makes you crazy. 99.9% fail or refuse to solve it.aplusclickQuestion

The man who counted

The man who counted

“In 1932 Malba Tahan published what would became one of the most successful books ever written in Brazil – O Homem que Calculava – The Man Who Counted.”

Math Magic: To Predict Unknown


One of the math magic is to predict unknown facts. Please find below several situations where math gives unexpected and useful answers.

1. Code Testers

John detected 2 errors and Mary – 3 errors in a code. There is one error in common. How many errors are still undetected?

It looks like a joke. However there is a mathematical solution of the problem that shows that the number of undetected errors is N. I am sure that you can easily to find the number N.

99.9% fail or refuse to solve it.


2. Lake Width


How estimate the width of a lake without crossing it? You just walk and make some measures at a lakeside.


ImageFor example in the situation shown at the picture the width of the lake is 200 meters.


3. How many fish are there in the lake?

Yesterday, I caught 30 fish of a certain size in the lake.
I marked and released them without any harm.
Today I also caught and released 80 fish of the same size and noticed that there were 6 marked fish in the second catch.

How many fish of the same size are there in the lake?

4.  Seller’s decision

ImageI sell my car. People come to my garage one at a time and make bids to buy it. I make an immediate decision whether to accept or reject an offer after receiving it. I decide to reject the first N offers, mark the highest price P, and accept the first offer that is greater than P.

What number N do you recommend me if I expect that 100 people can make a bid?

If N is small, I can accept a small amount of money.
If N is large, I can reject the best offer.

This is the famous problem of the optimal stopping theory (Secretary Problem).

5. Winning Strategy

In a game, Anna and Bill take 1, 2, or 3 coins on each turn. The player to take the last coin from the pile wins. If Anna goes first and there are 40 coins on the table, how many coins should she take to guarantee that she would win?

The truth is “She always loses if Bill knows the winning strategy.” Do you know Bill’s strategy?