<s>[INST] Range of $\sqrt{a^2 + (1 - b)^2} + \sqrt{b^2 + (1 - c)^2} + \sqrt{c^2 + (1 - d)^2} + \sqrt{d^2 + (1 - a)^2}$ on $0 \le a,$ $b,$ $c,$ $d \le 1.$ Let $0 \le a,$ $b,$ $c,$ $d \le 1.$ Find the possible values of the expression
$$\sqrt{a^2 + (1 - b)^2} + \sqrt{b^2 + (1 - c)^2} + \sqrt{c^2 + (1 - d)^2} + \sqrt{d^2 + (1 - a)^2}.$$

I tried to use some inequalities to find the bounds of the expression, but it didn't really work. Also, I don't know calculus yet, so please keep the responses and hints non-calc.
Thanks in advance!!
 [/INST] Denote the expression by $P(a, b, c, d)$.
First, using Minkowski inequality, we have
\begin{align*}
 P &\ge \sqrt{(a + b + c + d)^2 + (1 - b + 1 - c + 1 - d + 1 - a)^2}\\
 &= \sqrt{x^2 + (4 - x)^2}\\
 &= \sqrt{2(x - 2)^2 + 8}\\
 &\ge 2\sqrt 2
\end{align*}
where $x = a + b + c + d$.
Also, $P(1/2, 1/2, 1/2, 1/2) = 2\sqrt 2$. Thus, the minimum of $P$ is $2\sqrt 2$.
Second, using $x + y + z + t \le \sqrt{4(x^2 + y^2 + z^2 + t^2)}$ for all $x, y, z, t \ge 0$ (well-known, the so-called AM-QM), we have
\begin{align*}
 P &\le \sqrt{4[a^2 + (1 - b)^2 + b^2 + (1 - c)^2 + c^2 + (1 - d)^2 + d^2 + (1 - a)^2]}\\
 &= \sqrt{16 - 8a - 8b - 8c - 8d + 8a^2 + 8b^2 + 8c^2 + 8d^2}\\
 &\le 4
\end{align*}
where we have used $a \ge a^2$ etc.
Also, $P(1, 1, 1, 1) = 4$.
Thus, the maximum of $P$ is $4$.
Thus, the range of $P$ is $[2\sqrt 2, 4]$.
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