There are five coins. There are three real coins, one fake coin that weighs more than the real coin, and one fake coin that weighs less than the real coin. In three weighings, identify both counterfeit coins.

📜 Leaflet #balance

🎞 Toy models — small mathematics in a big world — Tadashi Tokieda

In 9 bags there are real coins (10 g each), and in one fake (11 g). With one weighing on a two-cup scale with an arrow, determine which bag the counterfeit coins are in. (The arrow shows how much the weight of coins on the "heavy" cup is greater than on the "light" one.)

📜 Leaflet #balance

There are 64 coins, all different in weight. For no more than 94 weighings, determine the lightest and heaviest coins.

📜 Leaflet #balance

At 4 o'clock in the afternoon, 6 seconds elapsed from the first to the last beat of the clock. How much time will pass from the first to the last beat at noon?

📜 Leaflet #time

On the clock that runs exactly, all the numbers have come off. Only the divisions without signatures remained. How do I know where to return each digit? (You don't have any other watches.)

📜 Leaflet #time

The driver of a long-range truck looked at the instruments of his car and saw that the speedometer shows 25952. "What a beautiful number I drove. Probably, the next beautiful number will not fall out soon," he thought. However, after an hour and twenty minutes, the following beautiful number appeared on the speedometer. How fast was the truck going?

📜 Leaflet #time

There are two hourglasses: one for 5 minutes and one for 8 minutes. How can they be used to time 7 minutes?

📜 Leaflet #time

Cut the dial into two parts so that
1) the sum of the numbers in each part was the same;
2) the sum of the digits in each part was the same.

📜 Leaflet #time

How many times a day do the hands of the clock form a right angle?

📜 Leaflet #time

Lucas's alarm clock is ruined: he is in a hurry for 4 minutes an hour. At 7 p.m., Lucas set the exact time on it and set the call to 7 a.m. What time will Lucas wake up?

📜 Leaflet #time

James has 10 sticks 50 cm long. He wants to cut them so that 50 sticks with a length of 10 cm are obtained.
How many cuts will he have to make?

📜 Leaflet #different

In a 7×7 square, paint over several cells so that there are exactly three painted cells in each row and in each column.

📜 Leaflet #different

How many three-digit numbers have 0 as the middle digit?

📜 Leaflet #different

On an island of knights and liars (knights always tell the truth and liars always lie) in some company each declared to the others: "There are two knights among you."
How many knights could there be in this company?

📜 Leaflet #different

The store sells chocolate in the form of letters of the English alphabet. The same letters have the same price, and different ones have different prices. It is known that the word ONE costs $6, the word TWO costs $9, and the word ELEVEN costs $16.
How much does the word TWELVE cost?

📜 Leaflet #different

Lucas borrowed $19 from James, pledging to return it within 4 months. Moreover, every month the amount of payment should grow, amount to an integer number of dollars and be completely divided by the amount of payment in the previous month.
How much will Lucas pay in the last month?

📜 Leaflet #different