$\left.\begin{array}{l}\text { The ratio of volumes } \\ {[\text { Arithmetic. Mental calculation, etc. ]] }\end{array}\right]$

When Gulliver arrived in Lilliput, he found that everything there was exactly 12 times shorter than in his homeland. Can you tell how many Lilliputian matchboxes would fit into Gulliver's matchbox?
Notice that 12 Lilliputian matchboxes should fit into a Gulliver's matchbox in width, 12 - in length, and 12 - in height.

## Solution

In a Gulliver's matchbox, 12 Lilliputian matchboxes should fit in width, 12 - in length, and 12 - in height. In total, $12 \cdot 12 \cdot 12=1728$ matchboxes.

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Outstanding student Polikarp composed a huge number by writing down natural numbers from 1 to 500: 123...1011... 499500. Poor student Kolya erased the first 500 digits of this number. What do you think, with which digit does the remaining number start?

## Hint

Kolya was erasing digits, while Polikarp was writing down numbers - single-digit, two-digit, and three-digit.

## Solution

Out of the 500 digits erased by Kolya, 9 digits will be used for single-digit numbers, so 491 digits will remain. For two-digit numbers, $90 \times 2=180$ digits will be used, so 311 digits will remain. From this number of digits, 103 three-digit numbers and two more digits from the 104th number will be obtained. This means that the digit we are interested in is the 3rd digit of the 104th three-digit number. This number is 203, so the desired digit is 3.