4. Dad, Masha, and Yasha are going to school. While Dad takes 3 steps, Masha takes 5 steps. While Masha takes 3 steps, Yasha takes 5 steps. Masha and Yasha counted that together they made 400 steps. How many steps did Dad take?
Answer: 90 steps.

Solution. 1st method. Let's call the distance equal to 3 steps of Masha and 5 steps of Yasha a Giant's step. While the Giant makes one step, Masha and Yasha together make 8 steps. Since they made 400 steps together, the Giant would have made 400:8=50 giant steps in this time. If the Giant made 50 steps, then Masha made 150 steps. Now let's count them in "fives." 150 is 30 times 5 steps. This means that the father made 30 times 3 steps, which is 90 steps.

2nd method. While Masha makes 3*5=15 steps, the father makes 3*3=9 steps, and Yasha makes 5*5=25 steps. Together, Masha and Yasha will make 15+25=40 steps in this time. And while they make 400 steps, the father will also make 10 times more steps, i.e., 9*10=90 steps.

In the museum, there are 16 halls arranged as shown in the diagram. In half of them, paintings are exhibited, and in the other half, sculptures. From any hall, you can go to any adjacent one (having a common wall). During any tour of the museum, the halls alternate: a hall with paintings - a hall with sculptures - a hall with paintings, and so on. The tour starts in hall A, where paintings are displayed, and ends in hall B.

a) Mark with crosses all the halls where paintings are displayed. Solution. See the diagram.

b) A tourist wants to visit as many halls as possible (travel from hall A to hall B), but visit each hall no more than once. What is the maximum number of halls he can visit? Draw one of his longest routes and prove that he could not visit more halls.

Answer: 15.

Solution. One of the possible routes is shown in the diagram. Let's prove that if the tourist wants to visit each hall no more than once, he cannot visit more than 15 halls. Note that the route starts in a hall with paintings (A) and ends in a hall with paintings (B). This means that the number of halls with paintings that the tourist has passed is one more than the number of halls with sculptures. Since the number of halls with paintings that the tourist could pass is no more than 8, the number of halls with sculptures is no more than 7. Therefore, the route cannot pass through more than

![](https://cdn.mathpix.com/cropped/2024_05_06_0b5079389ba7141ce391g-05.jpg?height=725&width=374&top_left_y=1017&top_left_x=1521)
15 halls.

## 6th Grade. Recommendations for Checking.

Each problem is scored out of 7 points. Each score is an integer from 0 to 7. Some guidelines for checking are provided below. Naturally, the creators cannot foresee all cases. When evaluating a solution, it should be determined whether the provided solution is generally correct (although it may have flaws) - in which case the solution should be scored no less than 4 points. Or if it is incorrect (although it may have significant progress) - in which case the score should be no higher than 3 points.