Knights and Liars

On the island "Lie-truth" there are living knights who always tell the truth, and liars who always lie.

A traveler met three islanders and asked each of them: "How many knights are among other two islanders?".

The first one answered: "None".
The second one said: "One".

What did the third one say?

Find the number

Find the smallest number for which you know the following:

When it divided by 2, it gives 1 in the remainder, when it divided by 3 it gives 2 in the remainder, when divided by 4 it gives the remainder 3, when divided by 5 it gives the remainder 4, and finally when dividing by 6 it gives 5 in the remainder. But on 7 this number is divided completely.

Eggs

Today's exercise is about a woman who was carrying eggs. Be careful and try to translate exercise to algebra first.

Elizabeth was carrying a basket of eggs for sale. David met her and by carelessness pushed her such that the basket fell to the ground and all the eggs were broken. 

David wanted to pay the woman the cost of broken eggs and asked how many of eggs were there. 

"I do not remember" Elisabeth said, "I only remember that when I rearranged the eggs by 2, there was only one egg left." In the same way, there was always one egg left when I rearranged them by 3, 4, 5 and  6. When I rearranged them by 7, then there was not a single egg left." 

The question is, how many eggs were there? Please state the smallest such number.

The Cunning Employee

Today, I want to share with you and ask to solve an old, interesting problem about how an employee cunningly deceived the store owner. Here is the problem:

The owner of the store arranged in his cellar a cabinet in the form of a square with nine compartments. The middle one he left free for empty bottles, and in the rest he arranged 60 bottles of wine so that in each corner compartment there were 6 bottles, and in each of the rest by 9 bottles. 

Thus, on each side of the square there were 21 bottles. The employee noticed that when the owner  was checking the number of bottles, he only counted the bottles on the sides of the square and made sure that on each side of the square there were 21 bottles. Then the cunning employee first time took away 4 bottles, and the rest arranged so that again the sum of the bottles turned 21 on each side. The owner counted the bottles in his usual way and thought that the bottles remained the same number and that the employee only moved them. The employee took advantage of the owner's mistake and again took away 4 bottles, placing the rest so that on each side of the square there were again 21 bottles. The same he kept repeating until it was possible. 

The question is, how many times did he take bottles and how many bottles did he take away?

A to Z activity

The Rabbit's Way

The basics of combinatorics can be studied from the early ages. The following activity is very helpful for the developing skills of identifying the possible options.

We will need the number of cards like the ones described above.

Process:

Ideally it would be great to have groups with 2 persons. The task is following: How can rabbit get carrots in Z, if the starting point is A. Note that, there is allowed to go right or upward. You need to find all the possible routes. You can use pencils with the different colors to show that routes. Also, it will be very helpful to record this info on the paper. For example, rurur (right-up-right-up-right).

You can ask some additional questions to analyze the activity or for the future development. For example: 

How many routes you get? 

What pattern do you see?

"Target" Number

Tasks and Preparation

The development of mathematical thinking is very important and need to be done from the early ages. Here is an example of the 20 min game-activity for the children ages  8, 9 which can be used for the developing skills to calculate the value of simple numerical expressions with one-digit numbers. It's an non-standard activity which requires a brain strain - the best way to train your brain. Another important example is the development of skills to interact with each other.

For this activity we will need labeled (or opaque) two boxes, one with the label ”one-digit numbers” (i.e. cards with labels 1, 2, …, 9) and the other one with “tens” (i.e. cards with 10, 20, …, 90). We will need to arrange tables for the group working. Ideally for a group of 4 or 5 children.

Process

Divide children into groups of 4 persons. Every person choose 3 cards randomly from the first box and one card “Target” number from the “tens” box. Children wrote in their notes the numbers that are on the table and put them again inside the box, for those who haven’t numbers. Then children are trying to get “Target” number with their 3 one-digit numbers. (Every number need to be used only once, also they need to find all the possible ways). For example, from 1st box we have 1, 3, 7 and from the second one 20. Then we have 3×7-1=20.

Hints

In how many ways did you get “Target” number? 

If someone could not get the desired “Target” number, work together and try to find solution.