2 New Year's tasks that will give your brain a boost
Miscellaneous / / January 03, 2022
One is solved in a minute, but the second will have to be thought over.
1. Family Dinner Problem
Every New Year, the Morozov family hosts a festive dinner. Dad, mom, their son Petya, uncle Kolya, aunt Tom and grandfather gather at the table.
Mom is responsible for organizing the party. Traditionally, a large rectangular table is set in the living room with a double sofa, one armchair and three chairs. Here's a top view of the room:
The most difficult thing in preparation is to come up with a seating plan, because each family member has his own wishes. Mom listened to them in advance and made a list:
- Grandpa has not seen Aunt Tom for a long time, so he wants to sit opposite her.
- Petya would like to sit next to his grandfather.
- Uncle Kolya does not want to sit at the head of the table.
- Aunt Toma's back hurtsso she wants to sit on the couch.
Dad always sits at the head of the table on the left side on the armchair, and mom is on the couch to his left.
Will mom be able to take into account the wishes of all family members? And where will each of them sit then?
Important: it is believed that relatives are sitting side by side, even if they are separated by a corner of the table.
It is immediately known that dad is sitting on an armchair, and mom is on the couch to his left. Aunt Tom also wants to sit on the sofa, which means she will take second place on it.
Grandpa wants to sit opposite Aunt Toma, which means he will take a chair in the lower right corner of the table.
There are two empty seats at the table. Since Uncle Kolya does not want to sit at the head of the table, he will have a place to the right of dad. And Petya will sit opposite dad, next to grandfather.
Then the seating chart will look like this:
Mom will be able to seat the family members the way they wanted.
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2. Snowman's confectionery problem
The snowman working in the forest bakery is tired of counting the trifles of customers. Therefore, he decided to no longer accept coins in denominations of 1, 2 and 5 kopecks.
Snowman did not have time to redo the price tags. Instead, he decided to round the amount of each purchase to the nearest ten kopecks. For example, if a product costs 5 rubles and 23 kopecks, he will take 5 rubles and 20 kopecks for it.
The Bear, the Hare, the Squirrel and the Hedgehog passed by the Snowman's pastry shop. The bear decided to treat his friends to fresh pastries. It didn't take long to persuade them.
From the menu, the Bear chose a cranberry pie for 15 rubles 88 kopecks, the Hare wanted a bag of honey cakes for 3 rubles 16 kopecks, Belka - a bag of nuts with condensed milk for 3 rubles 35 kopecks, and the Hedgehog preferred a waffle cake for 12 rubles 49 kopecks.
When the company was about to place an order, Snowman told them that he would not accept coins less than 10 kopecks for payment.
The bear quickly added up the prices of baked goods: the whole purchase would cost 34 rubles 88 kopecks. We'll have to round up this amount to 34 rubles 90 kopecks and overpay two kopecks.
The bear decided that he could break the order and save his money. How many purchases should you split it into to save as much as possible?
Just in case, let's remind the rules rounding off:
- If, when rounding, the extreme digit in the number is less than or equal to 4, then the digit following it does not change. For example, 6.72 is rounded to 6.70.
- If, when rounding numbers, the extreme digit in the number is greater than or equal to 5, then the digit following it is increased by one. For example, 3.48 is rounded to 3.50.
Obviously, you will only be able to save money if the price is rounded down. The highest extreme figure in the price at which it is rounded down is 4. That is, for one purchase, you can save a maximum of 4 kopecks.
The order of animals can be divided in only three ways: buy everything separately, make two or three purchases.
Let's consider all the scenarios:
1. If the Bear buys all the baked goods separately, he will not save. The prices of all the necessary goods end with a number greater than 4, that is, all four purchases will have to be rounded up. This option is not suitable.
2. If the Bear makes three purchases, then he will need to pay for two goods separately, and for two - together. We know that buying any of the goods separately, you will have to round up the price and overpay. This means that you can only save money when buying a pair. It turns out that with any combination of goods, using this method, the Bear will save less than 4 kopecks.
3. To save as much as possible with two purchases, you need to split the products into pairs so that the sum of their prices ends in 4. Then the Bear will save a maximum of 8 kopecks. You can do this if you buy gingerbread with a pie, and a cake with nuts:
- 3 rubles 16 kopecks + 15 rubles 88 kopecks = 19 rubles 4 kopecks = 19 rubles;
- 12 rubles 49 kopecks + 3 rubles 35 kopecks = 15 rubles 84 kopecks = 15 rubles 80 kopecks.
Only two purchases will have to be paid: 19 rubles + 15 rubles 80 kopecks = 34 rubles 80 kopecks. This amount is 8 kopecks less than the original price of all baked goods.
Answer: The bear needs to split the order into two purchases, then he will save 8 kopecks.
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Set of puzzles based on problems from the German math competition for schoolchildren Mathe im Advent 2021.
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A hedonist with a huge passion for food. I baked my first biscuit at the age of five, since then I have been cooking every day and I get real pleasure from it. I am fond of intellectual games, with great interest I study scientific works on history and medicine, and at the same time I follow pop culture, fashion and beauty trends.
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