4. Find the minimum value of the sum

$$
\left|x-1^{2}\right|+\left|x-2^{2}\right|+\left|x-3^{2}\right|+\ldots+\left|x-10^{2}\right|
$$
4. Grouping the terms, we get $\left(\left|x-1^{2}\right|+\left|x-10^{2}\right|\right)+\left(\left|x-2^{2}\right|+\left|x-9^{2}\right|\right)+\left(\left|x-3^{2}\right|+\left|x-8^{2}\right|\right)+\left(\left|x-4^{2}\right|+\left|x-7^{2}\right|\right)+\left(\left|x-5^{2}\right|+\left|x-6^{2}\right|\right)$. It is known that the pair $|x-a|+|x-b|$ takes the minimum value when $a \leq x \leq b$ and this value is equal to $|b-a|$. Therefore, the minimum value of the first bracket $\left(\left|x-1^{2}\right|+\left|x-10^{2}\right|\right)$ will be $10^{2}-1^{2}$, the second bracket respectively $9^{2}-2^{2}$, the third $8^{2}-3^{2}$, and so on. However, the intervals $5^{2} \leq x \leq 6^{2}, 4^{2} \leq x \leq 7^{2}, \ldots$ are nested within each other, since the numbers $1^{2}<2^{2}<3^{2} \ldots<10^{2}$. Therefore, the interval $\left[1,10^{2}\right]$ contains all other intervals, on each of which all terms of the form $|x-a|+|x-b|$ simultaneously take the minimum value. Then our expression takes the minimum value, equal to $10^{2}-1^{2}+9^{2}-2^{2}+8^{2}-3^{2}+7^{2}-4^{2}+6^{2}-5^{2}$, (i.e., the sum of the lengths of the nested intervals). We find this sum: $11 \cdot 9+11 \cdot 7+11 \cdot 5+11 \cdot 3+11 \cdot 1=11(9+7+5+3+1)=$ $=11 \cdot 25=275$ Answer: $\{275\}$