Problem No. 22. Lines on a plane

Let's draw lines!

Is it possible that 7 lines on a plane have 30 intersection points in total? Can they have twenty intersection points?

Student tip

Find the maximum number of intersection points. Start with two lines, add one at a time, until you reveal the pattern.

Problem No. 21. Missing numbers again

Here is a new brain teaser!

Find the missing numbers, if the tables follow the same rule.

6	17	2		3	21	1
8		22				12
5	13			6	9	4

Teacher tip

Brain teasers can be used as "worm ups" at the beginning of the class.

Problem No. 20 A cup of water

"Please give me a cup of water!"

You have a 7 cup bottle and a 30 cup bottle. How can you measure one cup of water?

Parent tip

You can make your own 7 cup and 30 cup bottles. Take two bottles, measure the wanted amount of water and mark the bottles, very carefully, at their maximal levels, the first one as the 7 cup bottle, and the other one as the 30 cup bottle. Then you can all have fun, you and your children, with pouring the water in and out of the bottles.

Problem No. 19. Divisibility

For which values of \(n\), is \(2n+5\) divisible by 7? And for which values, is it divisible by 30?

Student tip

Check with the reminders of \(n\) when dividing by 7. A number is divisible by 30, if it is divisible by 2, 3 and 5.

Problem No. 18. Min-max sum

What is the minimum and what is the maximum sum of 7 numbers which product is equal to 30?

Teacher tip

Suitable for exercising the factorization of numbers.

Problem No. 17. Cookies

There are 7 different types of cookies in Christine's Bakery that Mary loves. Mary wants to buy 30 cookies. If we suppose that there are enough cookies of each type she loves (at least 30 cookies of each type), in how many different ways can she buy her cookies?

Student tip

Recall how to count all possible ways to select items from a collection, such that the order of selection does not matter.

Problem No. 16. Points on a plane

7 different colors are used for coloring 30 points on a plane. Prove that, regardless the way you color them, there will always be at least five points colored in the same color.

Parent tip

You can enjoy a fun family time with your children coloring the points in different colors and different ways.

Problem No. 15. Jenn's blog

Jenn has a blog site. She started blogging on July 30 at 7.30 am. She is posting her posts at exactly 7 hours and 30 minutes apart. When will she post her 730th post?

Teacher tip

Suitable for practicing calendar calculations.

Problem No. 14. Two sisters

There have been two sisters. When the older sister was asked about their age, she said: "We had 730 months in total, 7 years and 30 months ago. Also, you have to know that I am 7 years and 30 months older than my sister." How old is each of the sisters?

Student tip

First find their age, 7 years and 30 months ago.

Problem No. 13. Sequence of numbers

Continue the following number sequence:

7, 3, 0, 3, 3, 6, 9, 5, …

Parent tip

Children love finding solutions together with their parents. Pretend that it is a difficult problem.

Problem No. 12. Sum of squares

Find all representations of the number 730 as a sum of two squares? And as a sum of three squares?

Teacher tip

Suitable for implementing the elimination and reminder methods while practicing the square numbers up to \(30^2\).

Problem No. 11. Urns and balls

There are 7 urns with 30 balls each. In each urn there is equal number of white and black balls. Randomly, we take out one ball from the first urn and put it into the second urn. Then, we take one ball from the second urn and put it into the third urn. We continue this process, until we take a ball from the seventh urn and put it into the first urn. Then, we take a ball from the first urn. What are the chances that the ball is black?

Student tip

First, find the probabilities for taking a black ball from each urn in first 7 steps.

Problem No. 10. Socks in a drawer

You have 7 pairs of black socks in your drawer, among 30 pairs in total. The rest of them are white socks. What is the smallest number of socks you should take out from the drawer (with your eyes closed) to be sure that you have a matching pair of white socks?

Parent tip

You can give this problem during the house chores. It is the fun way to implement the mathematical thinking outside the classroom. Children should know that math is real, fun and useful.

Problem No. 9. Points on a circle

How many points can be drawn on a circle such that each two adjacent points are at the same distance, and each three adjacent points form an angle of \(7^o 30'\).

Student tip

You need to recall on the properties of inscribed angles in a circle.

Problem No. 8. Seven 7s

How can you make 30, using seven 7s and the basic math operations (brackets are allowed)? What is the smallest number of 7s you can use, to make 30?

Parent tip

Have fun with your kids by exploring and calculating different number sentences. Do not forget that the brackets can help a lot.

Problem No. 7. Coins on a table

There are 7 coins laid on a table with the heads up, and 30 coins laid with the tails up. Each coin has a head side and a tail side. How can you split the coins in two piles, with your eyes closed, so that the piles have the same number of coins with the tails up?

Teacher tip

Use logic puzzles at your classes to increase the motivation and to exercise the flexibility in mathematical thinking.

Problem No. 6. Clock hands revisited

How many times in a day the clock hands will form the same angle as the angle at 7.30?

Student tip

For a hint use Problem No. 3.

Problem No. 5. Missing numbers

Here is a 5 x 5 table filed with numbers according to some rule. What are the missing numbers?

Parent tip

Exploring patterns with your children can be a fun way to spend some quality time together.

Problem No. 4. Airplane fly

The travel by plane from Paris,France to New York City, USA, takes about 8 hours. If a passenger starts his journey from Paris at 7.30 pm local time, when is he supposed to arrive at New York City, local time?

Teacher tip

Problems with an extra-mathematical knowledge are suitable to demonstrate the application of math and its interdisciplinary character. Here, the students should be familiar with or should first learn about time zones.

Problem No. 3. Clock hands

It is 7.30 now. How much time will it pass until the clock hands form the same angle again?

Student tip

Suppose that minute and hour hands move at a constant velocity each.

Problem No. 2. Book

Ann usually has the following problem at 7.30 in the mornings: She wakes up and wants to continue reading the book that she had read the night before, but the lamp she uses for reading has depleted batteries and she does have neither spare batteries nor another lamp nearby. But, she always manages to solve the problem very quickly, without leaving either the room or the bed, and continues to read the book. How does she do it?

Parent tip

Suitable for family time, especially with younger children to exercise their listening skills.

Problem No. 1. Digital Clock

It is 7.30 now. How long should we wait until the digital clock uses the same digits to display the time?

Teacher tip

Good recreational problem for a "warm up" activity at the beginning of the class.