The approach of a lot of problems can be made of several manners. Some possible approaches are:
- The groping. It is a frequent method, that sometimes gives results (if one has luck).
- The use of knowledge (theorems, known previous analogous results).
- The" brutal strength": if the number of possibilities is limited, to examine, with the help of a computer, all cases.
- The approach by reasoning. This is particularly efficient for solving puzzles, because in a puzzle we have information and a question. Solving a puzzle is merely to try to use all the information in order to answer the question. This can be often done by a series of questions, answers and intermediate conclusions.
Let's give here some examples. According to my experience, these problems are difficult for common people.
Problem 1 (this is the oldest form I have found of the "Umbrella problem"): Four miners are at the same place, inside a mine. The mine is going to explode in precisely one hour. They don't move to the same speed, and are situated respectively to 5, 10, 20 and 25 minutes of the unique exit. They cannot move without light and they have only one lamp. Besides, they can move maximum simultaneously by two at a time (then the speed is the one of the slowest) .
How are they going all to run away?
The solution can be found by groping, or by examining all cases (they are not very numerous). It is a lot simpler to make the following reasoning:
Let's call the miners 5, 10, 20 and 25.
- Question: Where does the problem come from?
- Answer: From the two slow men 20 and 25.
- First conclusion: 20 and 25 must travel together.
- Question: Can they make the first travel together?
- Answer: No (the total time would be very long)
- Conclusion and solution of the problem: The two fast men 5 and 10 travel first together.
- Verification: Total time = 15 + 25 + 20 min. (or 20 + 25 + 15 min.) = 1 hour.
Problem 2. With matches, form the following motive:
This drawing consists of five squares (of side one match). While displacing two matches, form a drawing only including four squares (of side one match). The matches cannot be bent, nor cut, nor superimposed, nor burnt. All matches must be used for the construction of the drawing.
The solution can be found by groping. It generally takes a long time. What reasoning can one make to manage immediately to solve the problem?
- Question: What are all the information we have?
- Answer: We have matches, a drawing with 5 squares, and we must transform it in a drawing with four squares.
- Question: Is that all the information?
- Answer: No! We have a definite number of matches. Let's count: We have 16 matches.
- Conclusion: As we have to get four squares, they will be therefore necessarily disconnected (no side common to several squares). It gives us the solution:
Problem 3. We have a 6 x 6 board, so 36 squares. Let's call domino a rectangular piece of dimension 1 x 2. To cover the board, 18 dominoes are needed. Now, let's put out two opposite corners of the board. We have now 34 squares. Is it possible to cover them with 17 dominoes? Only the use of elementary maths is allowed. A child aged 8 must be able to immediately understand the solution.
If one gropes, we see there is no solution.
To prove it, using odd and even numbers is not difficult, but don't take into account all the information: This is a little complicate for a very young child.
- Question: Could we approach and solve the problem using only very elementary maths? What does a little boy knows for sure?
- Answer: He knows to count (not too large numbers)
- Question: Could the problem be transformed into a counting problem?
- Answer: Let's put colour on half of the squares, transforming the board into a part of a chessboard.
And we have the solution: We have 18 squares of one colour, 16 of another. And a domino always cover two squares of different colour.
- Verification: To show it to a child aged 8.
Problem 4. (This was question 8 of the second international contest, 2001)
H, I, N, O, S, ?
The contest was said to be culture fair. There were an English and a French version. The question was the same in both.
Let's try to solve it by questions / answers.
- Question: Is it a word or in connection with words or sentences?
- Answer: No (culture fair)
- Question: How to use all the information we have? What is the information given?
- Answer: We have 5 capital letters, written in alphabetic order. They have a geometrical property: they stay unchanged by a rotation of 180°.
- Question: What other letters have the same property?
- Answer and solution: Only X.
Problem 5. (This was question 8 of the third international contest, 2002 - 2003)
It was said that no special knowledge in maths where required, and that the questions are culture fair. So it can't be in connection with a word.
The difficulty is that we have only two numbers, and in the second three digits are missing.
So let's try to use all the information we have. What is the given information?
The first number is a 7 digits number. The used digits are 1 to 6.
The second number is a 6 digits number.
In the first number, there is no symmetry. For the second we know very few: three digits are given (twice 4 and a 5)
Up to know, the only observation that could be helpful is the 6. Let's try to use it. In the first number, the digits are from 1 to 6. The second number has 6 digits. So the digits of the first number may refer to the position of the digits in the second number.
What about just reading the first number? We obtain: The 2nd digit is 6, the 6th digit is 1, the 1st digit is 4, the fourth digit is 5, the fifth digit is 3 and the third digit is 4.
That's 464531 and we have the solution.