10 exciting problems from a Soviet mathematician

2024 Author: Malcolm Clapton | [email protected]. Last modified: 2023-12-17 03:44

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

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 the officer notices two boys in a boat near the shore. But the boat is so small that only one soldier or only two boys can cross it - no more! However, all the soldiers crossed the river in this particular boat. How?

The boys crossed the river. One of them stayed on the shore, while the other drove the boat to the soldiers and got out. A soldier got into 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 side and brought the boat back again, after which he got out, and the second soldier got into it and crossed.

Thus, after every two passes of the boat across the river and back, one soldier was ferried. This was repeated as many times as there were people in the detachment.

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2. How many parts?

In the lathe shop of the plant, parts are turned from lead blanks. From one workpiece - a part. The shavings resulting from the manufacture of six parts can be remelted and another blank can be prepared. How many parts can be made in this way from thirty-six lead blanks?

With insufficient attention to the condition of the problem, they argue as follows: thirty-six blanks are thirty-six parts; since the chips of every six workpieces give another new workpiece, then six new workpieces are formed from the chips of thirty-six workpieces - this is another six parts; total 36 + 6 = 42 parts.

At the same time, they forget that the shavings obtained from the last six blanks will also make up a new blank, that is, one more detail. Thus, there will be not 42, but 43 parts in total.

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3. At high tide

Not far from the shore there is a ship with a rope ladder lowered into the water along the side. The staircase has ten steps; distance between steps 30 cm. The lowest step touches the water surface.

The ocean is very calm today, but the tide begins, which raises the water every hour by 15 cm. How long will it take for the third step of the rope ladder to be covered with water?

When a task concerns any physical phenomenon, then all aspects of it should be taken into account so as not to get into a mess. So it is here.

None of the calculations will lead to the true result, if you do not take into account that with the water both the ship and the ladder will rise, so that in reality the water will never cover the third step.

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4. Ninety nine

How many plus signs (+) must be placed between the digits of 987 654 321 to add up to 99?

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

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5. For the Tsimlyansk hydroelectric complex

A team consisting of an experienced foreman and nine young workers took part in the fulfillment of an urgent order for the manufacture of measuring instruments for the Tsimlyansk hydroelectric complex.

During the day, each of the young workers assembled 15 instruments, and the foreman - 9 more instruments than the average of each of the ten members of the brigade. How many measuring instruments were installed by the team 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 were installed on average by each of the ten members of the team.

Having distributed equally among the nine young workers 9 devices, made additionally by the foreman, we learn that, on average, each member of the brigade mounted 15 + 1 = 16 devices. It follows that the foreman made 16 + 9 = 25 instruments, and the whole team (15 × 9) + 25 = 160 instruments.

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6. Try to weigh

The package contains 9 kg of cereals. Try using a weighing scale with 50 and 200 g weights to distribute all the cereals into two packages: one - 2 kg, the other - 7 kg. In this case, only 3 weighings are allowed.

First weighing: weigh the cereal into 2 equal parts (this can be done without weights), 4, 5 kg each. Second weighing: once again hang one of the resulting parts in half - 2, 25 kg each. Third weighing: weigh 250 g from one of these parts (using a weight). 2 kg remain.

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7. Smart kid

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

“I,” he said, “will keep only half of the apples I have, and I will divide the rest between you equally. After that, let the middle brother also keep half for himself, and give the rest of the apples to me and the older brother equally, and then let the older brother keep half of all the apples he has, and divide the rest between me and the middle brother equally.

The brothers, not suspecting treachery in such a proposal, agreed to satisfy the younger's desire. As a result… everyone had equal apples. How old was the baby and each of the other brothers?

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

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

Since everyone first received as many apples as they were three years ago, the youngest is now 7 years old, the middle brother is 10 years old, and the older one is 16.

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8. Crush into pieces

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

The parts you are looking for are 8, 12, 5, and 20.

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9. Planting trees

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

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

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

- Let's go, guys, help the fifth graders!

All agreed. We crossed to the other side of the street, planted 5 trees, paid off, therefore, the debt, and even managed to plant 5 trees, and all the work was finished.

“Even though you came before us, we still overtook you,” one sixth grader laughed, addressing the younger children.

- Just think, overtook! Only 5 trees, - someone objected.

- No, not by 5, but by 10, - the sixth graders rustled.

The controversy flared up. Some insist that it is 5, others are trying to somehow prove that it is 10. Who is right?

Sixth graders exceeded their task by 5 trees, and therefore fifth graders did not complete their task by 5 trees. Consequently, the elders planted 10 more trees than the younger ones.

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10. Four ships

4 motor ships are 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 will the ships come together again in this port for the first time?

The least common multiple of 4, 8, 12, and 16 is 48. Consequently, the ships will converge in 48 weeks, that is, on December 4.

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The problems for this collection were taken from the collection "Mathematical Ingenuity" by Boris Kordemsky, which was published by the publishing house "Alpina Publisher".