# 10 fascinating tasks from Soviet mathematics

Try to solve the puzzles from the popularizer of mathematics Boris Kordemsky, without using tips.

## 1. Crossing the river

A small military detachment approached the river through which it was necessary to cross. The bridge is broken and the river is deep. How to be? Suddenly an officer notices two boys in a boat by the shore. But the boat is so small that only one soldier or only two boys can cross it – no more! However, all soldiers crossed the river on this boat. How?

The boys moved the river. One of them remained on the shore, and the other drove the boat to the soldiers and crawled out. A soldier sat in the boat and crossed to the other side. The boy, who remained there, drove the boat back to the soldiers, took his comrade, took it to the other shore and again delivered the boat back, after which he got out, and the second soldier sat down and crossed it and crossed it and crossed it and crossed it.

Thus, after every two hangs of the boat across the river and back, one soldier was transported. This was repeated as many times as there was a person in the detachment.

## 2. How many details?

In the turning workshop of the plant, parts from lead blanks are pulled out. From one workpiece – part. Chips obtained during sewing of six parts can be re -melted and prepared another blank. How many parts can be made in this way from thirty -six lead blanks?

With an insufficiently careful attitude to the condition of the task, they argue as follows: thirty -six blanks are thirty -six details;Since chips of every six blanks give another new workpiece, six new blanks are formed from shakes of thirty -six blanks – these are six more details;total 36 + 6 = 42 parts.

At the same time, they forget that the chips received from the last six blanks will also make a new workpiece, that is, another detail. Thus, the total details will not be 42, but 43.

## 3. During the tide

Not far from the shore there is a ship with a rope ladder launched along the side. The stairs have ten steps;Distance between steps 30 cm. The lowest step touches the surface of the water.

The ocean is very calm today, but the tide begins, which lifts water in every hour by 15 cm. After what time the third step of the rope ladder will be covered with water?

. So here.

No calculations will lead to a true result, if you do not take into account that both the ship and the staircase will rise with the water, so that in reality the water will never cover the third step.

## 4. Ninety nine

How much do you need to put the “plus” signs (+) between the numbers of the number 987 654 321, so that 99?

Two solutions are possible: 9 + 8 + 7 + 65 + 4 + 3 + 2 + 1 = 99 or 9 + 8 + 7 + 6 + 5 + 43 + 21 = 99.

## 5. For the Tsimlyansk hydraulic engine

The brigade took part in the implementation of the urgent order for the manufacture of measuring instruments for the Tsimlyansk hydroelectric complex as part of an experienced team leader and nine young workers.

During the day, each of the young workers mounted 15 devices, and the foreman – 9 more devices than on average each of the ten members of the brigade. How many measuring devices were mounted by the brigade in one working day?

To solve the problem, you need to know the number of devices mounted by the foreman. And for this, in turn, you need to know how many devices on average were mounted by each of the ten members of the brigade.

Distributed equally between nine young workers of 9 devices manufactured additional by the foreman, we will learn that on average each member of the brigade mounted 15 + 1 = 16 devices. It follows that the foreman made 16 + 9 = 25 devices, and the entire team (15 × 9) + 25 = 160 devices.

There are 9 kg of cereals in the package. Try to distribute the entire cereal using two and 200 g using cup weights with weights of 50 and 200 g: in one – 2 kg, in the other – 7 kg. In this case, it is allowed to produce only 3 weighing.

First weighing: hang the cereal into 2 equal parts (this can be done without weights) 4.5 kg. Second weighing: one of the resulting parts to hang again in half again – 2.25 kg. Third weighing: from one of these parts, spread (using weight) 250 g. 2 kg will remain.

## 7. A smart baby

Three brothers received 24 apples, and everyone got as many apples as he was three years ago. The youngest, the boy is very smart, offered the brothers such an exchange of apples:

“I,” he said, “I will leave myself only half of the apples I have, and the rest I will divide between you. After that, let the middle brother also leaves half, and the rest of the apples will give me and the eldest brother equally, and then let her elder brother leave half of all his apples, and the rest will share between me and the middle brother equally.

The brothers, not suspecting the treachery of such a sentence, agreed to satisfy the desire of the younger. As a result … everyone had apples equally. How many years were the baby and each of the other brothers?

At the end of the exchange, each of the brothers had 8 apples each. Therefore, the elder before he gave half of the apples his brothers had 16 apples, and in the middle and younger – 4 apples each each.

Further, before the middle brother shared his apples, he had 8 apples, and the elder had 14 apples, the younger one had 2. Hence, before his younger brother shared his apples, he had 4 apples, the middle one had 7 apples and the elder had 13.

Since everyone first received as many apples as he was three years ago, the younger is now 7 years old, the middle brother is 10 years old, and the eldest 16.

## 8. Crush into pieces

Ground 45 to four parts so that if you add 2 to the first part, take 2 from the second, multiply the third by 2, and divide the fourth by 2, then all the results will be equal. You can do it?

The desired parts 8, 12, 5 and 20.

## 9. Tree planting

Fifth graders and sixth graders were instructed to plant trees on both sides of the street on an equal amount on each side.

In order not to hit the mud in front of sixth graders, the fifth graders went to work early and managed to plant 5 trees while the older guys came, but it turned out that they planted trees not on their side.

Five -graders had to go to their side and start work again. Sixth graders, of course, coped with the task earlier. Then the teacher suggested:

– Come on, guys, we will help the fifth graders!

Everyone agreed. We crossed to the other side of the street, planted 5 trees, gave them, which means they managed to plant 5 trees, and all the work was completed.

“Although you came before us, but still we overtook you,” one sixth grader laughed, turning to the younger guys.

– Just think, overtook! For 5 trees only,-someone objected.

– No, not at 5, but at 10, – sixth graders rustled.

The dispute flared up. Some insist that at 5, others try to somehow prove that for 10. Who is right?

Sixth graders exceeded their task for 5 trees, and therefore fifth graders did not fulfill their task for 5 trees. Therefore, the elders planted 10 more trees more than the younger.

## 10. Four motor shipers

4 heaters moored in the port. At noon on January 2, they simultaneously left the port. It is known that the first ship returns to this port every 4 weeks, the second – every 8 weeks, the third – after 12 weeks, and the fourth – after 16 weeks.

When the first time the boat will again converge together in this port?

The smallest common multiple numbers 4, 8, 12 and 16 – 48. Therefore, the ships will converge in 48 weeks, that is, December 4.

The tasks for this collection were taken from the collection “Mathematical Sattering” by Boris Kordemsky, who was published by Alpina Plisher Publishing House.

## 1. Crossing the river

A small military detachment approached the river through which it was necessary to cross. The bridge is broken and the river is deep. How to be? Suddenly an officer notices two boys in a boat by the shore. But the boat is so small that only one soldier or only two boys can cross it – no more! However, all soldiers crossed the river on this boat. How?

The boys moved the river. One of them remained on the shore, and the other drove the boat to the soldiers and crawled out. A soldier sat in the boat and crossed to the other side. The boy, who remained there, drove the boat back to the soldiers, took his comrade, took it to the other shore and again delivered the boat back, after which he got out, and the second soldier sat down and crossed it and crossed it and crossed it and crossed it.

Thus, after every two hangs of the boat across the river and back, one soldier was transported. This was repeated as many times as there was a person in the detachment.

## 2. How many details?

In the turning workshop of the plant, parts from lead blanks are pulled out. From one workpiece – part. Chips obtained during sewing of six parts can be re -melted and prepared another blank. How many parts can be made in this way from thirty -six lead blanks?

With an insufficiently careful attitude to the condition of the task, they argue as follows: thirty -six blanks are thirty -six details;Since chips of every six blanks give another new workpiece, six new blanks are formed from shakes of thirty -six blanks – these are six more details;total 36 + 6 = 42 parts.

At the same time, they forget that the chips received from the last six blanks will also make a new workpiece, that is, another detail. Thus, the total details will not be 42, but 43.

## 3. During the tide

Not far from the shore there is a ship with a rope ladder launched along the side. The stairs have ten steps;Distance between steps 30 cm. The lowest step touches the surface of the water.

The ocean is very calm today, but the tide begins, which lifts water in every hour by 15 cm. After what time the third step of the rope ladder will be covered with water?

. So here.

No calculations will lead to a true result, if you do not take into account that both the ship and the staircase will rise with the water, so that in reality the water will never cover the third step.

## 4. Ninety nine

How much do you need to put the “plus” signs (+) between the numbers of the number 987 654 321, so that 99?

Two solutions are possible: 9 + 8 + 7 + 65 + 4 + 3 + 2 + 1 = 99 or 9 + 8 + 7 + 6 + 5 + 43 + 21 = 99.

## 5. For the Tsimlyansk hydraulic engine

The brigade took part in the implementation of the urgent order for the manufacture of measuring instruments for the Tsimlyansk hydroelectric complex as part of an experienced team leader and nine young workers.

During the day, each of the young workers mounted 15 devices, and the foreman – 9 more devices than on average each of the ten members of the brigade. How many measuring devices were mounted by the brigade in one working day?

To solve the problem, you need to know the number of devices mounted by the foreman. And for this, in turn, you need to know how many devices on average were mounted by each of the ten members of the brigade.

Distributed equally between nine young workers of 9 devices manufactured additional by the foreman, we will learn that on average each member of the brigade mounted 15 + 1 = 16 devices. It follows that the foreman made 16 + 9 = 25 devices, and the entire team (15 × 9) + 25 = 160 devices.

There are 9 kg of cereals in the package. Try to distribute the entire cereal using two and 200 g using cup weights with weights of 50 and 200 g: in one – 2 kg, in the other – 7 kg. In this case, it is allowed to produce only 3 weighing.

First weighing: hang the cereal into 2 equal parts (this can be done without weights) 4.5 kg. Second weighing: one of the resulting parts to hang again in half again – 2.25 kg. Third weighing: from one of these parts, spread (using weight) 250 g. 2 kg will remain.

## 7. A smart baby

Three brothers received 24 apples, and everyone got as many apples as he was three years ago. The youngest, the boy is very smart, offered the brothers such an exchange of apples:

“I,” he said, “I will leave myself only half of the apples I have, and the rest I will divide between you. After that, let the middle brother also leaves half, and the rest of the apples will give me and the eldest brother equally, and then let her elder brother leave half of all his apples, and the rest will share between me and the middle brother equally.

The brothers, not suspecting the treachery of such a sentence, agreed to satisfy the desire of the younger. As a result … everyone had apples equally. How many years were the baby and each of the other brothers?

At the end of the exchange, each of the brothers had 8 apples each. Therefore, the elder before he gave half of the apples his brothers had 16 apples, and in the middle and younger – 4 apples each each.

Further, before the middle brother shared his apples, he had 8 apples, and the elder had 14 apples, the younger one had 2. Hence, before his younger brother shared his apples, he had 4 apples, the middle one had 7 apples and the elder had 13.

Since everyone first received as many apples as he was three years ago, the younger is now 7 years old, the middle brother is 10 years old, and the eldest 16.

## 8. Crush into pieces

Ground 45 to four parts so that if you add 2 to the first part, take 2 from the second, multiply the third by 2, and divide the fourth by 2, then all the results will be equal. You can do it?

The desired parts 8, 12, 5 and 20.

## 9. Tree planting

Fifth graders and sixth graders were instructed to plant trees on both sides of the street on an equal amount on each side.

In order not to hit the mud in front of sixth graders, the fifth graders went to work early and managed to plant 5 trees while the older guys came, but it turned out that they planted trees not on their side.

Five -graders had to go to their side and start work again. Sixth graders, of course, coped with the task earlier. Then the teacher suggested:

– Come on, guys, we will help the fifth graders!

Everyone agreed. We crossed to the other side of the street, planted 5 trees, gave them, which means they managed to plant 5 trees, and all the work was completed.

“Although you came before us, but still we overtook you,” one sixth grader laughed, turning to the younger guys.

– Just think, overtook! For 5 trees only,-someone objected.

– No, not at 5, but at 10, – sixth graders rustled.

The dispute flared up. Some insist that at 5, others try to somehow prove that for 10. Who is right?

Sixth graders exceeded their task for 5 trees, and therefore fifth graders did not fulfill their task for 5 trees. Therefore, the elders planted 10 more trees more than the younger.