[Classical combinatorics (miscellaneous).]
$[$ Inclusion-exclusion principle $]$

From the sequence of natural numbers, all numbers that are squares or cubes of integers have been erased. Which of the remaining numbers is in the hundredth position?
Consider the first hundred natural numbers. Among these numbers, there are ten squares (from 1 to $10^{2}=100$) and four cubes (from 1 to $4^{3}=64$). Note that two of these numbers, namely, 1 and 64, are both squares and cubes. Thus, from the first hundred, we have crossed out 12 numbers. Among the next twelve numbers, there are neither squares nor cubes ( $11^{2}=121,5^{3}=125$ ), therefore, the hundredth place among the remaining numbers is occupied by the number 112.

## Answer

112. 

## $\left.\begin{array}{lll}\text { Problem } 116782 & \text { Topics: }[ & \text { Arithmetic. Mental calculation, etc. } \\ \text { [ Text problems (miscellaneous). }\end{array}\right]$  [Union, intersection, and difference of sets ]

Twelve toddlers went out to play in the sandbox. Everyone who brought a bucket also brought a spade. Nine toddlers forgot their bucket at home, and two forgot their spade. How many fewer toddlers brought a bucket than those who brought a spade but forgot their bucket?

## Solution

3 toddlers brought a bucket, and therefore, a spade. 10 toddlers brought a spade. Thus, 7 toddlers brought a spade without a bucket. $7-3=4$.

## Answer

By 4.